diff --git a/README.md b/README.md
index 88fa0e4..c498019 100644
--- a/README.md
+++ b/README.md
@@ -1,3 +1,13 @@
+
+[![](https://img.shields.io/badge/docs-quarto-blue.svg?logo=data:image/png;base64,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)](https://peekxc.github.io/primate/)
+<!-- [![coverage_badge](https://img.shields.io/endpoint?url=https://gist.githubusercontent.com/peekxc/ef42349965f40edf4232737026690c5f/raw/coverage_info.json)](https://coveralls.io/github/peekxc/simplextree-py)  -->
+[![build_macos](https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_macos.yml?logo=apple&logoColor=white)](https://github.com/peekxc/primate/actions/workflows/build_macos.yml) 
+[![build_windows](https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_windows.yml?logo=windows&logoColor=white)](https://github.com/peekxc/primate/actions/workflows/build_windows.yml) 
+[![build_linux](https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_linux.yml?logo=linux&logoColor=white)](https://github.com/peekxc/primate/actions/workflows/build_linux.yml)
+<!-- [![PyPI Version](https://img.shields.io/pypi/v/simplextree)](https://pypi.org/project/simplextree) -->
+[![Python versions](https://img.shields.io/badge/python-%E2%89%A5%203.8-blue.svg)](https://github.com/peekxc/primate/actions)
+
+
 `primate`, short for **Pr**obabalistic **I**mplicit **Ma**trix **T**race **E**stimator, is a Python package that provides estimators of quantities derived from [matrix functions](https://en.wikipedia.org/wiki/Analytic_function_of_a_matrix); that is, matrices parameterized by functions:
 
 $$ f(A) \triangleq U f(\Lambda) U^{\intercal}, \quad \quad f : [a,b] \to \mathbb{R}$$
diff --git a/docs/advanced/cpp_integration.html b/docs/advanced/cpp_integration.html
index cd5c6d2..042aea7 100644
--- a/docs/advanced/cpp_integration.html
+++ b/docs/advanced/cpp_integration.html
@@ -65,7 +65,7 @@
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
 <link href="../advanced/pybind11_integration.html" rel="next">
-<link href="../theory/matrix_functions.html" rel="prev">
+<link href="../theory/lanczos.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
@@ -228,14 +228,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -984,8 +984,8 @@ <h2 class="anchored" data-anchor-id="other-concepts">Other Concepts</h2>
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../theory/matrix_functions.html" class="pagination-link  aria-label=" matrix="" functions"="">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Matrix functions</span>
+      <a href="../theory/lanczos.html" class="pagination-link  aria-label=" the="" lanczos="" method"="">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">The Lanczos Method</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
diff --git a/docs/advanced/pybind11_integration.html b/docs/advanced/pybind11_integration.html
index d277626..5ccd635 100644
--- a/docs/advanced/pybind11_integration.html
+++ b/docs/advanced/pybind11_integration.html
@@ -178,14 +178,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/advanced/slq_param.html b/docs/advanced/slq_param.html
index 5309a9a..bb0fea4 100644
--- a/docs/advanced/slq_param.html
+++ b/docs/advanced/slq_param.html
@@ -228,14 +228,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/basic/install.html b/docs/basic/install.html
index 9498b2e..fb0bd97 100644
--- a/docs/basic/install.html
+++ b/docs/basic/install.html
@@ -209,6 +209,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -349,12 +355,58 @@ <h1 class="title">Installation</h1>
 <!-- https://stackoverflow.com/questions/122906s39/quantifiable-metrics-benchmarks-on-the-usage-of-header-only-c-libraries/13593041#13593041 -->
 <section id="platform-support" class="level3">
 <h3 class="anchored" data-anchor-id="platform-support">Platform support</h3>
-<p>Platform-specific wheels are currently built with cibuildwheel and uploaded to PyPI. These enable <code>primate</code> to be installed on supported architecture / CPython implementations without compilation. The currently supported platforms include:</p>
-<p>If your platform isn’t on this list, feel free to make an issue requesting support for your platform.</p>
+<p>Platform-specific wheels are currently built with cibuildwheel and uploaded to PyPI. These enable <code>primate</code> to be installed on supported architecture / CPython implementations without compilation. As of 12/29/23, native <code>primate</code> wheels are buitl for the following platforms:</p>
+<table class="table">
+<thead>
+<tr class="header">
+<th>Platform</th>
+<th style="text-align: center;">3.8</th>
+<th style="text-align: center;">3.9</th>
+<th style="text-align: center;">3.10</th>
+<th style="text-align: center;">3.11</th>
+<th style="text-align: center;">3.12</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td><strong>Linux (manylinux x86_64)</strong></td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+</tr>
+<tr class="even">
+<td><strong>MacOS (x86_64)</strong></td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+</tr>
+<tr class="odd">
+<td><strong>MacOS ARM (arm64) <a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></strong></td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+</tr>
+<tr class="even">
+<td><strong>Windows (AMD64) <a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a></strong></td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+<td style="text-align: center;">✅</td>
+</tr>
+</tbody>
+</table>
+<p>Wheels are currently built with <a href="https://cibuildwheel.readthedocs.io/en/stable/">cibuildwheel</a>. Currently no support is offered for PyPy, 32-bit runtimes, or <a href="https://devguide.python.org/versions/#supported-versions">unsupported versions of CPython</a>. If your platform isn’t on this list, feel free to make an issue requesting support.</p>
 </section>
 <section id="compiling-from-source" class="level3">
 <h3 class="anchored" data-anchor-id="compiling-from-source">Compiling from source</h3>
-<p>To install the package from its source distribution, a C++20 compiler is required; the current builds are all built with some variant of <a href="">clang</a>. For platform- and compiler-specific settings, consult the build scripts and CI configuration files.</p>
+<p>To install the package from its source distribution, a C++20 compiler is required; the current builds are all built with some variant of <a href="https://clang.llvm.org/">clang</a>. For platform- and compiler-specific settings, consult the build scripts and CI configuration files.</p>
 </section>
 <section id="c-installation" class="level3">
 <h3 class="anchored" data-anchor-id="c-installation">C++ Installation</h3>
@@ -389,7 +441,14 @@ <h3 class="anchored" data-anchor-id="c-installation">C++ Installation</h3>
 
 </section>
 
-</main> <!-- /main -->
+
+<div id="quarto-appendix" class="default"><section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"><h2 class="anchored quarto-appendix-heading">Footnotes</h2>
+
+<ol>
+<li id="fn1"><p>Single-thread execution only; ARM-based OSX runners come stocked with Apple’s clang, which doesn’t natively ship with <code>libomp.dylib</code>, though this <a href="https://mac.r-project.org/openmp/">may be fixable</a>. Feel free to file an PR if you can get this working.<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+<li id="fn2"><p>Single-thread execution only; <code>primate</code> depends on OpenMP 4.5+, which isn’t supported on any Windows compiler I’m aware of.<a href="#fnref2" class="footnote-back" role="doc-backlink">↩︎</a></p></li>
+</ol>
+</section></div></main> <!-- /main -->
 <script id="quarto-html-after-body" type="application/javascript">
 window.document.addEventListener("DOMContentLoaded", function (event) {
   const toggleBodyColorMode = (bsSheetEl) => {
diff --git a/docs/basic/integration.html b/docs/basic/integration.html
index 3f05c32..f19fbdd 100644
--- a/docs/basic/integration.html
+++ b/docs/basic/integration.html
@@ -225,6 +225,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
diff --git a/docs/basic/performance.html b/docs/basic/performance.html
index 88a5a21..4e22577 100644
--- a/docs/basic/performance.html
+++ b/docs/basic/performance.html
@@ -207,6 +207,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -336,7 +342,7 @@ <h1 class="title">Performance</h1>
 <p><code>primate</code> provides a variety of efficient algorithms for estimating quantities derived from matrix functions. These algorithms are largely implemented in C++ to minimize overhead, and for some computational problems <code>primate</code> can out-perform the standard algorithms for estimating spectral quantities by several orders of magnitude. Nonetheless, there are some performance-related caveats to be aware of.</p>
 <!-- The main  -->
 <!-- To illustrate this, we give an example below using Toeplitz matrices.   -->
-<div id="370cac97" class="cell" data-execution_count="1">
+<div id="a5cf844d" class="cell" data-execution_count="1">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> scipy.linalg <span class="im">import</span> toeplitz</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.trace <span class="im">import</span> hutch </span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -380,7 +386,7 @@ <h1 class="title">Performance</h1>
 <span id="cb1-41"><a href="#cb1-41" aria-hidden="true" tabindex="-1"></a>a, b <span class="op">=</span> lanczos(T, deg<span class="op">=</span><span class="dv">499</span>, orth<span class="op">=</span><span class="dv">150</span>)</span>
 <span id="cb1-42"><a href="#cb1-42" aria-hidden="true" tabindex="-1"></a>np.<span class="bu">sum</span>(np.<span class="bu">abs</span>(eigvalsh_tridiagonal(a,b)))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
-<div id="c8032eb8" class="cell" data-execution_count="2">
+<div id="87320ce5" class="cell" data-execution_count="2">
 <div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> timeit </span>
 <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>timeit.timeit(<span class="kw">lambda</span>: hutch(A, maxiter<span class="op">=</span><span class="dv">20</span>, deg<span class="op">=</span><span class="dv">5</span>, fun<span class="op">=</span><span class="st">"log"</span>, quad<span class="op">=</span><span class="st">"fttr"</span>), number <span class="op">=</span> <span class="dv">1000</span>)</span>
 <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a>timeit.timeit(<span class="kw">lambda</span>: np.<span class="bu">sum</span>(np.log(np.linalg.eigvalsh(A))), number <span class="op">=</span> <span class="dv">1000</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
diff --git a/docs/basic/todo.html b/docs/basic/todo.html
index ef7d75f..6c6a3d0 100644
--- a/docs/basic/todo.html
+++ b/docs/basic/todo.html
@@ -210,14 +210,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -323,7 +323,7 @@
 
 
 
-<div id="7ecea2e7" class="cell" data-execution_count="1">
+<div id="7e43e6e6" class="cell" data-execution_count="1">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># a, b = 0.8, 2</span></span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="co"># x = np.random.uniform(low=0, high=10, size=40)</span></span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a><span class="co"># eps = np.random.normal(loc=0, scale=1.0, size=40)</span></span>
diff --git a/docs/basic/usage.html b/docs/basic/usage.html
index 4bd36f9..b40068d 100644
--- a/docs/basic/usage.html
+++ b/docs/basic/usage.html
@@ -226,6 +226,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -354,7 +360,7 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 
 <p>Below is a quick introduction to <code>primate</code>. For more introductory material, theor</p>
 <p>To do trace estimation, use functions in the <code>trace</code> module:</p>
-<div id="ee5cb1a1" class="cell" data-execution_count="2">
+<div id="0b0fe8a0" class="cell" data-execution_count="2">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> primate.trace <span class="im">as</span> TR</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.random <span class="im">import</span> symmetric</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>A <span class="op">=</span> symmetric(<span class="dv">150</span>)  <span class="co">## random positive-definite matrix </span></span>
@@ -364,12 +370,12 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 <span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"XTrace:       </span><span class="sc">{</span>TR<span class="sc">.</span>xtrace(A)<span class="sc">:6f}</span><span class="ss">"</span>)     <span class="co">## Epperly's algorithm</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>Actual trace: 75.697397
-Girard-Hutch: 75.981329
+Girard-Hutch: 75.386523
 XTrace:       75.697398</code></pre>
 </div>
 </div>
 <p>For matrix functions, you can either construct a <code>LinearOperator</code> directly via the <code>matrix_function</code> API, or supply a string to the parameter <code>fun</code> describing the spectral function to apply. For example, one might compute the log-determinant as follows:</p>
-<div id="00f65b7e" class="cell" data-execution_count="3">
+<div id="d4d6cebd" class="cell" data-execution_count="3">
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.operator <span class="im">import</span> matrix_function</span>
 <span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a>M <span class="op">=</span> matrix_function(A, fun<span class="op">=</span><span class="st">"log"</span>)</span>
 <span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -382,14 +388,14 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 <span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a><span class="co">## M = matrix_function(A, fun=np.log)</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>logdet(A):  -148.321844
-GR approx:  -147.250366
+GR approx:  -147.600540
 XTrace:     -148.315937</code></pre>
 </div>
 </div>
 <p>Note in the above example you can supply to <code>fun</code> either string describing a built-in spectral function or an arbitrary <code>Callable</code>. The former is preferred when possible, as function evaluations will generally be faster and <code>hutch</code> can also be parallelized. Multi-threaded execution of e.g.&nbsp;<code>hutch</code> with arbitrary functions is not currently allowed due to the GIL, though there are options available, see <a href="../advanced/cpp_integration.html">the integration docs</a> for more details.</p>
 <p>For ‘plain’ operators, <code>XTrace</code> should recover the exact trace (up to roundoff error). For matrix functions <span class="math inline">f(A)</span>, there will be some inherent inaccuracy as the underlying matrix-vector multiplication is approximated with the Lanczos method.</p>
 <p>In general, the amount of accuracy depends both on the Lanczos parameters and the type of matrix function. Spectral functions that are difficult or impossible to approximate via low-degree polynomials, for example, may suffer more from inaccuracy issues than otherwise. For example, consider the example below that computes that rank:</p>
-<div id="da5ae5db" class="cell" data-execution_count="4">
+<div id="82aa268f" class="cell" data-execution_count="4">
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="co">## Make a rank-deficient operator</span></span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>ew <span class="op">=</span> np.sort(ew)</span>
 <span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>ew[:<span class="dv">30</span>] <span class="op">=</span> <span class="fl">0.0</span></span>
@@ -401,25 +407,25 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 <span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"XTrace:     </span><span class="sc">{</span>TR<span class="sc">.</span>xtrace(M)<span class="sc">}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>Rank:       120
-GR approx:  144.8700408935547
+GR approx:  145.9887237548828
 XTrace:     143.97018151807674</code></pre>
 </div>
 </div>
 <p>This is not so much a fault of <code>hutch</code> or <code>xtrace</code> as much as it is the choice of approximation and Lanczos parameters. The <code>sign</code> function has a discontinuity at 0, is not smooth, and is difficult to approximate with low-degree polynomials. One workaround to handle this issue is relax the sign function with a low-degree “soft-sign” function: <span class="math display"> \mathcal{S}_\lambda(x) = \sum\limits_{i=0}^q \left( x(1 - x^2)^i \prod_{j=1}^i \frac{2j - 1}{2j} \right)</span></p>
 <p>Visually, the soft-sign function looks like this:</p>
-<div id="d2cdfe20" class="cell" data-execution_count="5">
+<div id="f4ca7472" class="cell" data-execution_count="5">
 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.special <span class="im">import</span> soft_sign, figure_fun</span>
 <span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>show(figure_fun(<span class="st">"smoothstep"</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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@@ -444,7 +450,7 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 </div>
 </div>
 <p>It’s been shown that there is a low degree polynomial <span class="math inline">p^\ast</span> that uniformly approximates <span class="math inline">\mathcal{S}_\lambda</span> up to a small error on the interval <span class="math inline">[-1,1]</span>. Since <code>matrix_function</code> uses a low degree Krylov subspace to approximate the action <span class="math inline">v \mapsto f(A)v</span>, one way to improve the accuracy of rank estimation is to replace <span class="math inline">\mathrm{sign} \mapsto \mathcal{S}_{\lambda}</span> for some choice of <span class="math inline">q \in \mathbb{Z}_+</span> (this function is available in <code>primate</code> under the name <code>soft_sign</code>):</p>
-<div id="c5dce435" class="cell" data-execution_count="6">
+<div id="803b72cb" class="cell" data-execution_count="6">
 <div class="sourceCode cell-code" id="cb8"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.special <span class="im">import</span> soft_sign</span>
 <span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> q <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">0</span>, <span class="dv">50</span>, <span class="dv">5</span>):</span>
 <span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>  M <span class="op">=</span> matrix_function(A, fun<span class="op">=</span>soft_sign(q<span class="op">=</span>q))</span>
@@ -470,7 +476,7 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 \end{cases}
 </span></p>
 <p>In the above example, the optimal cutoff <span class="math inline">\lambda_{\text{min}}</span> is given by the smallest non-zero eigenvalue. Since the trace estimators all stochastic to some degree, we set the cutoff to slightly less than this value:</p>
-<div id="7601340c" class="cell" data-execution_count="7">
+<div id="d2c5680c" class="cell" data-execution_count="7">
 <div class="sourceCode cell-code" id="cb10"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>lambda_min <span class="op">=</span> <span class="bu">min</span>(ew[ew <span class="op">!=</span> <span class="fl">0.0</span>])</span>
 <span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Smallest non-zero eigenvalue: </span><span class="sc">{</span>lambda_min<span class="sc">:.6f}</span><span class="ss">"</span>)</span>
 <span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -483,7 +489,7 @@ <h1 class="title"><code>primate</code> usage - quickstart</h1>
 </div>
 </div>
 <p>Indeed, this works! Of course, here we’ve used the fact that we know the optimal cutoff value, but this can also be estimated with the <code>lanczos</code> method itself.</p>
-<div id="51e3f97a" class="cell" data-execution_count="8">
+<div id="6a04722a" class="cell" data-execution_count="8">
 <div class="sourceCode cell-code" id="cb12"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.diagonalize <span class="im">import</span> lanczos</span>
 <span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> scipy.linalg <span class="im">import</span> eigvalsh_tridiagonal</span>
 <span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a>a,b <span class="op">=</span> lanczos(A)</span>
diff --git a/docs/imate_compare.html b/docs/imate_compare.html
index 95c1561..2916a93 100644
--- a/docs/imate_compare.html
+++ b/docs/imate_compare.html
@@ -194,14 +194,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="./theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="./theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="./theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="./theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/index.html b/docs/index.html
index 2c90d82..5f60667 100644
--- a/docs/index.html
+++ b/docs/index.html
@@ -190,6 +190,12 @@
   <a href="./theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="./theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -316,6 +322,7 @@ <h1 class="title">Package overview</h1>
 </header>
 
 
+<p><a href="https://peekxc.github.io/primate/"><img src="https://img.shields.io/badge/docs-quarto-blue.svg?logo=data:image/png;base64,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.png" class="img-fluid"></a> <!-- [![coverage_badge](https://img.shields.io/endpoint?url=https://gist.githubusercontent.com/peekxc/ef42349965f40edf4232737026690c5f/raw/coverage_info.json)](https://coveralls.io/github/peekxc/simplextree-py)  --> <a href="https://github.com/peekxc/primate/actions/workflows/build_macos.yml"><img src="https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_macos.yml?logo=apple&amp;logoColor=white" class="img-fluid" alt="build_macos"></a> <a href="https://github.com/peekxc/primate/actions/workflows/build_windows.yml"><img src="https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_windows.yml?logo=windows&amp;logoColor=white" class="img-fluid" alt="build_windows"></a> <a href="https://github.com/peekxc/primate/actions/workflows/build_linux.yml"><img src="https://img.shields.io/github/actions/workflow/status/peekxc/primate/build_linux.yml?logo=linux&amp;logoColor=white" class="img-fluid" alt="build_linux"></a> <!-- [![PyPI Version](https://img.shields.io/pypi/v/simplextree)](https://pypi.org/project/simplextree) --> <a href="https://github.com/peekxc/primate/actions"><img src="https://img.shields.io/badge/python-%E2%89%A5%203.8-blue.svg" class="img-fluid" alt="Python versions"></a></p>
 <p><code>primate</code>, short for <strong>Pr</strong>obabalistic <strong>I</strong>mplicit <strong>Ma</strong>trix <strong>T</strong>race <strong>E</strong>stimator, is a Python package that provides estimators of quantities derived from <a href="https://en.wikipedia.org/wiki/Analytic_function_of_a_matrix">matrix functions</a>; that is, matrices parameterized by functions:</p>
 <p><span class="math display"> f(A) \triangleq U f(\Lambda) U^{\intercal}, \quad \quad f : [a,b] \to \mathbb{R}</span></p>
 <p>Estimator approximations are obtained via the <em>Lanczos</em><a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a> and <em>stochastic Lanczos quadrature</em><a href="#fn2" class="footnote-ref" id="fnref2" role="doc-noteref"><sup>2</sup></a> methods, which are well-suited for sparse or structured operators supporting fast <span class="math inline">v \mapsto Av</span> actions.</p>
diff --git a/docs/reference/diagonalize.lanczos.html b/docs/reference/diagonalize.lanczos.html
index 4c31a39..3b895a7 100644
--- a/docs/reference/diagonalize.lanczos.html
+++ b/docs/reference/diagonalize.lanczos.html
@@ -191,6 +191,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -321,8 +327,8 @@ <h1>diagonalize.lanczos</h1>
 <p>This function implements Paiges A27 variant (1) of the Lanczos method for tridiagonalizing linear operators, with additional modifications to support varying degrees of re-orthogonalization. In particular, <code>orth=0</code> corresponds to no re-orthogonalization, <code>orth &lt; deg</code> corresponds to partial re-orthogonalization, and <code>orth &gt;= deg</code> corresponds to full re-orthogonalization.</p>
 <section id="notes" class="level2">
 <h2 class="anchored" data-anchor-id="notes">Notes</h2>
-<p>The Lanczos method iteratively builds a tridiagonal matrix <code>T</code> of a symmetric <code>A</code> via an orthogonal change-of-basis <code>Q</code>: <span class="math display"> Q^T A Q  = T </span> Unlike other Lanczos implementations (e.g.&nbsp;SciPy’s <code>eigsh</code>), which includes e.g.&nbsp;sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes <code>deg</code> steps of the Lanczos method with the supplied <code>v0</code> and returns the resulting tridiagonal matrix.</p>
-<p>Diagonalizing <code>T</code> via e.g.&nbsp;<code>scipy.linalg.eigh_tridiagonal</code> yields Rayleigh-Ritz approximations of the eigenvalues of <code>A</code>, though note no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing <code>orth</code> and <code>deg</code>. Supplying either negative values or values larger than <code>deg</code> for <code>orth</code> will result in full re-orthogonalization, though note the number of matvecs scales linearly with <code>deg</code> and the number of inner-products scales quadratically with <code>orth</code>.</p>
+<p>The Lanczos method builds a tridiagonal <code>T</code> from a symmetric <code>A</code> via an orthogonal change-of-basis <code>Q</code>: <span class="math display"> Q^T A Q  = T </span> Unlike other Lanczos implementations (e.g.&nbsp;SciPy’s <code>eigsh</code>), which includes e.g.&nbsp;sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes <code>deg</code> steps of the Lanczos method with the supplied <code>v0</code> and returns the resulting tridiagonal matrix <code>T</code>.</p>
+<p>Rayleigh-Ritz approximations of the eigenvalues of <code>A</code> can be further obtained by diagonalizing <code>T</code> via any symmetric tridiagonal eigenvalue solver, <code>scipy.linalg.eigh_tridiagonal</code> though note unlike <code>eigsh</code> no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing <code>orth</code> and <code>deg</code>. Supplying either negative values or values larger than <code>deg</code> for <code>orth</code> will result in full re-orthogonalization, though note the number of matvecs scales linearly with <code>deg</code> and the number of inner-products scales quadratically with <code>orth</code>.</p>
 </section>
 <section id="parameters" class="level2">
 <h2 class="anchored" data-anchor-id="parameters">Parameters</h2>
diff --git a/docs/reference/get_include.html b/docs/reference/get_include.html
index ab12690..58d9681 100644
--- a/docs/reference/get_include.html
+++ b/docs/reference/get_include.html
@@ -173,11 +173,17 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Lanczos</span></a>
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -235,12 +241,12 @@
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
   <a href="../reference/trace.xtrace.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Xtrace</span></a>
+ <span class="menu-text">XTrace</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../reference/primate.operator.matrix_function" class="sidebar-item-text sidebar-link">
+  <a href="../reference/operator.matrix_function.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Matrix Function</span></a>
   </div>
 </li>
diff --git a/docs/reference/index.html b/docs/reference/index.html
index 4e4a6ab..687cada 100644
--- a/docs/reference/index.html
+++ b/docs/reference/index.html
@@ -194,14 +194,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/reference/operator.matrix_function.html b/docs/reference/operator.matrix_function.html
index 412e7dc..87388d2 100644
--- a/docs/reference/operator.matrix_function.html
+++ b/docs/reference/operator.matrix_function.html
@@ -194,14 +194,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -305,6 +305,7 @@ <h2 id="toc-title">On this page</h2>
   <ul>
   <li><a href="#primate.operator.matrix_function" id="toc-primate.operator.matrix_function" class="nav-link active" data-scroll-target="#primate.operator.matrix_function">operator.matrix_function</a>
   <ul class="collapse">
+  <li><a href="#parameters" id="toc-parameters" class="nav-link" data-scroll-target="#parameters">Parameters</a></li>
   <li><a href="#returns" id="toc-returns" class="nav-link" data-scroll-target="#returns">Returns</a></li>
   </ul></li>
   </ul>
@@ -320,7 +321,7 @@ <h2 id="toc-title">On this page</h2>
 <h1>operator.matrix_function</h1>
 <p><code>operator.matrix_function(A, fun='identity', deg=20, rtol=None, orth=0, **kwargs)</code></p>
 <p>Constructs a <code>LinearOperator</code> approximating the action of the matrix function <code>fun(A)</code>.</p>
-<p>This function uses the Lanczos quadrature method to approximate the action of matrix function: <span class="math display"> v \mapsto f(A)v, \quad f(A) = U f(\lambda) U^T </span> The resulting operator may be used in conjunction with other trace and vector-based estimation methods, such as <code>hutch</code> or <code>xtrace</code>.</p>
+<p>This function uses the Lanczos method to approximate the action of matrix function: <span class="math display"> v \mapsto f(A)v, \quad f(A) = U f(\lambda) U^T </span> The resulting operator may be used in conjunction with other estimation methods, such as <code>hutch</code> or <code>xtrace</code>.</p>
 <p>The weights of the quadrature may be computed using either the Golub-Welsch (GW) or Forward Three Term Recurrence algorithms (FTTR) (see the <code>quad</code> parameter). For a description of the other parameters, see the Lanczos function.</p>
 <div class="callout callout-style-default callout-note callout-titled">
 <div class="callout-header d-flex align-content-center">
@@ -335,6 +336,63 @@ <h1>operator.matrix_function</h1>
 <p>To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require <span class="math inline">O(\mathrm{deg}^2)</span> time to execute, the former requires <span class="math inline">O(\mathrm{deg}^2)</span> space but is highly accurate, while the latter uses only <span class="math inline">O(1)</span> space at the cost of stability. If <code>deg</code> is large, <code>fttr</code> is preferred.</p>
 </div>
 </div>
+<section id="parameters" class="level2">
+<h2 class="anchored" data-anchor-id="parameters">Parameters</h2>
+<table class="table">
+<colgroup>
+<col style="width: 5%">
+<col style="width: 32%">
+<col style="width: 53%">
+<col style="width: 7%">
+</colgroup>
+<thead>
+<tr class="header">
+<th>Name</th>
+<th>Type</th>
+<th>Description</th>
+<th>Default</th>
+</tr>
+</thead>
+<tbody>
+<tr class="odd">
+<td><code>A</code></td>
+<td>scipy.sparse.linalg.LinearOperator or ndarray or sparray</td>
+<td>real symmetric operator.</td>
+<td><em>required</em></td>
+</tr>
+<tr class="even">
+<td><code>fun</code></td>
+<td>str or typing.Callable</td>
+<td>real-valued function defined on the spectrum of <code>A</code>.</td>
+<td><code>"identity"</code></td>
+</tr>
+<tr class="odd">
+<td><code>deg</code></td>
+<td>int</td>
+<td>Degree of the Krylov expansion.</td>
+<td><code>20</code></td>
+</tr>
+<tr class="even">
+<td><code>rtol</code></td>
+<td>float</td>
+<td>Relative tolerance to consider two Lanczos vectors are numerically orthogonal.</td>
+<td><code>1e-8</code></td>
+</tr>
+<tr class="odd">
+<td><code>orth</code></td>
+<td>int</td>
+<td>Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis.</td>
+<td><code>0</code></td>
+</tr>
+<tr class="even">
+<td><code>kwargs</code></td>
+<td>dict</td>
+<td>additional key-values to parameterize the chosen function ‘fun’.</td>
+<td><code>{}</code></td>
+</tr>
+</tbody>
+</table>
+</section>
 <section id="returns" class="level2">
 <h2 class="anchored" data-anchor-id="returns">Returns</h2>
 <table class="table">
diff --git a/docs/reference/random.normal.html b/docs/reference/random.normal.html
index 311ea2a..0831dcf 100644
--- a/docs/reference/random.normal.html
+++ b/docs/reference/random.normal.html
@@ -177,14 +177,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/reference/random.rademacher.html b/docs/reference/random.rademacher.html
index 5fbbc1e..1b45aef 100644
--- a/docs/reference/random.rademacher.html
+++ b/docs/reference/random.rademacher.html
@@ -175,6 +175,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
diff --git a/docs/reference/trace.hutch.html b/docs/reference/trace.hutch.html
index 6c4a816..1bd18a5 100644
--- a/docs/reference/trace.hutch.html
+++ b/docs/reference/trace.hutch.html
@@ -191,6 +191,12 @@
   <a href="../theory/intro.html" class="sidebar-item-text sidebar-link">
  <span class="menu-text">Introduction</span></a>
   </div>
+</li>
+          <li class="sidebar-item">
+  <div class="sidebar-item-container"> 
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
+  </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
@@ -301,6 +307,7 @@ <h2 id="toc-title">On this page</h2>
   <ul class="collapse">
   <li><a href="#parameters" id="toc-parameters" class="nav-link" data-scroll-target="#parameters">Parameters</a></li>
   <li><a href="#returns" id="toc-returns" class="nav-link" data-scroll-target="#returns">Returns</a></li>
+  <li><a href="#notes" id="toc-notes" class="nav-link" data-scroll-target="#notes">Notes:</a></li>
   <li><a href="#see-also" id="toc-see-also" class="nav-link" data-scroll-target="#see-also">See Also</a></li>
   <li><a href="#reference" id="toc-reference" class="nav-link" data-scroll-target="#reference">Reference</a></li>
   </ul></li>
@@ -315,7 +322,7 @@ <h2 id="toc-title">On this page</h2>
 
 <section id="primate.trace.hutch" class="level1">
 <h1>trace.hutch</h1>
-<p><code>trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='fttr', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)</code></p>
+<p><code>trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='golub_welsch', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)</code></p>
 <p>Estimates the trace of a symmetric <span class="math inline">A</span> or matrix function <span class="math inline">f(A)</span> via a Girard-Hutchinson estimator.</p>
 <p>This function uses up to <code>maxiter</code> random isotropic vectors to estimate of the trace of <span class="math inline">f(A)</span>, where: <span class="math display">\mathrm{tr}(f(A)) = \mathrm{tr}(U f(\Lambda) U^T) = \sum\limits_{i=1}^n f(\lambda_i) </span> The estimator is obtained by averaging quadratic forms <span class="math inline">v \mapsto v^T f(A)v</span>, rescaling as necessary. This estimator may be used to quickly approximate of a variety of quantities, such as the trace inverse, the log-determinant, the numerical rank, etc. See the <a href="https://peekxc.github.io/primate/">online documentation</a> for more details.</p>
 <div class="callout callout-style-default callout-note callout-titled">
@@ -338,7 +345,7 @@ <h2 class="anchored" data-anchor-id="parameters">Parameters</h2>
 <col style="width: 8%">
 <col style="width: 26%">
 <col style="width: 54%">
-<col style="width: 9%">
+<col style="width: 10%">
 </colgroup>
 <thead>
 <tr class="header">
@@ -407,7 +414,7 @@ <h2 class="anchored" data-anchor-id="parameters">Parameters</h2>
 <td><code>quad</code></td>
 <td>str</td>
 <td>Method used to obtain the weights of the Gaussian quadrature. See notes.</td>
-<td><code>'fttr'</code></td>
+<td><code>'golub_welsch'</code></td>
 </tr>
 <tr class="odd">
 <td><code>confidence</code></td>
@@ -476,11 +483,15 @@ <h2 class="anchored" data-anchor-id="returns">Returns</h2>
 <tbody>
 <tr class="odd">
 <td>typing.Union[float, tuple]</td>
-<td>Estimate the trace of <span class="math inline">f(A)</span>. If ‘info = True’, additional information about the computation is also returned.</td>
+<td>Estimate the trace of <span class="math inline">f(A)</span>. If <code>info = True</code>, additional information about the computation is also returned.</td>
 </tr>
 </tbody>
 </table>
 </section>
+<section id="notes" class="level2">
+<h2 class="anchored" data-anchor-id="notes">Notes:</h2>
+<p>To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require <span class="math inline">O(\mathrm{deg}^2)</span> time to execute, the former requires <span class="math inline">O(\mathrm{deg}^2)</span> space but is highly accurate, while the latter uses only <span class="math inline">O(1)</span> space at the cost of stability. If <code>deg</code> is large, <code>fttr</code> is preferred.</p>
+</section>
 <section id="see-also" class="level2">
 <h2 class="anchored" data-anchor-id="see-also">See Also</h2>
 <p>lanczos : the lanczos tridiagonalization algorithm.</p>
@@ -489,6 +500,7 @@ <h2 class="anchored" data-anchor-id="see-also">See Also</h2>
 <h2 class="anchored" data-anchor-id="reference">Reference</h2>
 <ol type="1">
 <li>Ubaru, S., Chen, J., &amp; Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.</li>
+<li>Hutchinson, Michael F. “A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines.” Communications in Statistics-Simulation and Computation 18.3 (1989): 1059-1076.</li>
 </ol>
 
 
diff --git a/docs/reference/trace.xtrace.html b/docs/reference/trace.xtrace.html
index 286f2eb..ec463f1 100644
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-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
diff --git a/docs/search.json b/docs/search.json
index f13d9b3..957ac6d 100644
--- a/docs/search.json
+++ b/docs/search.json
@@ -7,121 +7,113 @@
     "text": "get_include\nget_include()\nReturn the directory that contains the primate’s *.h header files.\nExtension modules that need to compile against primate should use this function to locate the appropriate include directory.\nNotes: When using distutils, for example in setup.py: python     import primate     ...     Extension('extension_name', ..., include_dirs=[primate.get_include()])     ... Or with meson-python, for example in meson.build: meson     ...     run_command(py, ['-c', 'import primate; print(primate.get_include())', check : true).stdout().strip()     ..."
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-    "text": "trace.sl_trace(A, fun='identity', maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], orth=0, confidence=0.95, pdf='rademacher', rng='lcg', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)\nEstimates the trace of a matrix function f(A) using stochastic Lanczos quadrature (SLQ).\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n= \"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n= 10\n\n\ndeg\nint\nDegree of the quadrature approximation.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See details.\n= None\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See details.\n= 1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n= \"confidence\"\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against when building the Krylov basis.\n0\n\n\nconfidence\nfloat\nConfidence level to Only used when stop = “confidence”.\n= 0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\nstr\nRandom number generator to use. Defaults to PCG64 generator.\n= \"pcg\"\n\n\nseed\nint\nSeed to initialize the entropy source. Use non-negative integers for reproducibility.\n= -1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Use values &lt;= 0 to maximize the number of threads.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\n= False\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\nfloat\nEstimate of the trace of the matrix function f(A).\n\n\n(dict, optional)\nIf ‘info = True’, additional information about the computation.\n\n\n\n\n\n\nlanczos : the lanczos algorithm.\n\n\n\n[1] Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.",
+    "text": "trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='golub_welsch', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)\nEstimates the trace of a symmetric A or matrix function f(A) via a Girard-Hutchinson estimator.\nThis function uses up to maxiter random isotropic vectors to estimate of the trace of f(A), where: \\mathrm{tr}(f(A)) = \\mathrm{tr}(U f(\\Lambda) U^T) = \\sum\\limits_{i=1}^n f(\\lambda_i)  The estimator is obtained by averaging quadratic forms v \\mapsto v^T f(A)v, rescaling as necessary. This estimator may be used to quickly approximate of a variety of quantities, such as the trace inverse, the log-determinant, the numerical rank, etc. See the online documentation for more details.\n\n\n\n\n\n\nNote\n\n\n\nConvergence behavior is controlled by the stop parameter: “confidence” uses the central limit theorem to generate confidence intervals on the fly, which may be used in conjunction with atol and rtol to upper-bound the error of the approximation. Alternatively, when stop = “change”, the estimator is considered converged when the error between the last two iterates is less than atol (or rtol, respectively), similar to the behavior of scipy.integrate.quadrature.\n\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n10\n\n\ndeg\nint\nDegree of the quadrature approximation. Must be at least 1.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See notes.\nNone\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See notes.\n1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n\"confidence\"\n\n\nncv\nint\nNumber of Lanczos vectors to allocate. Must be at least 2.\n2\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against. Must be less than ncv.\n0\n\n\nquad\nstr\nMethod used to obtain the weights of the Gaussian quadrature. See notes.\n'golub_welsch'\n\n\nconfidence\nfloat\nConfidence level to consider estimator as converged. Only used when stop = “confidence”.\n0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘mt64’\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Set to &lt;= 0 to let OpenMP decide.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\nFalse\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\ntyping.Union[float, tuple]\nEstimate the trace of f(A). If info = True, additional information about the computation is also returned.\n\n\n\n\n\n\nTo compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require O(\\mathrm{deg}^2) time to execute, the former requires O(\\mathrm{deg}^2) space but is highly accurate, while the latter uses only O(1) space at the cost of stability. If deg is large, fttr is preferred.\n\n\n\nlanczos : the lanczos tridiagonalization algorithm.\n\n\n\n\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.\nHutchinson, Michael F. “A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines.” Communications in Statistics-Simulation and Computation 18.3 (1989): 1059-1076.",
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+    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n10\n\n\ndeg\nint\nDegree of the quadrature approximation. Must be at least 1.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See notes.\nNone\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See notes.\n1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n\"confidence\"\n\n\nncv\nint\nNumber of Lanczos vectors to allocate. Must be at least 2.\n2\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against. Must be less than ncv.\n0\n\n\nquad\nstr\nMethod used to obtain the weights of the Gaussian quadrature. See notes.\n'golub_welsch'\n\n\nconfidence\nfloat\nConfidence level to consider estimator as converged. Only used when stop = “confidence”.\n0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘mt64’\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Set to &lt;= 0 to let OpenMP decide.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\nFalse\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}",
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-    "text": "Type\nDescription\n\n\n\n\nfloat\nEstimate of the trace of the matrix function f(A).\n\n\n(dict, optional)\nIf ‘info = True’, additional information about the computation.",
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-    "text": "lanczos : the lanczos algorithm.",
+    "text": "To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require O(\\mathrm{deg}^2) time to execute, the former requires O(\\mathrm{deg}^2) space but is highly accurate, while the latter uses only O(1) space at the cost of stability. If deg is large, fttr is preferred.",
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-    "text": "[1] Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.",
+    "text": "lanczos : the lanczos tridiagonalization algorithm.",
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-    "text": "random.normal(size, rng='splitmix64', seed=-1, dtype=np.float32)\nGenerates random vectors from the standard normal distribution.\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nsize\nint or tuple\nOutput shape to generate.\nrequired\n\n\nrng\nstr = “splitmix64”\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint = -1\nSeed for the generator. Use -1 to for random (non-deterministic) behavior.\n-1\n\n\ndtype\ndtype = float32\nFloating point dtype for the output. Must be float32 or float64.\nnp.float32\n\n\n\n\n\n\n: Randomly generated matrix of shape size with entries in { -1, 1 }.",
+    "text": "Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.\nHutchinson, Michael F. “A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines.” Communications in Statistics-Simulation and Computation 18.3 (1989): 1059-1076.",
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-    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nsize\nint or tuple\nOutput shape to generate.\nrequired\n\n\nrng\nstr = “splitmix64”\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint = -1\nSeed for the generator. Use -1 to for random (non-deterministic) behavior.\n-1\n\n\ndtype\ndtype = float32\nFloating point dtype for the output. Must be float32 or float64.\nnp.float32",
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-    "text": "diagonalize.lanczos(A, v0=None, deg=None, rtol=1e-08, orth=0, sparse_mat=False, return_basis=False, seed=None, dtype=None)\nLanczos method for matrix tridiagonalization.\nThis function implements Paiges A27 variant (1) of the Lanczos method for tridiagonalizing linear operators, with additional modifications to support varying degrees of re-orthogonalization. In particular, orth=0 corresponds to no re-orthogonalization, orth &lt; deg corresponds to partial re-orthogonalization, and orth &gt;= deg corresponds to full re-orthogonalization.\n\n\nThe Lanczos method iteratively builds a tridiagonal matrix T of a symmetric A via an orthogonal change-of-basis Q:  Q^T A Q  = T  Unlike other Lanczos implementations (e.g. SciPy’s eigsh), which includes e.g. sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes deg steps of the Lanczos method with the supplied v0 and returns the resulting tridiagonal matrix.\nDiagonalizing T via e.g. scipy.linalg.eigh_tridiagonal yields Rayleigh-Ritz approximations of the eigenvalues of A, though note no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing orth and deg. Supplying either negative values or values larger than deg for orth will result in full re-orthogonalization, though note the number of matvecs scales linearly with deg and the number of inner-products scales quadratically with orth.\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nscipy.sparse.linalg.LinearOperator or ndarray or sparray\nSymmetric operator to tridiagonalize.\nrequired\n\n\nv0\nndarray\nInitial vector to orthogonalize against.\nNone\n\n\ndeg\nint\nSize of the Krylov subspace to expand.\nNone\n\n\nrtol\nfloat\nRelative tolerance to consider the invariant subspace as converged.\n1e-8\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against.\n0\n\n\nsparse_mat\nbool\nWhether to output the tridiagonal matrix as a sparse matrix.\nFalse\n\n\nreturn_basis\nbool\nIf True, returns the Krylov basis vectors Q.\nFalse\n\n\ndtype\ndtype\nThe precision dtype to specialize the computation.\nNone\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\ntuple\nA tuple (a,b) parameterizing the diagonal and off-diagonal of the tridiagonal matrix. If return_basis=True, the tuple (a,b), Q is returned, where Q represents an orthogonal basis for the degree-deg Krylov subspace.\n\n\n\n\n\n\nscipy.linalg.eigh_tridiagonal : Eigenvalue solver for real symmetric tridiagonal matrices. operator.matrix_function : Approximates the action of a matrix function via the Lanczos method.\n\n\n\n\nPaige, Christopher C. “Computational variants of the Lanczos method for the eigenproblem.” IMA Journal of Applied Mathematics 10.3 (1972): 373-381.",
+    "text": "operator.matrix_function(A, fun='identity', deg=20, rtol=None, orth=0, **kwargs)\nConstructs a LinearOperator approximating the action of the matrix function fun(A).\nThis function uses the Lanczos method to approximate the action of matrix function:  v \\mapsto f(A)v, \\quad f(A) = U f(\\lambda) U^T  The resulting operator may be used in conjunction with other estimation methods, such as hutch or xtrace.\nThe weights of the quadrature may be computed using either the Golub-Welsch (GW) or Forward Three Term Recurrence algorithms (FTTR) (see the quad parameter). For a description of the other parameters, see the Lanczos function.\n\n\n\n\n\n\nNote\n\n\n\nTo compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require O(\\mathrm{deg}^2) time to execute, the former requires O(\\mathrm{deg}^2) space but is highly accurate, while the latter uses only O(1) space at the cost of stability. If deg is large, fttr is preferred.\n\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nscipy.sparse.linalg.LinearOperator or ndarray or sparray\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\ndeg\nint\nDegree of the Krylov expansion.\n20\n\n\nrtol\nfloat\nRelative tolerance to consider two Lanczos vectors are numerically orthogonal.\n1e-8\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against when building the Krylov basis.\n0\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\nscipy.sparse.linalg.LinearOperator\na LinearOperator approximating the action of fun on the spectrum of A",
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-    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nA\nscipy.sparse.linalg.LinearOperator or ndarray or sparray\nSymmetric operator to tridiagonalize.\nrequired\n\n\nv0\nndarray\nInitial vector to orthogonalize against.\nNone\n\n\ndeg\nint\nSize of the Krylov subspace to expand.\nNone\n\n\nrtol\nfloat\nRelative tolerance to consider the invariant subspace as converged.\n1e-8\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against.\n0\n\n\nsparse_mat\nbool\nWhether to output the tridiagonal matrix as a sparse matrix.\nFalse\n\n\nreturn_basis\nbool\nIf True, returns the Krylov basis vectors Q.\nFalse\n\n\ndtype\ndtype\nThe precision dtype to specialize the computation.\nNone",
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-    "text": "This function implements the Lanczos method, or as Lanczos called it, the method of minimized iterations.",
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-    "text": "primate, short for Probabalistic Implicit Matrix Trace Estimator, is a Python package that provides estimators of quantities derived from matrix functions; that is, matrices parameterized by functions:\nf(A) \\triangleq U f(\\Lambda) U^{\\intercal}, \\quad \\quad f : [a,b] \\to \\mathbb{R}\nEstimator approximations are obtained via the Lanczos1 and stochastic Lanczos quadrature2 methods, which are well-suited for sparse or structured operators supporting fast v \\mapsto Av actions.\nNotable features of primate include:\nprimate was partially inspired by the imate package—for a comparison of the two, see here.",
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+    "text": "Footnotes\n\n\nBefore v0.2, much of primate’s code was essentially ported and refactored from imate. The code for v0.2+ has been re-written using the Eigen template C++ library.↩︎\nprimate does not provide native GPU-implemented Linear operators. However, there is nothing preventing one from using e.g. CUDA- or ROCm-based GPU-based tensor libraries to accelerate matrix-vector products. Indeed, primate was designed to work with essentially any operator matching the interface.↩︎\nSee imates documentation for the list of supported functions.↩︎",
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+    "text": "Below is a quick introduction to primate. For more introductory material, theor\nTo do trace estimation, use functions in the trace module:\n\nimport primate.trace as TR\nfrom primate.random import symmetric\nA = symmetric(150)  ## random positive-definite matrix \n\nprint(f\"Actual trace: {A.trace():6f}\")        ## Actual trace\nprint(f\"Girard-Hutch: {TR.hutch(A):6f}\")      ## Monte-carlo tyoe estimator\nprint(f\"XTrace:       {TR.xtrace(A):6f}\")     ## Epperly's algorithm\n\nActual trace: 75.697397\nGirard-Hutch: 75.386523\nXTrace:       75.697398\n\n\nFor matrix functions, you can either construct a LinearOperator directly via the matrix_function API, or supply a string to the parameter fun describing the spectral function to apply. For example, one might compute the log-determinant as follows:\n\nfrom primate.operator import matrix_function\nM = matrix_function(A, fun=\"log\")\n\new = np.linalg.eigvalsh(A)\nprint(f\"logdet(A):  {np.sum(np.log(ew)):6f}\")\nprint(f\"GR approx:  {TR.hutch(M):6f}\")\nprint(f\"XTrace:     {TR.xtrace(M):6f}\")\n\n## Equivalently could've used: \n## M = matrix_function(A, fun=np.log)\n\nlogdet(A):  -148.321844\nGR approx:  -147.600540\nXTrace:     -148.315937\n\n\nNote in the above example you can supply to fun either string describing a built-in spectral function or an arbitrary Callable. The former is preferred when possible, as function evaluations will generally be faster and hutch can also be parallelized. Multi-threaded execution of e.g. hutch with arbitrary functions is not currently allowed due to the GIL, though there are options available, see the integration docs for more details.\nFor ‘plain’ operators, XTrace should recover the exact trace (up to roundoff error). For matrix functions f(A), there will be some inherent inaccuracy as the underlying matrix-vector multiplication is approximated with the Lanczos method.\nIn general, the amount of accuracy depends both on the Lanczos parameters and the type of matrix function. Spectral functions that are difficult or impossible to approximate via low-degree polynomials, for example, may suffer more from inaccuracy issues than otherwise. For example, consider the example below that computes that rank:\n\n## Make a rank-deficient operator\new = np.sort(ew)\new[:30] = 0.0\nA = symmetric(150, ew = ew, pd = False)\nM = matrix_function(A, fun=np.sign)\n\nprint(f\"Rank:       {np.linalg.matrix_rank(A)}\")\nprint(f\"GR approx:  {TR.hutch(M)}\")\nprint(f\"XTrace:     {TR.xtrace(M)}\")\n\nRank:       120\nGR approx:  145.9887237548828\nXTrace:     143.97018151807674\n\n\nThis is not so much a fault of hutch or xtrace as much as it is the choice of approximation and Lanczos parameters. The sign function has a discontinuity at 0, is not smooth, and is difficult to approximate with low-degree polynomials. One workaround to handle this issue is relax the sign function with a low-degree “soft-sign” function:  \\mathcal{S}_\\lambda(x) = \\sum\\limits_{i=0}^q \\left( x(1 - x^2)^i \\prod_{j=1}^i \\frac{2j - 1}{2j} \\right)\nVisually, the soft-sign function looks like this:\n\nfrom primate.special import soft_sign, figure_fun\nshow(figure_fun(\"smoothstep\"))\n\n\n  \n\n\n\n\n\nIt’s been shown that there is a low degree polynomial p^\\ast that uniformly approximates \\mathcal{S}_\\lambda up to a small error on the interval [-1,1]. Since matrix_function uses a low degree Krylov subspace to approximate the action v \\mapsto f(A)v, one way to improve the accuracy of rank estimation is to replace \\mathrm{sign} \\mapsto \\mathcal{S}_{\\lambda} for some choice of q \\in \\mathbb{Z}_+ (this function is available in primate under the name soft_sign):\n\nfrom primate.special import soft_sign\nfor q in range(0, 50, 5):\n  M = matrix_function(A, fun=soft_sign(q=q))\n  print(f\"XTrace S(A) for q={q}: {TR.xtrace(M):6f}\")\n\nXTrace S(A) for q=0: 72.776775\nXTrace S(A) for q=5: 109.325087\nXTrace S(A) for q=10: 114.843157\nXTrace S(A) for q=15: 116.952146\nXTrace S(A) for q=20: 118.027760\nXTrace S(A) for q=25: 118.657212\nXTrace S(A) for q=30: 119.055340\nXTrace S(A) for q=35: 119.319911\nXTrace S(A) for q=40: 119.501839\nXTrace S(A) for q=45: 119.630120\n\n\nIf the type of operator A is known to typically have a large spectral gap, another option is to compute the numerical rank by thresholding values above some fixed value \\lambda_{\\text{min}}. This is equivalent to applying the following spectral function:\n S_{\\lambda_{\\text{min}}}(x) =\n\\begin{cases}\n1 & \\text{ if } x \\geq \\lambda_{\\text{min}} \\\\\n0 & \\text{ otherwise }\n\\end{cases}\n\nIn the above example, the optimal cutoff \\lambda_{\\text{min}} is given by the smallest non-zero eigenvalue. Since the trace estimators all stochastic to some degree, we set the cutoff to slightly less than this value:\n\nlambda_min = min(ew[ew != 0.0])\nprint(f\"Smallest non-zero eigenvalue: {lambda_min:.6f}\")\n\nstep = lambda x: 1 if x &gt; (lambda_min * 0.90) else 0\nM = matrix_function(A, fun=step, deg=50)\nprint(f\"XTrace S_t(A) for t={lambda_min*0.90:.4f}: {TR.xtrace(M):6f}\")\n\nSmallest non-zero eigenvalue: 0.191916\nXTrace S_t(A) for t=0.1727: 120.000000\n\n\nIndeed, this works! Of course, here we’ve used the fact that we know the optimal cutoff value, but this can also be estimated with the lanczos method itself.\n\nfrom primate.diagonalize import lanczos\nfrom scipy.linalg import eigvalsh_tridiagonal\na,b = lanczos(A)\nrr = eigvalsh_tridiagonal(a,b) # Rayleigh-Ritz values\ntol = 10 * np.finfo(A.dtype).resolution\nprint(f\"Approx. cutoff: {np.min(rr[rr &gt; tol]):.6f}\")\n\nApprox. cutoff: 0.191916"
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+    "text": "primate provides a variety of efficient algorithms for estimating quantities derived from matrix functions. These algorithms are largely implemented in C++ to minimize overhead, and for some computational problems primate can out-perform the standard algorithms for estimating spectral quantities by several orders of magnitude. Nonetheless, there are some performance-related caveats to be aware of.\n\n\n\nfrom scipy.linalg import toeplitz\nfrom primate.trace import hutch \n\nc = np.random.uniform(size=100, low=0, high=1)\nT = toeplitz(c)\n\n# np.sum(np.reciprocal(np.linalg.eigvalsh(T)))\nnp.sum(np.abs(np.linalg.eigvalsh(T)))\nest, info = hutch(T, fun=np.abs, maxiter=200, pdf='rademacher', rng=\"mt\", deg=20, seed=-1, plot=True)\n\nest, info = hutch(T, fun=np.abs, maxiter=800, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=True)\nest, info = hutch(C, fun=np.abs, maxiter=800, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=True)\n\n\n\nnp.sum(np.linalg.svd(T)[1])\n\nZ = np.zeros(T.shape)\nC = np.block([[Z, T], [T.T, Z]])\nhutch(C, fun=np.abs, maxiter=200, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=False)\n\n\nhutch(T, fun=np.abs, maxiter=200, pdf='rademacher', deg=50, seed=-1, plot=False)\n\np = info['figure']\n\nfrom primate.trace import xtrace\nfrom primate.operator import matrix_function\nM = matrix_function(T, \"abs\")\nxtrace(M)\n\ns = info['samples']\nv = np.random.choice([-1, 1], size=T.shape[0])\nT @ v\n\ns[np.abs(s) &lt;= np.linalg.norm(T)].mean()\n\n\nfrom primate.diagonalize import lanczos\nfrom scipy.linalg import eigvalsh_tridiagonal\na, b = lanczos(T, deg=499, orth=150)\nnp.sum(np.abs(eigvalsh_tridiagonal(a,b)))\n\n\nimport timeit \ntimeit.timeit(lambda: hutch(A, maxiter=20, deg=5, fun=\"log\", quad=\"fttr\"), number = 1000)\ntimeit.timeit(lambda: np.sum(np.log(np.linalg.eigvalsh(A))), number = 1000)"
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-    "text": "primate relies on BLAS libraries\n\npipx run cibuildwheel –platform linux",
+    "text": "# a, b = 0.8, 2\n# x = np.random.uniform(low=0, high=10, size=40)\n# eps = np.random.normal(loc=0, scale=1.0, size=40)\n# y = (a * x + b) + eps\n\n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# show(p)\n\n# X = np.c_[x,y]\n# from scipy.linalg import lstsq\n# from scipy.optimize import least_squares, leastsq, minimize, minimize_scalar\n# def L(beta: tuple):\n#   b, a = beta\n#   return np.linalg.norm(y - a * x + b)**2\n\n# def L(beta: tuple):\n#   b, a = beta\n#   return (y - (a * x + b))\n\n\n\n# res = minimize_scalar(L, x0=(0,1))\n# b_opt, a_opt = res.x\n# L([a_opt,b_opt])\n\n# res = least_squares(L, x0=(0,1))\n# b_opt, a_opt = res.x\n# f_opt = lambda x: x * a_opt + b_opt \n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# p.line(x=[0, 10], y=[b_opt, f_opt(10)])\n# show(p)\n\n# ## Normal equations...\n# XI = np.c_[np.ones(X.shape[0]), X]\n\n# c, b_opt, a_opt = (np.linalg.inv((XI.T @ XI)) @ XI.T) @ y\n# f_opt = lambda x: x * a_opt + b_opt \n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# p.line(x=[0, 10], y=[b_opt, f_opt(10)])\n# show(p)"
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+    "text": "To clarify that that means, here’s an abstract presentation of the generic SLQ procedure:\n\n\n\\begin{algorithm} \\caption{Stochastic Lanczos Quadrature} \\begin{algorithmic} \\Input Symmetric operator ($A \\in \\mathbb{R}^{n \\times n}$) \\Require Number of queries ($n_v$), Degree of quadrature ($k$) \\Function{SLQ}{$A$, $n_v$, $k$} \\State $\\Gamma \\gets 0$ \\For{$j = 1, 2, \\dots, n_v$} \\State $v_i \\sim \\mathcal{D}$ where $\\mathcal{D}$ satisfies $\\mathbb{E}(v v^\\top) = I$ \\State $T^{(j)}(\\alpha, \\beta)$ $\\gets$ $\\mathrm{Lanczos}(A,v_j,k+1)$ \\State $[\\Theta, Y] \\gets \\mathrm{eigh\\_tridiag}(T^{(j)}(\\alpha, \\beta))$ \\State $\\tau_i \\gets \\langle e_1, y_i \\rangle$ \\State &lt; Do something with the node/weight pairs $(\\theta_i, \\tau_i^2)$ &gt; \\EndFor \\EndFunction \\end{algorithmic} \\end{algorithm}"
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+    "text": "primate contains a variety of methods for estimating quantities from matrix functions. One popular such quantity is the trace of f(A), for some generic f: [a,b] \\to \\mathbb{R} defined on the spectrum of A:  \\mathrm{tr}(f(A)) \\triangleq \\mathrm{tr}(U f(\\Lambda) U^{\\intercal}), \\quad \\quad f : [a,b] \\to \\mathbb{R}\nMany quantities of common interest can be expressed in as traces of matrix functions with the right choice of f. A few such function specializations include:\n\\begin{align*}\nf &= \\mathrm{id} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{tr}(A) \\\\\nf &= f^{-1} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{tr}(A^{-1}) \\\\\nf &= \\log \\quad &\\Longleftrightarrow& \\quad  &\\mathrm{logdet}(A) \\\\\nf &= \\mathrm{exp} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{exp}(A) \\\\\nf &= \\mathrm{sign} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{rank}(A)  \\\\\n&\\vdots \\quad &\\hphantom{\\Longleftrightarrow}& \\quad & \\vdots &\n\\end{align*}\nIn this introduction, the basics of randomized trace estimation are introduced, with a focus on how to use matrix-free algorithms estimate traces of matrix functions.",
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-    "text": "To get started calling any matrix-free function provided by primate, such sl_trace or lanczos, simply pass any type with a .shape() and .matvec() member functions defined like so:\nIt’s up to you to ensure shape() yields the correct size; primate will supply vectors to input of size .shape().second (number of columns) and guarantees the pointer to the output will be at least shape().first (number of rows), no more.",
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+    "text": "Basics: Trace computation\n\nSuppose we wanted to compute the trace of some matrix A. By the very definition of the trace, we have a variety of means to do it: \\mathrm{tr}(A) \\triangleq \\sum\\limits_{i=1}^n A_{ii} = \\sum\\limits_{i=1}^n \\lambda_i = \\sum\\limits_{i=1}^n e_i^T A e_i \nwhere, in the right-most sum, we’re using e_i to represent the ith identity vector:  A_{ii} = e_i^T A e_i, \\quad e_i = [0, \\dots, 0, \\underbrace{1}_{i}, 0, \\dots, 0] \nLet’s see some code. First, we start with a simple (random) positive-definite matrix A \\in \\mathbb{R}^{n \\times n}.\n\nfrom primate.random import symmetric\nA = symmetric(150, pd = True)\n\nBy default, symmetric normalizes A such that the eigenvalues are uniformly distributed in the interval [0, 1]. For reference, the spectrum of A looks as follows:\n\n\n\n  \n\n\n\n\n\nNow, using numpy, we can verify all of these trace definitions are indeed equivalent:\n\n\nimport numpy as np\neye = lambda i: np.ravel(np.eye(1, 150, k=i))\nprint(f\"Direct: {np.sum(A.diagonal()):.8f}\")\nprint(f\"Eigen:  {np.sum(np.linalg.eigvalsh(A)):.8f}\")\nprint(f\"Matvec: {sum([eye(i) @ A @ eye(i) for i in range(150)]):.8f}\")\n\nDirect: 75.69739746\nEigen:  75.69739746\nMatvec: 75.69739746\n\n\nWhile (1) trivially yields \\mathrm{tr}(A) in (optimal) O(n) time, there exist situations where the diagonal entries of A are not available—particularly in the large-scale regime. In contrast, though both (2) and (3) are less efficient, they are nonetheless viable alternatives to obtain \\mathrm{tr}(A).",
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+    "text": "Implicit trace estimation\nThe implicit trace estimation problem can be stated as follows:\n\nGiven access to a square matrix A \\in \\mathbb{R}^{n \\times n} via its matrix-vector product operator x \\mapsto Ax, estimate its trace \\mathrm{tr}(A) = \\sum\\limits_{i=1}^n A_{ii}\n\nNote we no longer assume we have direct access to the diagonal here, leaving us with the Eigen or Matvec methods. The Eigen method is expensive, and though the Matvec method solves the problem exactly, its pretty inefficient from a computational standpoint—most of the entries of e_i are zero, thus most inner product computations do not contribute to the trace estimate at all.\nOne idea, attributed to A. Girard, is to replace e_i with random zero-mean vectors, e.g. random sign vectors v \\in \\{-1, +1\\}^{n} or random normal vectors v \\sim \\mathcal{N}(\\mu=0, \\sigma=1):\n\\mathtt{tr}(A) \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nIt was shown more formally later by M.F. Huchinson that if these random vectors v \\in \\mathbb{R}^n satisfy \\mathbb{E}[vv^T] = I then this approach indeed forms an unbiased estimator of \\mathrm{tr}(A). To see this, its sufficient to combine the linearity of expectation, the cyclic-property of the trace function, and the aforementioned isotropy conditions: \\mathtt{tr}(A) = \\mathtt{tr}(A I) = \\mathtt{tr}(A \\mathbb{E}[v v^T]) = \\mathbb{E}[\\mathtt{tr}(Avv^T)] = \\mathbb{E}[\\mathtt{tr}(v^T A v)] = \\mathbb{E}[v^T A v]\nNaturally we expect the approximation to gradually improve as more vectors are sampled, converging as n_v \\to \\infty. Let’s see if we can check this: we start by collecting a few sample quadratic forms\n\ndef isotropic(n):\n  yield from [np.random.choice([-1,+1], size=A.shape[1]) for i in range(n)]\nestimates = np.array([v @ A @ v for v in isotropic(500)])\n\nPlotting both the estimator formed by averaging the first k samples along with its absolute error, we get the following figures for k = 50 and k = 500, respectively:\n\n\n\n  \n\n\n\n\n\nSure enough, the estimator quickly gets within an error rate of about 1-5% away from the actual trace, getting generally closer as more samples are collected. On the other hand, improvement is not gaurenteed, and securing additional precision much less than 1% becomes increasingly difficult due to the randomness and variance of the sample estimates.",
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+    "text": "Estimator Implementation\nRather than manually averaging samples, trace estimates can be obtained via the hutch function:\n\nfrom primate.trace import hutch\nprint(f\"Trace estimate: {hutch(A, maxiter=50)}\")\n\nTrace estimate: 76.84235846113064\n\n\nThough its estimation is identical to averaging quadratic forms v^T A v of isotropic vectors as above, hutch comes with a few extra features to simplify its usage. In particular, all v \\mapsto v^T A v evaluations are done in parallel, there are a variety of isotropic distributions and random number generators (RNGs) to choose from, and the estimator may be suppleid various convergence criteria to allow for early-stopping; see the hutch documentation page for more details.",
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+    "text": "Stochastic Lanczos Quadrature\nWhile there are some real-world applications for estimating \\mathrm{tr}(A), in many setting the trace of a given matrix is not a very informative quantity. On the other hand, as shown in the beginning, there are many situations where we might be interested in estimating the trace of a matrix function f(A). It is natural to consider extending the Hutchinson estimator above with something like:\n \\mathrm{tr}(f(A)) \\approx \\frac{n}{n_v} \\sum\\limits_{i=1}^{n_v} v_i^T f(A) v_i \nOf course, the remaining difficult lies in computing quadratic forms v \\mapsto v^T f(A) v efficiently. The approach taken by Ubaru, Saad, and Seghouane (2017) is to notice that sums of these quantities can transformed into a Riemann-Stieltjes integral:\n v^T f(A) v = v^T U f(\\Lambda) U^T v = \\sum\\limits_{i}^n f(\\lambda_i) \\mu_i^2 = \\int\\limits_{a}^b f(t) \\mu(t)\nwhere scalars \\mu_i \\in \\mathbb{R} constitute a cumulative, piecewise constant function \\mu : \\mathbb{R} \\to \\mathbb{R}_+:\n\n\\mu_i = (U^T v)_i, \\quad \\mu(t) = \\begin{cases}\n0, & \\text{if } t &lt; a = \\lambda_1 \\\\\n\\sum_{j=1}^{i-1} \\mu_j^2, & \\text{if } \\lambda_{i-1} \\leq t &lt; \\lambda_i, i = 2, \\dots, n \\\\\n\\sum_{j=1}^n \\mu_j^2, & \\text{if } b = \\lambda_n \\leq t\n\\end{cases}\n\nAs with any definite integral, one would ideally like to approximate its value via some quadrature rule. An exemplary such approximation is the m-point Gaussian quadrature rule:\n \\int\\limits_{a}^b f(t) \\mu(t) \\approx \\sum\\limits_{k=1}^m \\omega_k f(\\eta_k) \nwith weights \\{\\omega_k\\} and nodes \\{ \\eta_k\\}. On one hand, Gaussian Quadrature (GQ) is just one of many quadrature rules that could be used to approximate v^T A v. On the other hand, GQ is often preferred over other rules due to its exactness on polynomials up to degree 2m - 1.\nOf course, both the weights and nodes of GQ rule are unknown for the integral above. This is in strong contrast to the typical setting, wherein \\omega is assumed uniform or otherwise known ahead-of-time. A surprising and non-trivial fact is that both the weights \\{\\omega_k\\} and nodes \\{ \\eta_k\\} of the m-point GQ above are directly computable the degree m matrix T_m produced by the Lanczos method:\n Q_m^T A Q_m = T_m = Y \\Theta Y^T \\quad \\Leftrightarrow \\quad (\\eta_i, \\omega_i) = (\\theta_i, \\tau_i), \\; \\tau_i = (e_1^T y_i)^2 \nIn other words, the Ritz values \\{\\theta_i\\} of T_m are exactly the nodes of GQ quadrature rule applied to \\mu(t), while the first components of the Ritz vectors \\tau_i (squared) are exactly the weights. For more information about this non-trivial fact, see Golub and Meurant (2009).",
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+    "text": "Example: Logdet computation\nPutting all of the theory above together, a simple way to estimate the trace of a matrix function is to average Lanczos quadrature estimates:\n\\mathrm{tr}(f(A)) \\approx \\frac{n}{n_v} \\sum\\limits_{i=1}^{n_v} e_1^T f(T_m) e_1  = \\frac{n}{n_v} \\sum\\limits_{j=1}^{n_v} \\left ( \\sum\\limits_{i=1}^m \\tau_i^{(j)} f(\\theta_i)^{(j)} \\right )\nTo enable this kind of estimator in hutch, simply set the fun argument to either the name of a built-in function or an arbitrary Callable. For example, to approximate the log-determinant:\n\n## Log-determinant\nlogdet_test = hutch(A, fun=\"log\", maxiter=25)\nlogdet_true = np.sum(np.log(np.linalg.eigvalsh(A)))\n\nprint(f\"Logdet (exact):  {logdet_true}\")\nprint(f\"Logdet (approx): {logdet_test}\")\n\nLogdet (exact):  -148.3218440234385\nLogdet (approx): -148.15107274186403\n\n\nHere we see that using n / 6 evaluatons of e_1^T f(T_m) e_1, we get a decent approximation of logdet(A). To get a better idea of the quality of this estimate, you can set verbose=True:\n\nest = hutch(A, fun=\"log\", maxiter=25, seed=5, verbose=True)\n\nGirard-Hutchinson estimator (fun=log, deg=20, quad=golub_welsch)\nEst: -144.400 +/- 6.05 (95% CI), CV: 2%, Evals: 25 [R] (seed: 5)\n\n\nThe first line of the statement contains fixed about the estimator, including the type of estimator (Girard-Hutchinson), the function applied to the spectrum (log), the degree of the Lanczos quadrature approximation (20), and the quadrature method used (fttr).\nThe second line prints the runtime information about the samples, such as the final trace estimate, its margin of error and coefficient of variation (aka the relative std. deviation), and the number of samples used + their distribution prefix (‘R’ for rademacher).\nAnother way to get an idea of how well the estimator is converging is to pass the plot=True option. Here, we pass the same seed for reproducibility with the above call.\n\nest = hutch(A, fun=\"log\", maxiter=100, seed=5, plot=True)\n\n\n  \n\n\n\n\n\nSimilar information as reported by the verbose=True flag is shown cumulatively applied to every sample. The left plot shows the sample point estimates alongside yellow bands denoting the margin of error associated with the 95% confidence interval (CI) at every sample index. The right plot show convergence information, e.g. the running coefficient of variation is shown on the top-right and an upper bound on the absolute error is derived from the CI is shown on the bottom right.\nWhile the margin of error is typically a valid, conservative estimate of estimators error, in some situations these bounds may not be valid as they are based solely on the CLT implications for normally distributed random variables. In particular, there is an inherent amount of error associated with the Lanczos-based quadrature approximations that is not taken into account in the error bound, which may invalidate the corresponding CI’s.",
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-    "text": "The LinearOperator concept\nprimate’s generic API is enabled through C++20 concepts. Thus, the more exact statement is that any type respecting the LinearOperator concept shown below can be passed:\nusing FP = std::floating_point; \ntemplate &lt; typename T, FP float_t = typename T::value_type &gt;\nconcept LinearOperator = requires(T A, const float_t* v, float_t* o) {\n  { A.matvec(v, o) }; // o = A v\n  { A.shape() } -&gt; std::convertible_to&lt; std::pair&lt; size_t, size_t &gt; &gt;;\n};\nAn instance A of type T is said to support the LinearOperator concept if it has:\n\nA method Av \\mapsto o, with signature A.matvec(const float_t* v, float_t* o)\nA method yielding (\\mathrm{card}(o), \\mathrm{card}(v)), with signatureA.shape() -&gt; pair&lt; ... &gt;\n\nshape() should yield a pair (n,m) representing the sizes of the output and input vectors, respectively. This corresponds to the number of rows and columns in the matrix setting.",
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-    "text": "Other Concepts\nDepending on the problem at hand, the supplied operator may need to meet other constraints. Here’s a short list additional operator concepts:\n\n\n\n\n\n\n\n\n\nConcept\nSupports\nSignature\nRequires\n\n\n\n\nLinearOperator\nA v \\mapsto o\nA.matvec(v, o)\nNA\n\n\nAdjointOperator\nA^T v \\mapsto o\nA.rmatvec(v, o)\nLinearOperator\n\n\nAdditiveOperator\no \\gets o + \\alpha Av\nA.matvec_add(v, alpha, o)\nLinearOperator\n\n\nAffineOperator\nSets t s.t. A + tB\nA.set_parameter(t)\nLinearOperator\n\n\n\nRespecting these constraints is opt-in: if your operator is symmetric and you only need access to the Lanczos method, then any satisfying the LinearOperator concept is sufficient.",
+    "text": "Other Concepts\nDepending on the problem at hand, the supplied operator may need to meet other constraints. Here’s a short list additional operator concepts:\n\n\n\n\n\n\n\n\n\nConcept\nSupports\nSignature\nRequires\n\n\n\n\nLinearOperator\nA v \\mapsto o\nA.matvec(v, o)\nNA\n\n\nAdjointOperator\nA^T v \\mapsto o\nA.rmatvec(v, o)\nLinearOperator\n\n\nAdditiveOperator\no \\gets o + \\alpha Av\nA.matvec_add(v, alpha, o)\nLinearOperator\n\n\nAffineOperator\nSets t s.t. A + tB\nA.set_parameter(t)\nLinearOperator\n\n\nQuadOperator\nv^T A v\nA.quad(v)\nNA\n\n\n\nThe exported methods in primate only need the minimum constraints to be satisfied to compile: if you need access to the Lanczos method, then just supporting the LinearOperator concept is sufficient. On the other hand, adding support for other constraints can optimize the efficiency of certain methods; for example, the hutch method technically only requires a LinearOperator to do trace estimation (via matvec calls), but will also compile and prefer calling quad with a QuadOperator as input. In such a situaton, if your operator has an efficient quadratic form v \\mapsto v^T A v, then implementing quad may improve the performance of hutch.",
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-    "text": "primate offers an extensible implementation of the stochastic Lanczos method (SLQ). There are many algorithms named the “stochastic Lanczos quadrature” in the literature; though each is typically related, they often have distinct goals. Pseudocode for a generic form of SLQ is given below:\nThis guide walks through the SLQ method implemented in primate, which can be specialized for different purposes.",
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     "text": "SLQ as a function template\nBelow is the full signature of the SLQ function template:\n// Stochastic Lanczos quadrature method\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid slq (\n  const Matrix& A,                    // Any *LinearOperator*\n  const function&lt; F(int,F*,F*) &gt;& f,  // Generic function\n  const function&lt; bool(int) &gt;& stop,  // Early-stop function\n  const int nv,                       // Num. of sample vectors\n  const Distribution dist,            // Sample vector distribution\n  RBG& rng,                           // Random bit generator\n  const int lanczos_degree,           // Krylov subspace degree\n  const F lanczos_rtol,               // Lanczos residual tolerance\n  const int orth,                     // Add. vectors to orthogonalize\n  const int ncv,                      // Num. of Lanczos vectors\n  const int num_threads,              // # threads to allocate \n  const int seed                      // Seed for RNG \n)\nMany of the runtime arguments are documented in the lanczos or sl_trace docs; the compile-time (template) parameters are:\n\nThe floating point type (e.g. float, double, long double)\nThe operator type (e.g. Eigen::MatrixXf, torch::Tensor, LinOp)\nThe multi-threaded random number generator (e.g. ThreadedRNG64)\n\nNote any type combination satisfying these concepts (e.g. std::floating_point, LinearOperator) generates a function specialized of said types at compile-time—this is known as template instantiation.",
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     "text": "Generality via function passing\nGiven a valid set of parameters, the main body of the SLQ looks something like this:\n  bool stop_flag = false;\n  #pragma omp parallel shared(stop_flag)\n  {\n    // &lt; allocations for Q, alpha, beta, etc. &gt; \n    int tid = omp_get_thread_num(); // thread-id \n    \n    #pragma omp for\n    for (i = 0; i &lt; nv; ++i){\n      if (stop_flag){ continue; }\n      generate_isotropic&lt; F &gt;(...); // populates q\n      lanczos_recurrence&lt; F &gt;(...); // populates alpha + beta\n      lanczos_quadrature&lt; F &gt;(...); // populates nodes + weights\n      f(i, q, Q, nodes, weights);   // Run user-supplied function \n      #pragma omp critical\n      {\n        stop_flag = stop(i);        // Checks for early-stopping\n      }\n    } // end for\n  } // end parallel \nThere are two functions that can be used for generalizing SLQ for different purposes.\nThe first generic function f can read, save, or modify the information available from the iteration index i, the isotropic vector q, the Lanczos vectors Q, and/or the quadrature information nodes, weights. Note this function is run in the parallel section.\nThe second is a boolean-valued function stop which can be used to stop the iteration early, for example if convergence has been achieved according to some rule. Since this is run in the critical section, it is called sequentially.",
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     "text": "Using SLQ to estimate \\mathrm{tr}(f(A))\nThe SLQ method is often used to estimate the trace of an arbitrary matrix function:\n \\mathrm{tr}(f(A)), \\quad \\text{ where } f(A) = U f(\\Lambda) U^T \nIt’s has been shown1 that the information obtained by the Lanczos method is sufficient to obtained a Gaussian quadrature approximation of the empirical spectral measure of A. By sampling zero-mean vectors satisfying \\mathbb{E}[v v^T] = I, one can obtain estimates of the trace above: \\operatorname{tr}(f(A)) \\approx \\frac{n}{\\mathrm{n}_{\\mathrm{v}}} \\sum_{l=1}^{\\mathrm{n}_{\\mathrm{v}}}\\left(\\sum_{k=0}^m\\left(\\tau_k^{(l)}\\right)^2 f\\left(\\theta_k^{(l)}\\right)\\right)\nIt turns out averaging these trace estimates yields unbiased, Girard-Hutchinson estimator of the trace. To see why this estimator is unbiased, note that:  \\mathtt{tr}(A) = \\mathbb{E}[v^T A v] \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nThus, all we need to do to estimate the trace of a matrix function is multiply and sum the quadrature nodes and weights output by SLQ.",
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     "text": "sl_trace method\nTo see how these formulas are actually implemented with the generic SLQ implementation, here’s an abbreviated form of the sl_trace function implemented by primate:\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid sl_trace(\n  const Matrix& A, const std::function&lt; F(F) &gt; sf, RBG& rbg, \n  const int nv, const int dist, const int engine_id, const int seed,\n  const int deg, const float lanczos_rtol, const int orth, const int ncv,\n  const F atol, const F rtol\n  F* estimates\n){  \n  using VectorF = Eigen::Array&lt; F, Dynamic, 1&gt;;\n\n  // Parameterize the trace function (runs in parallel)\n  auto trace_f = [&](int i, F* q, F* Q, F* nodes, F* weights){\n    Map&lt; VectorF &gt; nodes_v(nodes, deg, 1);     // no-op\n    Map&lt; VectorF &gt; weights_v(weights, deg, 1); // no-op\n    nodes_v.unaryExpr(sf);\n    estimates[i] = (nodes_v * weights_v).sum();\n  };\n  \n  // Convergence checking like scipy.integrate.quadrature\n  int n = 0;\n  F mu_est = 0.0, mu_pre = 0.0;\n  const auto early_stop = [&](int i) -&gt; bool {\n    ++n; // Number of estimates\n    mu_est = (1.0 / F(n)) * (estimates[i] + (n - 1) * mu_pre); \n    bool atol_check = abs(mu_est - mu_pre) &lt;= atol;\n    bool rtol_check = abs(mu_est - mu_pre) / mu_est &lt;= rtol; \n    mu_pre = mu_est; \n    return atol_check || rtol_check;\n  };\n\n  // Execute the stochastic Lanczos quadrature with the trace function \n  slq&lt; float &gt;(A, trace_f, early_stop, ...);\n}\nAs before, two functions are used to parameterize the slq method.\nThe first (trace_f) applies an arbitrary spectral function sf to the Rayleigh-Ritz values obtained by the Lanczos tridiagonalization of A(or equivalently, the nodes of the Gaussian quadrature). These are the \\theta’s in the pseudocode above. When multiplied by the weights of the quadrature, the corresponding sum forms an estimate of the trace of the matrix function.\nThe second function early_stop is used to check for convergence of the estimator. First, it uses the trace estimate x_n to update the sample mean \\mu_n via the formula:\n \\mu_n = n^{-1} [x_n + (n - 1)\\mu_{n-1}] \nThen, much in the same way the quadrature function from scipy.integrate approximates a definite integral, it checks for convergence using the absolute and relative tolerances supplied by the user. Returning true signals convergence, stopping the iteration early.",
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-    "text": "To demonstrate the SLQ method in Python, we start with a simple symmetric matrix A \\in \\mathbb{R}^{n \\times n}.\n\nimport numpy as np\nfrom primate.random import symmetric\nA = symmetric(150, psd = True)\n\nThis generates a random positive semi-definite matrix with eigenvalues in the interval [0, 1].\n\nfrom primate.trace import sl_trace\ntrace_estimate = sl_trace(A)\nprint(A.trace()) \nprint(trace_estimate)\n\n78.57591588717017\n78.428444\n\n\n\n# tr_est = np.mean(estimates)\n# print(f\"Error: {abs(tr_est - A.trace()):.5}\")\n# print(f\"Samples std. deviation: {estimates.std(ddof=1)}\")\n# print(f\"Estimator standard error: {estimates.std(ddof=1)/np.sqrt(len(estimates))}\")",
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+    "text": "Footnotes\n\n\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.↩︎",
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-    "text": "If you’re using pybind11, you can easily incorporate your own custom linear operator / matrix function pair using primates binding headers.",
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-    "text": "Example: Log determinant\nFor explanatory purposes, the following code outline how to call the trace estimator to compute the log determinant using a custom user-implemented operator LinOp:\n#include &lt;cmath&gt;                              // std::log\n#include &lt;_linear_operator/linear_operator.h&gt; // LinearOperator\n#include &lt;_lanczos/lanczos.h&gt;                 // sl_trace\n#include \"LinOp.h\"                            // custom class\n\nvoid slq_log_det(LinOp A, ...){ \n  static_assert(LinearOperator&lt; LinOp &gt;);  // Constraint check\n  const auto matrix_func = std::log;       // any invocable\n  auto rbg = ThreadedRNG64();              // default RNG\n  auto estimates = vector&lt; float &gt;(n, 0);  // output estimates\n  sl_trace&lt; float &gt;(                       // specific precision\n    A, matrix_func, rbg,                   // main arguments\n    ...,                                   // other inputs \n    estimates.data()                       // output \n  );\n}",
+    "text": "If you’re using pybind11, you can easily incorporate your own custom linear operator / matrix function pair using primates binding headers.\nTODO",
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-    "text": "primate is a standard PEP-517 package, and thus can be installed via pip:\npip install &lt; primate source directory &gt;\nCurrently the package must be built from source via cloning the repository. PYPI support is planned.",
+    "text": "In the introduction, the basics of implicit trace estimation were introduced, wherein it shown both in theory and with code using primate how to estimate the trace of matrix functions:\nf(A) = U f(\\Lambda) U^T, \\quad f: [a,b] \\to \\mathbb{R}\nIn particular, the introduction briefly covered how the Lanczos method is intimately connected to Gaussian Quadrature, and how this connection enables fast randomized trace estimation of f(A).\nIn this post, I’ll cover how to approximate the action v \\mapsto f(A)v for any function f with primate with its matrix_function API, and how to compose this functionality with other trace estimators.",
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-    "text": "random.rademacher(size, rng='splitmix64', seed=-1, dtype=np.float32)\nGenerates random vectors from the rademacher distribution.\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nsize\nint or tuple\nOutput shape to generate.\nrequired\n\n\nrng\nstr = “splitmix64”\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint = -1\nSeed for the generator. Use -1 to for random (non-deterministic) behavior.\n-1\n\n\ndtype\ndtype = float32\nFloating point dtype for the output. Must be float32 or float64.\nnp.float32\n\n\n\n\n\n\nnp.narray Randomly generated matrix of shape size with entries in { -1, 1 }.",
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+    "text": "Matrix function approximation\nIf A \\in \\mathbb{R}^{n \\times n} and n is large, obtaining f(A) \\in \\mathbb{R}^{n \\times n} explicitly can be very expensive. One way to sidestep this difficulty is to approximate v \\mapsto f(A)v using the degree-k Lanczos expansion:\n Q^T A Q = T \\quad \\Leftrightarrow \\quad f(A)v \\approx \\lVert x \\rVert \\cdot Q f(T) e_1 \nIt’s been shown by (Musco, Musco, and Sidford 2018) that this approximation has the following guarantee:\n\\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| \\leq 2\\|\\mathbf{x}\\| \\cdot \\min _{\\substack{\\text { polynomial } p \\\\ \\text { with degree }&lt;k}}\\left(\\max _{x \\in\\left[\\lambda_{\\min }(\\mathbf{A}), \\lambda_{\\max }(\\mathbf{A})\\right]}|f(x)-p(x)|\\right)\nIn other words, up to a factor of 2, the error \\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| is bounded by the uniform error of the best polynomial approximation to f with degree &lt; k. For general matrix functions, this implies that finite-precision Lanczos essentially matches strongest known exact arithmetic bounds.\nThis suggests an idea: can we convert any matrix-like object into a LinearOperator that transparently converts matrix-vector products Av into products with matrix-functions f(A)v?",
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-    "text": "Functions for estimating the trace of matrices and matrix functions.\n\n\n\ntrace.hutch\nEstimates the trace of a symmetric A or matrix function f(A) via a Girard-Hutchinson estimator.\n\n\ntrace.xtrace\n\n\n\n\n\n\n\nMatrix operators\n\n\n\noperator.matrix_function\nConstructs a LinearOperator approximating the action of the matrix function fun(A).\n\n\n\n\n\n\nRandomized module\n\n\n\nrandom.rademacher\nGenerates random vectors from the rademacher distribution.\n\n\nrandom.normal\nGenerates random vectors from the standard normal distribution.\n\n\n\n\n\n\nDiagonalization methods\n\n\n\ndiagonalize.lanczos\nLanczos method for matrix tridiagonalization.\n\n\n\n\n\n\nMiscellenous functions\n\n\n\nget_include\nReturn the directory that contains the primate’s *.h header files.",
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-    "text": "Functions for estimating the trace of matrices and matrix functions.\n\n\n\ntrace.hutch\nEstimates the trace of a symmetric A or matrix function f(A) via a Girard-Hutchinson estimator.\n\n\ntrace.xtrace",
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-    "text": "Randomized module\n\n\n\nrandom.rademacher\nGenerates random vectors from the rademacher distribution.\n\n\nrandom.normal\nGenerates random vectors from the standard normal distribution.",
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-    "text": "Miscellenous functions\n\n\n\nget_include\nReturn the directory that contains the primate’s *.h header files.",
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+    "section": "Establishing a baseline",
+    "text": "Establishing a baseline\nprimate allows the user to construct a LinearOperator directly from an existing matrix-like object A and callable fun through its matrix_function API.\n\nAs a baseline example, consider the action that adds an \\epsilon amount of mass to the diagonal of A:\nv \\mapsto (A + \\epsilon I)v\nFor any fixed \\epsilon, imitating this matrix action can be done in three different ways:\n\nObtain u = Av and then add u \\gets u + \\epsilon \\cdot v\nForm (A + \\epsilon I) and explicitly carry out the multiplication\nMultiply v by f(A) induced by f(\\lambda) = \\lambda + \\epsilon\n\nLets see what the code to accomplish this using (3) looks like:\n\nfrom primate.random import symmetric\nfrom primate.operator import matrix_function\n\n## Random matrix + input vector\nA = symmetric(150, pd = True)\nv = np.random.uniform(size=A.shape[1])\n\n## Ground truth v |-&gt; f(A)v\nLambda, U = np.linalg.eigh(A)\nv_truth = (U @ np.diag(Lambda + 0.10) @ U.T) @ v\n\n## Lanczos approximation\nM = matrix_function(A, fun = lambda x: x + 0.10)\nv_test = np.ravel(M @ v)\n\nvn, vn_approx = np.linalg.norm(v_truth), np.linalg.norm(v_test)\nprint(f\"f(A)v norm:  {vn:.6f}\")\nprint(f\"Approx norm: {vn_approx:.6f}\")\nprint(f\"Max diff:    {np.max(np.abs(v_truth - np.ravel(v_test))):.6e}\")\nprint(f\"cosine sim:  {np.dot(v_test, v_truth) / (vn * vn_approx):6e}\")\n\nf(A)v norm:  4.392660\nApprox norm: 4.392660\nMax diff:    5.218048e-15\ncosine sim:  1.000000e+00\n\n\nObserve M matches the ground truth v \\mapsto (A + \\epsilon I)v. In this way, one benefit of using matrix_function is that it allows one to approximate f(A) by thinking only about what is happening at the spectral level (as opposed to the matrix level). Of course, this example is a bit non-convincing as there are simpler ways of achieving the same result as shown by (1) and (2), for example:\n\nnp.allclose(A @ v + 0.10 * v, v_truth)\n\nTrue\n\n\nBaseline established.",
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-    "text": "trace.sl_gauss(A, n=150, deg=20, pdf='rademacher', rng='pcg', seed=-1, orth=0, num_threads=0)\nStochastic Gaussian quadrature approximation.\nComputes a set of sample nodes and weights for the degree-k orthogonal polynomial approximating the cumulative spectral measure of A. This function can be used to approximate the spectral density of A, or to approximate the spectral sum of any function applied to the spectrum of A.\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nn\nint\nNumber of random vectors to sample for the quadrature estimate.\n150\n\n\ndeg\nint\nDegree of the quadrature approximation.\n20\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘lcg64’, ‘mt64’\nRandom number generator to use (PCG64 by default).\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against when building the Krylov basis.\n0\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Setting num_threads &lt; 1 to let OpenMP decide.\n0\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\nfloat\nEstimate of the trace of the matrix function f(A).\n\n\n(dict, optional)\nIf ‘info = True’, additional information about the computation.",
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+    "section": "When v \\mapsto f(A)v is not known",
+    "text": "When v \\mapsto f(A)v is not known\nOn the other hand, there are plenty of situations where one doesn’t have access to these simpler expressions. For example, consider the map:\nv \\mapsto (A + \\epsilon I)^{-1} v\nSuch expressions pop up in a variety of settings, such as in Tikhonov regularization, in Schatten-norm estimation Ubaru, Saad, and Seghouane (2017), in the Cholesky factorization of PSD matrices, and so on. Unlike the previous setting, we cannot readily access v \\mapsto f(A)v unless we explicitly compute the full spectral decomposition of A or the inverse of A, both of which are expensive to obtain.\n\n## Alternative: v_truth = np.linalg.inv((A + 0.10 * np.eye(A.shape[0]))) @ v\nv_truth = (U @ np.diag(np.reciprocal(Lambda + 0.10)) @ U.T) @ v\n\n## Lanczos approximation\nM = matrix_function(A, fun = lambda x: 1.0 / (x + 0.10))\nv_test = np.ravel(M @ v)\n\nvn, vn_approx = np.linalg.norm(v_truth), np.linalg.norm(v_test)\nprint(f\"f(A)v norm:  {vn:.6f}\")\nprint(f\"Approx norm: {vn_approx:.6f}\")\nprint(f\"Max diff:    {np.max(np.abs(v_truth - np.ravel(v_test))):.6e}\")\nprint(f\"cosine sim:  {np.dot(v_test, v_truth) / (vn * vn_approx):6e}\")\n\nf(A)v norm:  32.982583\nApprox norm: 32.982583\nMax diff:    1.626085e-05\ncosine sim:  1.000000e+00\n\n\nThere is a larger degree of error compared to the base as evidenced by the \\lVert \\cdot \\rVert_\\infty-normed difference between v_truth and v_test, however this is to be expected, as in general approximating the action v \\mapsto A^{-1} v will always be more difficult that v \\mapsto A v, even if A is well-conditioned.",
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-    "text": "trace\n\n\n\n\n\nName\nDescription\n\n\n\n\nhutch\nEstimates the trace of a matrix A or matrix function f(A) via a Girard-Hutchinson estimator.\n\n\nxtrace"
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-    "text": "primate supports a variety of matrix-types of the box, including numpy ndarray’s, compressed sparse matrices (a lá SciPy), and LinearOperators—the latter enables the use of matrix free operators.\nOutside of the natively types above, the basic requirements for any operator A to be used with e.g. the Lanczos method in primate are:\n\nA method A.matvec(input: ndarray) -&gt; ndarray implementing v \\mapsto Av\nAn attribute A.shape -&gt; tuple[int, int] giving the output/input dimensions of A\n\n\n\nHere’s an example of a simple operator representing a Diagonal matrix, which inherits a .matvec() method by following the subclassing rules of SciPy’s LinearOperator:\nimport numpy as np \nfrom numpy.typing import ArrayLike\nfrom scipy.sparse.linalg import LinearOperator \n\nclass DiagonalOp(LinearOperator):\n  diag: np.ndarray = None\n  \n  def __init__(self, d: ArrayLike, dtype = None):\n    self.diag = np.array(d)\n    self.shape = (len(d), len(d))\n    self.dtype = np.dtype('float32') if dtype is None else dtype\n\n  def _matvec(self, x: ArrayLike) -&gt; np.ndarray:\n    out = self.diag * np.ravel(x)\n    return out.reshape(x.shape)",
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+    "text": "Back to trace estimation\nThere are several use-cases wherein one might be interested in the output f(A)v itself—see Musco, Musco, and Sidford (2018) and references within for some examples. One use-case is the extension of existing matvec-dependent implicit trace estimators to matrix-function trace estimators.\n\nfrom primate.trace import hutch, xtrace\nM = matrix_function(A, fun=\"log\")\nprint(f\"Logdet exact:  {np.sum(np.log(np.linalg.eigvalsh(A))):6e}\")\nprint(f\"Logdet Hutch:  {hutch(A, fun='log'):6e}\")\nprint(f\"Logdet XTrace: {xtrace(M):6e}\")\n\nLogdet exact:  -1.483218e+02\nLogdet Hutch:  -1.475913e+02\nLogdet XTrace: -1.483155e+02\n\n\nAs with the hutch estimators applied to matrix functions, note that the action v \\mapto f(A)v is subject to the approximation errors studied by Musco, Musco, and Sidford (2018), making such extensions limited functions that are well-approximated by Lanczos.",
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-    "text": "The Lanczos Iteration\nThe Lanczos method exposes the spectrum of A by successively projecting onto Krylov subspaces. That is, given a symmetric A \\in \\mathbb{R}^{n \\times n} with eigenvalues \\lambda_1 \\geq \\lambda_2 &gt; \\dots \\geq \\lambda_r &gt; 0 and a vector v \\in \\mathbb{R} \\setminus \\{0\\}, the order-j Krylov subspaces / Krylov matrices of the pair (A, v) are given by:\n\n\\mathcal{K}_j(A, v) := \\mathrm{span}\\{ v, Av, A^2 v, \\dots, A^{j-1}v \\}, \\quad \\quad K_j(A, v) = [ v \\mid Av \\mid A^2 v \\mid \\dots \\mid A^{j-1}]\n\nKrylov subspaces arise naturally from using the minimal polynomial of A to express A^{-1} in terms of powers of A: if A is nonsingular and its minimal polynomial has degree m, then A^{-1}v \\in K_m(A, v) and K_m(A, v) is an invariant subspace.\nThe spectral theorem implies that since A is symmetric, it is orthogonally diagonalizable: thus, \\Lambda(A) may be obtained by generating an orthonormal basis for \\mathcal{K}_n(A, v). To do this, the Lanczos method constructs successive QR factorizations of K_j(A,v) = Q_j R_j for each j = 1, 2, \\dots, n. Due to A’s symmetry and the orthogonality of Q_j, we have q_k^T A q_l = q_l^T A^T q_k = 0 for k &gt; l + 1, implying T_j = Q_j^T A Q_j has a tridiagonal structure:\n\\begin{equation}\n    T_j = \\begin{bmatrix}\n    \\alpha_1 & \\beta_2 & & & \\\\\n    \\beta_2 & \\alpha_2 & \\beta_3 & & \\\\\n     & \\beta_3 & \\alpha_3 & \\ddots & \\\\\n    & & \\ddots & \\ddots & \\beta_{j} \\\\\n    & & & \\beta_{j} & \\alpha_{j}\n    \\end{bmatrix}, \\; \\beta_j &gt; 0, \\; j = 1, 2, \\dots, n\n\\end{equation}\n\nGiven an initial pair (A, q_1) satisfying \\lVert q_1 \\rVert = 1, one can restrict and project A to its j-th Krylov subspace T_j via: \n\\begin{equation}\n    A Q_j = Q_j T_j + \\beta_{j+1} q_{j+1} e_{j}^T \\quad\\quad (\\beta_j &gt; 0)\n\\end{equation}\n where Q_j = [\\, q_1 \\mid q_2 \\mid \\dots \\mid q_j \\,] is an orthonormal set of vectors mutually orthogonal to q_{j+1}. Equating the j-th columns on each side of the above and rearranging the terms yields the famed three-term recurrence: \\begin{equation}\n     \\beta_{j} \\, q_{j+1} = A q_j - \\alpha_j \\, q_j - \\beta_{j\\text{-}1} \\, q_{j\\text{-}1}  \n\\end{equation}\n where \\alpha_j = q_j^T A q_j, \\beta_j = \\lVert r_j \\rVert_2, r_j = (A - \\alpha_j I)q_j - \\beta_{j\\text{-}1} q_j, and q_{j+1} = r_j / \\beta_j. The equation above is a variable-coefficient second-order linear difference equation, and such equations have unique solutions: if (q_{j\\text{-}1}, \\beta_j, q_j) are known, then (\\alpha_j, \\beta_{j+1}, q_{j+1}) are completely determined. This sequential process which iteratively builds T_j via this three-term recurrence is what is known as the Lanczos iteration.",
+    "section": "The “iteration” part",
+    "text": "The “iteration” part\nLanczos originally referred to his algorithm as the method of minimized iterations, and indeed nowadays it is often called an iterative method. Where’s the iterative component?\nIf you squint hard enough, you can deduce that for every j \\in [1, n): A Q_j = Q_j T_j + \\beta_{j+1} q_{j+1} e_{j}^T Equating the j-th columns on each side of the equation and rearranging yields a three-term recurrence: \n\\begin{align*}\nA q_j &= \\beta_{j\\text{-}1} q_{j\\text{-}1} + \\alpha_j q_j + \\beta_j q_{j+1} \\\\\n\\Leftrightarrow \\beta_{j} \\, q_{j+1} &= A q_j - \\alpha_j \\, q_j - \\beta_{j\\text{-}1} \\, q_{j\\text{-}1}  \n\\end{align*}\n\nThe equation above is a variable-coefficient second-order linear difference equation, and it is known such equations have unique solutions; they are given below: \n\\alpha_j = q_j^T A q_j, \\;\\; \\beta_j = \\lVert r_j \\rVert_2, \\;\\; q_{j+1} = r_j / \\beta_j\n\n\n\\text{where  } r_j = (A - \\alpha_j I)q_j - \\beta_{j\\text{-}1} q_j\n\nIn other words, if (q_{j\\text{-}1}, \\beta_j, q_j) are known, then (\\alpha_j, \\beta_{j+1}, q_{j+1}) are completely determined. In theory, this means we can iteratively generate both Q and T using just a couple vectors at a time—no need to explicitly call to the QR algorithm as shown above. Pretty nifty, eh!",
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-    "text": "Uniqueness of T\nUnfortunately, unlike the spectral decomposition A = V \\Lambda V^T—which identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M—there is no canonical choice of T_j due to the arbitrary choice of v. However, there is a connection between the iterates K_j(A,v) and the full tridiagonalization of A: if Q^T A Q = T is tridiagonal and Q= [\\, q_1 \\mid q_2 \\mid \\dots \\mid q_n \\,] is an n \\times n orthogonal matrix Q Q^T = I_n = [e_1, e_2, \\dots, e_n], then we have: \n\\begin{equation}\n    K_n(A, q_1) = Q Q^T K_n(A, q_1) = Q[ \\, e_1 \\mid T e_1 \\mid T^2 e_1 \\mid \\dots \\mid T^{n-1} e_1 \\, ]\n\\end{equation}\n is the QR factorization of K_n(A, q_1). Thus, tridiagonalizing A with respect to a unit-norm q_1 determines Q. Indeed, the Implicit Q Theorem asserts that if an upper Hessenburg matrix T \\in \\mathbb{R}^{n \\times n} has only positive elements on its first subdiagonal and there exists an orthogonal matrix Q such that Q^T A Q = T, then Q and T are uniquely determined by (A, q_1).",
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-    "text": "SLQ as a function template\nBelow is the full signature of the SLQ function template:\n// Stochastic Lanczos quadrature method\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid slq (\n  const Matrix& A,                    // Any *LinearOperator*\n  const function&lt; F(int,F*,F*) &gt;& f,  // Generic function\n  const function&lt; bool(int) &gt;& stop,  // Early-stop function\n  const int nv,                       // Num. of sample vectors\n  const Distribution dist,            // Sample vector distribution\n  RBG& rng,                           // Random bit generator\n  const int lanczos_degree,           // Krylov subspace degree\n  const F lanczos_rtol,               // Lanczos residual tolerance\n  const int orth,                     // Add. vectors to orthogonalize\n  const int ncv,                      // Num. of Lanczos vectors\n  const int num_threads,              // # threads to allocate \n  const int seed                      // Seed for RNG \n)\nMany of the runtime arguments are documented in the lanczos or sl_trace docs; the compile-time (template) parameters are:\n\nThe floating point type (e.g. float, double, long double)\nThe operator type (e.g. Eigen::MatrixXf, torch::Tensor, LinOp)\nThe multi-threaded random number generator (e.g. ThreadedRNG64)\n\nNote any type combination satisfying these concepts (e.g. std::floating_point, LinearOperator) generates a function specialized of said types at compile-time—this is known as template instantiation."
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-    "text": "Generality via function passing\nGiven a valid set of parameters, the main body of the SLQ looks something like this:\n  bool stop_flag = false;\n  #pragma omp parallel shared(stop_flag)\n  {\n    // &lt; allocations for Q, alpha, beta, etc. &gt; \n    int tid = omp_get_thread_num(); // thread-id \n    \n    #pragma omp for\n    for (i = 0; i &lt; nv; ++i){\n      if (stop_flag){ continue; }\n      generate_isotropic&lt; F &gt;(...); // populates q\n      lanczos_recurrence&lt; F &gt;(...); // populates alpha + beta\n      lanczos_quadrature&lt; F &gt;(...); // populates nodes + weights\n      f(i, q, Q, nodes, weights);   // Run user-supplied function \n      #pragma omp critical\n      {\n        stop_flag = stop(i);        // Checks for early-stopping\n      }\n    } // end for\n  } // end parallel \nThere are two functions that can be used for generalizing SLQ for different purposes.\nThe first generic function f can read, save, or modify the information available from the iteration index i, the isotropic vector q, the Lanczos vectors Q, and/or the quadrature information nodes, weights. Note this function is run in the parallel section.\nThe second is a boolean-valued function stop which can be used to stop the iteration early, for example if convergence has been achieved according to some rule. Since this is run in the critical section, it is called sequentially."
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-    "text": "Using SLQ to estimate \\mathrm{tr}(f(A))\nThe SLQ method is often used to estimate the trace of an arbitrary matrix function:\n \\mathrm{tr}(f(A)), \\quad \\text{ where } f(A) = U f(\\Lambda) U^T \nIt’s has been shown1 that the information obtained by the Lanczos method is sufficient to obtained a Gaussian quadrature approximation of the empirical spectral measure of A. By sampling zero-mean vectors satisfying \\mathbb{E}[v v^T] = I, one can obtain estimates of the trace above: \\operatorname{tr}(f(A)) \\approx \\frac{n}{\\mathrm{n}_{\\mathrm{v}}} \\sum_{l=1}^{\\mathrm{n}_{\\mathrm{v}}}\\left(\\sum_{k=0}^m\\left(\\tau_k^{(l)}\\right)^2 f\\left(\\theta_k^{(l)}\\right)\\right)\nIt turns out averaging these trace estimates yields unbiased, Girard-Hutchinson estimator of the trace. To see why this estimator is unbiased, note that:  \\mathtt{tr}(A) = \\mathbb{E}[v^T A v] \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nThus, all we need to do to estimate the trace of a matrix function is multiply and sum the quadrature nodes and weights output by SLQ."
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-    "text": "sl_trace method\nTo see how these formulas are actually implemented with the generic SLQ implementation, here’s an abbreviated form of the sl_trace function implemented by primate:\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid sl_trace(\n  const Matrix& A, const std::function&lt; F(F) &gt; sf, RBG& rbg, \n  const int nv, const int dist, const int engine_id, const int seed,\n  const int deg, const float lanczos_rtol, const int orth, const int ncv,\n  const F atol, const F rtol\n  F* estimates\n){  \n  using VectorF = Eigen::Array&lt; F, Dynamic, 1&gt;;\n\n  // Parameterize the trace function (runs in parallel)\n  auto trace_f = [&](int i, F* q, F* Q, F* nodes, F* weights){\n    Map&lt; VectorF &gt; nodes_v(nodes, deg, 1);     // no-op\n    Map&lt; VectorF &gt; weights_v(weights, deg, 1); // no-op\n    nodes_v.unaryExpr(sf);\n    estimates[i] = (nodes_v * weights_v).sum();\n  };\n  \n  // Convergence checking like scipy.integrate.quadrature\n  int n = 0;\n  F mu_est = 0.0, mu_pre = 0.0;\n  const auto early_stop = [&](int i) -&gt; bool {\n    ++n; // Number of estimates\n    mu_est = (1.0 / F(n)) * (estimates[i] + (n - 1) * mu_pre); \n    bool atol_check = abs(mu_est - mu_pre) &lt;= atol;\n    bool rtol_check = abs(mu_est - mu_pre) / mu_est &lt;= rtol; \n    mu_pre = mu_est; \n    return atol_check || rtol_check;\n  };\n\n  // Execute the stochastic Lanczos quadrature with the trace function \n  slq&lt; float &gt;(A, trace_f, early_stop, ...);\n}\nAs before, two functions are used to parameterize the slq method.\nThe first (trace_f) applies an arbitrary spectral function sf to the Rayleigh-Ritz values obtained by the Lanczos tridiagonalization of A(or equivalently, the nodes of the Gaussian quadrature). These are the \\theta’s in the pseudocode above. When multiplied by the weights of the quadrature, the corresponding sum forms an estimate of the trace of the matrix function.\nThe second function early_stop is used to check for convergence of the estimator. First, it uses the trace estimate x_n to update the sample mean \\mu_n via the formula:\n \\mu_n = n^{-1} [x_n + (n - 1)\\mu_{n-1}] \nThen, much in the same way the quadrature function from scipy.integrate approximates a definite integral, it checks for convergence using the absolute and relative tolerances supplied by the user. Returning true signals convergence, stopping the iteration early."
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-    "text": "To clarify that that means, here’s an abstract presentation of the generic SLQ procedure:\n\n\n\\begin{algorithm} \\caption{Stochastic Lanczos Quadrature} \\begin{algorithmic} \\Input Symmetric operator ($A \\in \\mathbb{R}^{n \\times n}$) \\Require Number of queries ($n_v$), Degree of quadrature ($k$) \\Function{SLQ}{$A$, $n_v$, $k$} \\State $\\Gamma \\gets 0$ \\For{$j = 1, 2, \\dots, n_v$} \\State $v_i \\sim \\mathcal{D}$ where $\\mathcal{D}$ satisfies $\\mathbb{E}(v v^\\top) = I$ \\State $T^{(j)}(\\alpha, \\beta)$ $\\gets$ $\\mathrm{Lanczos}(A,v_j,k+1)$ \\State $[\\Theta, Y] \\gets \\mathrm{eigh\\_tridiag}(T^{(j)}(\\alpha, \\beta))$ \\State $\\tau_i \\gets \\langle e_1, y_i \\rangle$ \\State &lt; Do something with the node/weight pairs $(\\theta_i, \\tau_i^2)$ &gt; \\EndFor \\EndFunction \\end{algorithmic} \\end{algorithm}",
+    "section": "Wait, isn’t T arbitrary?",
+    "text": "Wait, isn’t T arbitrary?\nUnfortunately—and unlike the spectral decomposition2—there is no canonical choice of T. Indeed, as T is a family with n - 1 degrees of freedom and v \\in \\mathbb{R}^n was chosen arbitrarily, there are infinitely many essentially distinct such decompositions.\nNot all hope is lost though, as it turns out that T is actually fully characterized by v. To see this, notice that since Q is an orthogonal matrix, we have:\n Q Q^T = I_n = [e_1, e_2, \\dots, e_n] \nBy extension, given an initial pair (A, q_1) satisfying \\lVert q_1 \\rVert = 1, the following holds:\n\nK_n(A, q_1) = Q Q^T K_n(A, q_1) = Q[ \\, e_1 \\mid T e_1 \\mid T^2 e_1 \\mid \\dots \\mid T^{n-1} e_1 \\, ]\n\n…this is actually a QR factorization, which is essentially unique! Indeed, the Implicit Q Theorem asserts that if an upper Hessenburg matrix T \\in \\mathbb{R}^{n \\times n} has only positive elements on its first subdiagonal and there exists an orthogonal matrix Q such that Q^T A Q = T, then Q and T are uniquely determined3 by (A, q_1).",
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-    "text": "Why care?\nComputing the eigen-decomposition A = U \\Lambda U^T for general symmetric A \\in \\mathbb{R}^{n \\times n} is essentially bounded above by \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx is the matrix-multiplication constant. This translates to an effective \\Omega(n^3) time bound if we exclude the Strassen-model for matrix multiplication (since it is not practical anyways). However, if one can show that v \\mapsto Av \\approx O(n), then there is a simple way of obtaining \\Lambda(A) in O(n^2) time and O(n) space!",
+    "section": "Beating the complexity bounds",
+    "text": "Beating the complexity bounds\nElegant and as theoretically founded as the Lanczos method may be, is it efficient in practice?\nLet’s start by establishing a baseline on its complexity:\n\nTheorem 1 (Parlett 1994) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos method computes \\Lambda(A) in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic\n\nAs its clear from the theorem, if we specialize it such that r = n and \\eta = \\nu = n, then the Lanczos method requires just O(n^2) time and O(n) space to execute. In other words, the Lanczos method drops both the time and space complexity4 of obtaining spectral information by order of magnitude over similar eigen-algorithms that decompose A directly.\nTo see why this is true, note that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then carrying out the recurrence clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nNow, if we need to store all of Y or Q explicitly, we clearly need O(n^2) space to do so. However, if we only need the eigen-values \\Lambda(A) (and not their eigen-vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!",
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-    "text": "Quadratic time and linear space? How?\nUnless you know the tricks, its not obvious at all decompositions above takes just O(n^2) time and O(n) space to obtain. So… how does the complexity argument play out?\n\nThe cubic bound\nFirst, its important to establish that computing the eigen-decomposition A = U \\Lambda U^T for general symmetric A \\in \\mathbb{R}^{n \\times n} is essentially bounded by \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx 2.37\\dots is the matrix-multiplication constant. Conceptually, if we exclude the Strassen-model for matrix multiplication (since it is not practical anyways), this translates to an effective \\Omega(n^3) time bound.\n\n\nSolving the right problem\nIn some applications, the eigenvectors are not needed all at once (or at all, even). One of the main draws to the Lanczos method is its efficiency: if one can perform v \\mapsto Av quickly—say, in \\approx O(n) time—then the Lanczos method can construct \\Lambda(A) in just O(n^2) time and O(n) space! Moreover, entire method is matrix free as the only input to the algorithm is a (fast) matrix-vector product v \\mapsto Av: one need not store A explicitly to do this for many special types of linear operators.\nIf you squint hard enough, you can deduce that since A Q_j = Q_j T_j + \\beta_{j+1} q_{j+1} e_{j}^T, every symmetric A \\in \\mathbb{R}^{n \\times n} expanded this way admits a three-term recurrence: \n\\begin{align*}\nA q_j &= \\beta_{j\\text{-}1} q_{j\\text{-}1} + \\alpha_j q_j + \\beta_j q_{j+1} \\\\\n\\Leftrightarrow \\beta_{j} \\, q_{j+1} &= A q_j - \\alpha_j \\, q_j - \\beta_{j\\text{-}1} \\, q_{j\\text{-}1}  \n\\end{align*}\n\nThe equation above is a variable-coefficient second-order linear difference equation, and it is known such equations have unique solutions: \n\\alpha_j = q_j^T A q_j, \\;\\; \\beta_j = \\lVert r_j \\rVert_2, \\;\\; q_{j+1} = r_j / \\beta_j\n\n\n\\text{where  } r_j = (A - \\alpha_j I)q_j - \\beta_{j\\text{-}1} q_j\n\nIn other words, if (q_{j\\text{-}1}, \\beta_j, q_j) are known, then (\\alpha_j, \\beta_{j+1}, q_{j+1}) are completely determined. This fact is fantastic from a computational point of view: no explicit call to the QR algorithm necessary2!\nNote that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then the above procedure clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nMoreover, if we only need the eigen-values \\Lambda(A) ( and not their vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!\nThe description above is essentially the proof of the following Theorem:\n\nTheorem 1 (Parlett 1994, Simon 1984) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos iteration computes \\Lambda(A) = \\{ \\lambda_1, \\lambda_2, \\dots, \\lambda_r \\} in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic",
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+    "text": "Footnotes\n\n\nA variant of the Lanczos method is actually at the heart scipy.sparse.linalg’s default eigsh solver (which is a port of ARPACK).↩︎\nThe spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M↩︎\nThe spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M↩︎\nFor general A \\in \\mathbb{R}^{n \\times n}, computing the spectral-decomposition is essentially bounded by the matrix-multiplication time: \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx 2.37\\dots is the matrix multiplication constant. If we exclude the Strassen model for computation, we get effectively a \\Omega(n^3) time and \\Omega(n^2) space bound.↩︎",
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-    "text": "primate is a standard PEP-517 package, and thus can be installed via pip:\npip install &lt; primate source directory &gt;\nCurrently the package must be built from source via cloning the repository. PYPI support is planned.\n\n\nPlatform support\nPlatform-specific wheels are currently built with cibuildwheel and uploaded to PyPI. These enable primate to be installed on supported architecture / CPython implementations without compilation. The currently supported platforms include:\nIf your platform isn’t on this list, feel free to make an issue requesting support for your platform.\n\n\nCompiling from source\nTo install the package from its source distribution, a C++20 compiler is required; the current builds are all built with some variant of clang. For platform- and compiler-specific settings, consult the build scripts and CI configuration files.\n\n\nC++ Installation\nprimate’s C++ interface is header-only, making it easy to compile your own extension modules. The simplest way to link these headers is to add primate as a dependency to your package and use the get_include() function to find the appropriate directory.\n\nsetuptoolsmeson-pythongit submodule\n\n\n# setup.py\nimport primate as pm\n...\nExtension('extension_name', ..., include_dirs=[pm.get_include()])\n...\n\n\n# meson.build\n...\nprimate_include_dirs = run_command(py, \n  ['-c', 'import primate as pm; print(pm.get_include())']\n).stdout().strip()\n...\n\n\nAssuming your headers are located in extern, from your git repository, you can use:\ngit submodule add https://github.com/peekxc/primate extern/primate\ngit submodule update --init\nFrom here, you can now include extern/primate/include into your C++ source files, or you can add this directory to the search path used other various build tools, such as CMake or Meson.",
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-    "text": "Surpassing the cubic bound\nComputing the eigen-decomposition A = U \\Lambda U^T for general symmetric A \\in \\mathbb{R}^{n \\times n} is essentially bounded by \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx 2.37\\dots is the matrix-multiplication constant. Conceptually, if we exclude the Strassen-model for matrix multiplication (since it is not practical anyways), this translates to an effective \\Omega(n^3) time bound. Not great!\nOne of the main draws to the Lanczos method is its efficiency: if one can perform v \\mapsto Av quickly—say, in \\approx O(n) time—then the Lanczos method can construct \\Lambda(A) in just O(n^2) time and O(n) space! Moreover, entire method is matrix free as the only input to the algorithm is a (fast) matrix-vector product v \\mapsto Av: one need not store A explicitly to do this for many special types of linear operators.",
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-    "text": "Footnotes\n\n\nMusco, Cameron, Christopher Musco, and Aaron Sidford. (2018) “Stability of the Lanczos method for matrix function approximation.”↩︎\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature.↩︎\nThis includes std::function’s, C-style function pointers, functors, and lambda expressions.↩︎",
+    "text": "Footnotes\n\n\nSingle-thread execution only; ARM-based OSX runners come stocked with Apple’s clang, which doesn’t natively ship with libomp.dylib, though this may be fixable. Feel free to file an PR if you can get this working.↩︎\nSingle-thread execution only; primate depends on OpenMP 4.5+, which isn’t supported on any Windows compiler I’m aware of.↩︎",
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-    "text": "Footnotes\n\n\nA variant of the Lanczos method is actually at the heart scipy.sparse.linalg’s default eigsh solver (which is a port of ARPACK).↩︎\nThe spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M↩︎\nThe spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M↩︎\nFor general A \\in \\mathbb{R}^{n \\times n}, computing the spectral-decomposition is essentially bounded by the matrix-multiplication time: \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx 2.37\\dots is the matrix multiplication constant. If we exclude the Strassen model for computation, we get effectively a \\Omega(n^3) time and \\Omega(n^2) space bound.↩︎",
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+    "text": "primate supports a variety of matrix-types of the box, including numpy ndarray’s, compressed sparse matrices (a lá SciPy), and LinearOperators—the latter enables the use of matrix free operators.\nOutside of the natively types above, the basic requirements for any operator A to be used with e.g. the Lanczos method in primate are:\n\nA method A.matvec(input: ndarray) -&gt; ndarray implementing v \\mapsto Av\nAn attribute A.shape -&gt; tuple[int, int] giving the output/input dimensions of A\n\n\n\nHere’s an example of a simple operator representing a Diagonal matrix, which inherits a .matvec() method by following the subclassing rules of SciPy’s LinearOperator:\nimport numpy as np \nfrom numpy.typing import ArrayLike\nfrom scipy.sparse.linalg import LinearOperator \n\nclass DiagonalOp(LinearOperator):\n  diag: np.ndarray = None\n  \n  def __init__(self, d: ArrayLike, dtype = None):\n    self.diag = np.array(d)\n    self.shape = (len(d), len(d))\n    self.dtype = np.dtype('float32') if dtype is None else dtype\n\n  def _matvec(self, x: ArrayLike) -&gt; np.ndarray:\n    out = self.diag * np.ravel(x)\n    return out.reshape(x.shape)",
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-    "text": "Footnotes\n\n\nBefore v0.2, much of primate’s code was essentially ported and refactored from imate. The code for v0.2+ has been re-written using the Eigen template C++ library.↩︎\nprimate does not provide native GPU-implemented Linear operators. However, there is nothing preventing one from using e.g. CUDA- or ROCm-based GPU-based tensor libraries to accelerate matrix-vector products. Indeed, primate was designed to work with essentially any operator matching the interface.↩︎\nSee imates documentation for the list of supported functions.↩︎",
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+    "text": "C++ usage\nSimilarly, to get started calling any matrix-free function provided by primate on the C++ side, such hutch or lanczos, simply pass any type with .shape() and .matvec() member functions:\nclass LinOp {\n  int nrow, ncol;\n  \n  LinOp(int nr, int nc) : nrow(nr), ncol(nc) {}\n  \n  void matvec(const float* input, float* output) const {\n    ... // implementation details \n  }\n\n  void shape() const { return std::make_pair(nrow, ncol); }\n}\nIt’s up to you to ensure shape() yields the correct size; primate will supply vectors to input of size .shape().second (number of columns) and guarantees the pointer to the output will be at least shape().first (number of rows), no more.\nTo read more about how semantics extend to the C++ side as well via C++20 concepts—see the C++ integration guide. If you’re using pybind11 and you want to extend primate’s Python API to work natively with linear operator implemented in C++, see the pybind11 integration guide.",
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-    "text": "This guide walks through how to parameterize the SLQ method implemented in primate on the C++ side to approximate some spectral quantity of interest.",
+    "text": "primate, short for Probabalistic Implicit Matrix Trace Estimator, is a Python package that provides estimators of quantities derived from matrix functions; that is, matrices parameterized by functions:\nf(A) \\triangleq U f(\\Lambda) U^{\\intercal}, \\quad \\quad f : [a,b] \\to \\mathbb{R}\nEstimator approximations are obtained via the Lanczos1 and stochastic Lanczos quadrature2 methods, which are well-suited for sparse or structured operators supporting fast v \\mapsto Av actions.\nNotable features of primate include:\nprimate was partially inspired by the imate package—for a comparison of the two, see here.",
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-    "text": "SLQ as a function template\nBelow is the full signature of the SLQ function template:\n// Stochastic Lanczos quadrature method\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid slq (\n  const Matrix& A,                    // Any *LinearOperator*\n  const function&lt; F(int,F*,F*) &gt;& f,  // Generic function\n  const function&lt; bool(int) &gt;& stop,  // Early-stop function\n  const int nv,                       // Num. of sample vectors\n  const Distribution dist,            // Sample vector distribution\n  RBG& rng,                           // Random bit generator\n  const int lanczos_degree,           // Krylov subspace degree\n  const F lanczos_rtol,               // Lanczos residual tolerance\n  const int orth,                     // Add. vectors to orthogonalize\n  const int ncv,                      // Num. of Lanczos vectors\n  const int num_threads,              // # threads to allocate \n  const int seed                      // Seed for RNG \n)\nMany of the runtime arguments are documented in the lanczos or sl_trace docs; the compile-time (template) parameters are:\n\nThe floating point type (e.g. float, double, long double)\nThe operator type (e.g. Eigen::MatrixXf, torch::Tensor, LinOp)\nThe multi-threaded random number generator (e.g. ThreadedRNG64)\n\nNote any type combination satisfying these concepts (e.g. std::floating_point, LinearOperator) generates a function specialized of said types at compile-time—this is known as template instantiation.",
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+    "text": "Footnotes\n\n\nMusco, Cameron, Christopher Musco, and Aaron Sidford. (2018) “Stability of the Lanczos method for matrix function approximation.”↩︎\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature.↩︎\nThis includes std::function’s, C-style function pointers, functors, and lambda expressions.↩︎",
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-    "text": "Generality via function passing\nGiven a valid set of parameters, the main body of the SLQ looks something like this:\n  bool stop_flag = false;\n  #pragma omp parallel shared(stop_flag)\n  {\n    // &lt; allocations for Q, alpha, beta, etc. &gt; \n    int tid = omp_get_thread_num(); // thread-id \n    \n    #pragma omp for\n    for (i = 0; i &lt; nv; ++i){\n      if (stop_flag){ continue; }\n      generate_isotropic&lt; F &gt;(...); // populates q\n      lanczos_recurrence&lt; F &gt;(...); // populates alpha + beta\n      lanczos_quadrature&lt; F &gt;(...); // populates nodes + weights\n      f(i, q, Q, nodes, weights);   // Run user-supplied function \n      #pragma omp critical\n      {\n        stop_flag = stop(i);        // Checks for early-stopping\n      }\n    } // end for\n  } // end parallel \nThere are two functions that can be used for generalizing SLQ for different purposes.\nThe first generic function f can read, save, or modify the information available from the iteration index i, the isotropic vector q, the Lanczos vectors Q, and/or the quadrature information nodes, weights. Note this function is run in the parallel section.\nThe second is a boolean-valued function stop which can be used to stop the iteration early, for example if convergence has been achieved according to some rule. Since this is run in the critical section, it is called sequentially.",
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-    "text": "Using SLQ to estimate \\mathrm{tr}(f(A))\nThe SLQ method is often used to estimate the trace of an arbitrary matrix function:\n \\mathrm{tr}(f(A)), \\quad \\text{ where } f(A) = U f(\\Lambda) U^T \nIt’s has been shown1 that the information obtained by the Lanczos method is sufficient to obtained a Gaussian quadrature approximation of the empirical spectral measure of A. By sampling zero-mean vectors satisfying \\mathbb{E}[v v^T] = I, one can obtain estimates of the trace above: \\operatorname{tr}(f(A)) \\approx \\frac{n}{\\mathrm{n}_{\\mathrm{v}}} \\sum_{l=1}^{\\mathrm{n}_{\\mathrm{v}}}\\left(\\sum_{k=0}^m\\left(\\tau_k^{(l)}\\right)^2 f\\left(\\theta_k^{(l)}\\right)\\right)\nIt turns out averaging these trace estimates yields unbiased, Girard-Hutchinson estimator of the trace. To see why this estimator is unbiased, note that:  \\mathtt{tr}(A) = \\mathbb{E}[v^T A v] \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nThus, all we need to do to estimate the trace of a matrix function is multiply and sum the quadrature nodes and weights output by SLQ.",
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+    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nsize\nint or tuple\nOutput shape to generate.\nrequired\n\n\nrng\nstr = “splitmix64”\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint = -1\nSeed for the generator. Use -1 to for random (non-deterministic) behavior.\n-1\n\n\ndtype\ndtype = float32\nFloating point dtype for the output. Must be float32 or float64.\nnp.float32",
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-    "text": "sl_trace method\nTo see how these formulas are actually implemented with the generic SLQ implementation, here’s an abbreviated form of the sl_trace function implemented by primate:\ntemplate&lt; std::floating_point F, LinearOperator Matrix, ThreadSafeRBG RBG &gt;\nvoid sl_trace(\n  const Matrix& A, const std::function&lt; F(F) &gt; sf, RBG& rbg, \n  const int nv, const int dist, const int engine_id, const int seed,\n  const int deg, const float lanczos_rtol, const int orth, const int ncv,\n  const F atol, const F rtol\n  F* estimates\n){  \n  using VectorF = Eigen::Array&lt; F, Dynamic, 1&gt;;\n\n  // Parameterize the trace function (runs in parallel)\n  auto trace_f = [&](int i, F* q, F* Q, F* nodes, F* weights){\n    Map&lt; VectorF &gt; nodes_v(nodes, deg, 1);     // no-op\n    Map&lt; VectorF &gt; weights_v(weights, deg, 1); // no-op\n    nodes_v.unaryExpr(sf);\n    estimates[i] = (nodes_v * weights_v).sum();\n  };\n  \n  // Convergence checking like scipy.integrate.quadrature\n  int n = 0;\n  F mu_est = 0.0, mu_pre = 0.0;\n  const auto early_stop = [&](int i) -&gt; bool {\n    ++n; // Number of estimates\n    mu_est = (1.0 / F(n)) * (estimates[i] + (n - 1) * mu_pre); \n    bool atol_check = abs(mu_est - mu_pre) &lt;= atol;\n    bool rtol_check = abs(mu_est - mu_pre) / mu_est &lt;= rtol; \n    mu_pre = mu_est; \n    return atol_check || rtol_check;\n  };\n\n  // Execute the stochastic Lanczos quadrature with the trace function \n  slq&lt; float &gt;(A, trace_f, early_stop, ...);\n}\nAs before, two functions are used to parameterize the slq method.\nThe first (trace_f) applies an arbitrary spectral function sf to the Rayleigh-Ritz values obtained by the Lanczos tridiagonalization of A(or equivalently, the nodes of the Gaussian quadrature). These are the \\theta’s in the pseudocode above. When multiplied by the weights of the quadrature, the corresponding sum forms an estimate of the trace of the matrix function.\nThe second function early_stop is used to check for convergence of the estimator. First, it uses the trace estimate x_n to update the sample mean \\mu_n via the formula:\n \\mu_n = n^{-1} [x_n + (n - 1)\\mu_{n-1}] \nThen, much in the same way the quadrature function from scipy.integrate approximates a definite integral, it checks for convergence using the absolute and relative tolerances supplied by the user. Returning true signals convergence, stopping the iteration early.",
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-    "text": "Footnotes\n\n\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.↩︎",
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-    "section": "But wait, isn’t T arbitrary?",
-    "text": "But wait, isn’t T arbitrary?\nUnfortunately, there is no canonical choice of T_j. Indeed, as T_n is a family with n - 1 degrees of freedom and v \\in \\mathbb{R}^n was chosen arbitrarily, there are infinitely many essentially distinct such decompositions. In contrast, the spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M.\nNot all hope is lost though. Notice that since Q is an orthogonal matrix, thus we have:\n Q Q^T = I_n = [e_1, e_2, \\dots, e_n] \nBy extension, given an initial pair (A, q_1) satisfying \\lVert q_1 \\rVert = 1, we have:\n\nK_n(A, q_1) = Q Q^T K_n(A, q_1) = Q[ \\, e_1 \\mid T e_1 \\mid T^2 e_1 \\mid \\dots \\mid T^{n-1} e_1 \\, ]\n\nThis is actually QR factorization! Indeed, the Implicit Q Theorem asserts that if an upper Hessenburg matrix T \\in \\mathbb{R}^{n \\times n} has only positive elements on its first subdiagonal and there exists an orthogonal matrix Q such that Q^T A Q = T, then Q and T are uniquely determined3 by (A, q_1).\nThus, tridiagonalizing A with respect to an arbitrary q_1 \\in \\mathbb{R}^n satisfying \\lVert q_1\\rVert = 1 determines Q.",
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-    "text": "primate’s generic API is enabled through function templates specialized using C++20 concepts. In other words, a function F requiring concept C will compile with any type T so long as T respects the constraints imposed by C. For example, generically, any type T respecting the LinearOperatorconcept shown below can be passed to the Lanczos method:\nIn english, an instance A of type T is said to support the LinearOperator concept if it has:\nshape() should return a pair (n,m) representing the sizes of the output and input vectors, respectively. Note in the matrix setting this corresponds to the number of rows and columns.",
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-    "text": "primate’s generic API is enabled through C++20 concepts. Thus, the more exact statement is that any type respecting the LinearOperator concept shown below can be passed:\nusing FP = std::floating_point; \ntemplate &lt; typename T, FP float_t = typename T::value_type &gt;\nconcept LinearOperator = requires(T A, const float_t* v, float_t* o) {\n  { A.matvec(v, o) }; // o = A v\n  { A.shape() } -&gt; std::convertible_to&lt; std::pair&lt; size_t, size_t &gt; &gt;;\n};\nAn instance A of type T is said to support the LinearOperator concept if it has:\n\nA method Av \\mapsto o, with signature A.matvec(const float_t* v, float_t* o)\nA method yielding (\\mathrm{card}(o), \\mathrm{card}(v)), with signatureA.shape() -&gt; pair&lt; ... &gt;\n\nshape() should yield a pair (n,m) representing the sizes of the output and input vectors, respectively. This corresponds to the number of rows and columns in the matrix setting.",
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-    "text": "Other Concepts\nDepending on the problem at hand, the supplied operator may need to meet other constraints. Here’s a short list additional operator concepts:\n\n\n\n\n\n\n\n\n\nConcept\nSupports\nSignature\nRequires\n\n\n\n\nLinearOperator\nA v \\mapsto o\nA.matvec(v, o)\nNA\n\n\nAdjointOperator\nA^T v \\mapsto o\nA.rmatvec(v, o)\nLinearOperator\n\n\nAdditiveOperator\no \\gets o + \\alpha Av\nA.matvec_add(v, alpha, o)\nLinearOperator\n\n\nAffineOperator\nSets t s.t. A + tB\nA.set_parameter(t)\nLinearOperator\n\n\nQuadOperator\nv^T A v\nA.quad(v)\nNA\n\n\n\nThe exported methods in primate only need the minimum constraints to be satisfied to compile: if you need access to the Lanczos method, then just supporting the LinearOperator concept is sufficient. On the other hand, adding support for other constraints can optimize the efficiency of certain methods; for example, the hutch method technically only requires a LinearOperator to do trace estimation (via matvec calls), but will also compile and prefer calling quad with a QuadOperator as input. In such a situaton, if your operator has an efficient quadratic form v \\mapsto v^T A v, then implementing quad may improve the performance of hutch.",
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-    "text": "Complexities\nElegant as the Lanczos method may be, what does it net us? Well, for starters, the Lanczos method can drop the complexity of obtaining the spectral decomposition by order of magnitude.\n\nTheorem 1 (Parlett 1994, Simon 1984) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos method computes \\Lambda(A) in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic\n\nFor general A \\in \\mathbb{R}^{n \\times n}, computing the spectral-decomposition is essentially bounded by the matrix-multiplication time: \\Theta(n^\\omega) time and \\Theta(n^2) space, where \\omega \\approx 2.37\\dots is the matrix multiplication constant. If we exclude the Strassen model for computation, we get effectively a \\Omega(n^3) time and \\Omega(n^2) space bound. In contrast, when both r = n and \\eta = \\nu = n, the Theorem above implies the spectral decomposition takes just O(n^2) time and O(n) space to obtain. Unless you know the tricks, this feels like a contradiction. So… how does the complexity argument play out?\n\nSolving the right problem\n\n\nIf you squint hard enough, you can deduce that since A Q_j = Q_j T_j + \\beta_{j+1} q_{j+1} e_{j}^T, every symmetric A \\in \\mathbb{R}^{n \\times n} expanded this way admits a three-term recurrence: \n\\begin{align*}\nA q_j &= \\beta_{j\\text{-}1} q_{j\\text{-}1} + \\alpha_j q_j + \\beta_j q_{j+1} \\\\\n\\Leftrightarrow \\beta_{j} \\, q_{j+1} &= A q_j - \\alpha_j \\, q_j - \\beta_{j\\text{-}1} \\, q_{j\\text{-}1}  \n\\end{align*}\n\nThe equation above is a variable-coefficient second-order linear difference equation, and it is known such equations have unique solutions, which are given by: \n\\alpha_j = q_j^T A q_j, \\;\\; \\beta_j = \\lVert r_j \\rVert_2, \\;\\; q_{j+1} = r_j / \\beta_j\n\n\n\\text{where  } r_j = (A - \\alpha_j I)q_j - \\beta_{j\\text{-}1} q_j\n\nIn other words, if (q_{j\\text{-}1}, \\beta_j, q_j) are known, then (\\alpha_j, \\beta_{j+1}, q_{j+1}) are completely determined. This fact is fantastic from a computational point of view: no explicit call to the QR algorithm necessary2!\nNote that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then the above procedure clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nMoreover, if we only need the eigen-values \\Lambda(A) ( and not their vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!\n\n\nBut wait, isn’t T arbitrary?\nUnfortunately, there is no canonical choice of T_j. Indeed, as T_n is a family with n - 1 degrees of freedom and v \\in \\mathbb{R}^n was chosen arbitrarily, there are infinitely many essentially distinct such decompositions. In contrast, the spectral decomposition A = U \\Lambda U^T identifies a diagonalizable A with its spectrum \\Lambda(A) up to a change of basis A \\mapsto M^{-1} A M.\nNot all hope is lost though. Notice that since Q is an orthogonal matrix, thus we have:\n Q Q^T = I_n = [e_1, e_2, \\dots, e_n] \nBy extension, given an initial pair (A, q_1) satisfying \\lVert q_1 \\rVert = 1, we have:\n\nK_n(A, q_1) = Q Q^T K_n(A, q_1) = Q[ \\, e_1 \\mid T e_1 \\mid T^2 e_1 \\mid \\dots \\mid T^{n-1} e_1 \\, ]\n\nThis is actually QR factorization! Indeed, the Implicit Q Theorem asserts that if an upper Hessenburg matrix T \\in \\mathbb{R}^{n \\times n} has only positive elements on its first subdiagonal and there exists an orthogonal matrix Q such that Q^T A Q = T, then Q and T are uniquely determined3 by (A, q_1).",
+    "text": "diagonalize.lanczos(A, v0=None, deg=None, rtol=1e-08, orth=0, sparse_mat=False, return_basis=False, seed=None, dtype=None)\nLanczos method for matrix tridiagonalization.\nThis function implements Paiges A27 variant (1) of the Lanczos method for tridiagonalizing linear operators, with additional modifications to support varying degrees of re-orthogonalization. In particular, orth=0 corresponds to no re-orthogonalization, orth &lt; deg corresponds to partial re-orthogonalization, and orth &gt;= deg corresponds to full re-orthogonalization.\n\n\nThe Lanczos method builds a tridiagonal T from a symmetric A via an orthogonal change-of-basis Q:  Q^T A Q  = T  Unlike other Lanczos implementations (e.g. SciPy’s eigsh), which includes e.g. sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes deg steps of the Lanczos method with the supplied v0 and returns the resulting tridiagonal matrix T.\nRayleigh-Ritz approximations of the eigenvalues of A can be further obtained by diagonalizing T via any symmetric tridiagonal eigenvalue solver, scipy.linalg.eigh_tridiagonal though note unlike eigsh no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing orth and deg. Supplying either negative values or values larger than deg for orth will result in full re-orthogonalization, though note the number of matvecs scales linearly with deg and the number of inner-products scales quadratically with orth.\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nscipy.sparse.linalg.LinearOperator or ndarray or sparray\nSymmetric operator to tridiagonalize.\nrequired\n\n\nv0\nndarray\nInitial vector to orthogonalize against.\nNone\n\n\ndeg\nint\nSize of the Krylov subspace to expand.\nNone\n\n\nrtol\nfloat\nRelative tolerance to consider the invariant subspace as converged.\n1e-8\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against.\n0\n\n\nsparse_mat\nbool\nWhether to output the tridiagonal matrix as a sparse matrix.\nFalse\n\n\nreturn_basis\nbool\nIf True, returns the Krylov basis vectors Q.\nFalse\n\n\ndtype\ndtype\nThe precision dtype to specialize the computation.\nNone\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\ntuple\nA tuple (a,b) parameterizing the diagonal and off-diagonal of the tridiagonal matrix. If return_basis=True, the tuple (a,b), Q is returned, where Q represents an orthogonal basis for the degree-deg Krylov subspace.\n\n\n\n\n\n\nscipy.linalg.eigh_tridiagonal : Eigenvalue solver for real symmetric tridiagonal matrices. operator.matrix_function : Approximates the action of a matrix function via the Lanczos method.\n\n\n\n\nPaige, Christopher C. “Computational variants of the Lanczos method for the eigenproblem.” IMA Journal of Applied Mathematics 10.3 (1972): 373-381.",
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+    "text": "The Lanczos method builds a tridiagonal T from a symmetric A via an orthogonal change-of-basis Q:  Q^T A Q  = T  Unlike other Lanczos implementations (e.g. SciPy’s eigsh), which includes e.g. sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes deg steps of the Lanczos method with the supplied v0 and returns the resulting tridiagonal matrix T.\nRayleigh-Ritz approximations of the eigenvalues of A can be further obtained by diagonalizing T via any symmetric tridiagonal eigenvalue solver, scipy.linalg.eigh_tridiagonal though note unlike eigsh no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing orth and deg. Supplying either negative values or values larger than deg for orth will result in full re-orthogonalization, though note the number of matvecs scales linearly with deg and the number of inner-products scales quadratically with orth.",
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-    "text": "Examples\n  ::: {#af54fa09 .cell execution_count=1}\n  ``` {.python .cell-code}\n          print(\"hello\")\n  ```\n  \n  ::: {.cell-output .cell-output-stdout}\n  ```\n  hello\n  ```\n  :::\n  :::",
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-    "text": "The Lanczos method iteratively builds a tridiagonal matrix T of a symmetric A via an orthogonal change-of-basis Q:  Q^T A Q  = T  Unlike other Lanczos implementations (e.g. SciPy’s eigsh), which includes e.g. sophisticated restarting, deflation, and selective-reorthogonalization steps, this method simply executes deg steps of the Lanczos method with the supplied v0 and returns the resulting tridiagonal matrix.\nDiagonalizing T via e.g. scipy.linalg.eigh_tridiagonal yields Rayleigh-Ritz approximations of the eigenvalues of A, though note no checking is performed for ‘ghost’ or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing orth and deg. Supplying either negative values or values larger than deg for orth will result in full re-orthogonalization, though note the number of matvecs scales linearly with deg and the number of inner-products scales quadratically with orth.",
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-    "text": "trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='fttr', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)\nEstimates the trace of a matrix A or matrix function f(A) via a Girard-Hutchinson estimator.\nThis function uses up to maxiter random isotropic vectors to form an unbiased estimator of the trace of A. The estimator is obtained by averaging quadratic forms of A (or f(A)), rescaling as necessary.\n\n\nFor matrix functions, the Lanczos method up to degree deg is used to approximate the action of f(A). By default,\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n10\n\n\ndeg\nint\nDegree of the quadrature approximation. Must be at least 1.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See notes.\nNone\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See notes.\n1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n\"confidence\"\n\n\nncv\nint\nNumber of Lanczos vectors to allocate. Must be at least 2.\n2\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against. Must be less than ncv.\n0\n\n\nquad\nstr\nMethod used to obtain the weights of the Gaussian quadrature. See notes.\n'fttr'\n\n\nconfidence\nfloat\nConfidence level to consider estimator as converged. Only used when stop = “confidence”.\n0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘lcg64’, ‘mt64’\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Set to &lt;= 0 to let OpenMP decide.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\nFalse\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\nfloat\nEstimate of the trace of A, if fun = \"identity\", otherwise estimates the trace of f(A).\n\n\n(dict, optional)\nIf ‘info = True’, additional information about the computation.\n\n\n\n\n\n\nlanczos : the lanczos algorithm.\n\n\n\n[1] Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099."
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-    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n10\n\n\ndeg\nint\nDegree of the quadrature approximation. Must be at least 1.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See notes.\nNone\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See notes.\n1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n\"confidence\"\n\n\nncv\nint\nNumber of Lanczos vectors to allocate. Must be at least 2.\n2\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against. Must be less than ncv.\n0\n\n\nquad\nstr\nMethod used to obtain the weights of the Gaussian quadrature. See notes.\n'fttr'\n\n\nconfidence\nfloat\nConfidence level to consider estimator as converged. Only used when stop = “confidence”.\n0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘lcg64’, ‘mt64’\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Set to &lt;= 0 to let OpenMP decide.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\nFalse\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}"
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-    "text": "[1] Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099."
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-    "text": "Name\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nn\nint\nNumber of random vectors to sample for the quadrature estimate.\n150\n\n\ndeg\nint\nDegree of the quadrature approximation.\n20\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘lcg64’, ‘mt64’\nRandom number generator to use (PCG64 by default).\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against when building the Krylov basis.\n0\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Setting num_threads &lt; 1 to let OpenMP decide.\n0",
+    "text": "Functions for estimating the trace of matrices and matrix functions.\n\n\n\ntrace.hutch\nEstimates the trace of a symmetric A or matrix function f(A) via a Girard-Hutchinson estimator.\n\n\ntrace.xtrace\n\n\n\n\n\n\n\nMatrix operators\n\n\n\noperator.matrix_function\nConstructs a LinearOperator approximating the action of the matrix function fun(A).\n\n\n\n\n\n\nRandomized module\n\n\n\nrandom.rademacher\nGenerates random vectors from the rademacher distribution.\n\n\nrandom.normal\nGenerates random vectors from the standard normal distribution.\n\n\n\n\n\n\nDiagonalization methods\n\n\n\ndiagonalize.lanczos\nLanczos method for matrix tridiagonalization.\n\n\n\n\n\n\nMiscellenous functions\n\n\n\nget_include\nReturn the directory that contains the primate’s *.h header files.",
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-    "text": "The “iteration” part\nLanczos originally referred to his algorithm as the method of minimized iterations, and indeed nowadays it is often called an iterative method. Where’s the iterative component?\nIf you squint hard enough, you can deduce that for every j \\in [1, n): A Q_j = Q_j T_j + \\beta_{j+1} q_{j+1} e_{j}^T Equating the j-th columns on each side of the equation and rearranging yields a three-term recurrence: \n\\begin{align*}\nA q_j &= \\beta_{j\\text{-}1} q_{j\\text{-}1} + \\alpha_j q_j + \\beta_j q_{j+1} \\\\\n\\Leftrightarrow \\beta_{j} \\, q_{j+1} &= A q_j - \\alpha_j \\, q_j - \\beta_{j\\text{-}1} \\, q_{j\\text{-}1}  \n\\end{align*}\n\nThe equation above is a variable-coefficient second-order linear difference equation, and it is known such equations have unique solutions; they are given below: \n\\alpha_j = q_j^T A q_j, \\;\\; \\beta_j = \\lVert r_j \\rVert_2, \\;\\; q_{j+1} = r_j / \\beta_j\n\n\n\\text{where  } r_j = (A - \\alpha_j I)q_j - \\beta_{j\\text{-}1} q_j\n\nIn other words, if (q_{j\\text{-}1}, \\beta_j, q_j) are known, then (\\alpha_j, \\beta_{j+1}, q_{j+1}) are completely determined. In theory, this means we can iteratively generate both Q and T using just a couple vectors at a time—no need to explicitly call to the QR algorithm as shown above. Pretty nifty, eh!",
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-    "text": "Complexity analysis\nElegant and as theoretically founded as the Lanczos method may be, is it efficient in practice?\nLet’s start by establishing a baseline on its complexity:\n\nTheorem 1 (Parlett 1994) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos method computes \\Lambda(A) in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic\n\nAs its clear from the theorem, if we specialize it such that r = n and \\eta = \\nu = n, then the Lanczos method requires just O(n^2) time and O(n) space to execute. In other words, the Lanczos method drops both the time and space complexity4 of obtaining spectral information by order of magnitude over similar eigen-algorithms that decompose A directly.\nTo see why this is true, note that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then carrying out the recurrence clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nNow, if we need to store all of Y or Q explicitly, we clearly need O(n^2) space to do so. However, if we only need the eigen-values \\Lambda(A) (and not their eigen-vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!",
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-    "text": "Rayleigh-Ritz approximations\nSuppose instead of constructing the full T \\in \\mathbb{R}^{n \\times n}, we stop at the j^{\\text{th}} iteration, where 1 \\leq j &lt; n.\n\nT_j = \\mathrm{tridiag}\\Bigg(\n\\begin{array}{ccccccccc}\n& \\beta_2 & & \\beta_3 & & \\cdot & & \\beta_j & \\\\\n\\alpha_1 & & \\alpha_2 & & \\cdot & & \\cdot & & \\alpha_j \\\\\n& \\beta_2 & & \\beta_3 & & \\cdot & & \\beta_j &\n\\end{array}\n\\Bigg)\n\nIt is natural to assume that the eigenvalues of T_j approximate some of eigenvalues of \\Lambda(A). This intuition not only turns out to be true, but in fact the eigenvalues of T_j are known to be optimal approximations of \\Lambda(A) under many appealing notions of optimality. Thus, if we need only to approximate the spectrum of A, we may potentially do so using just j &lt;&lt; n iterations; indeed, this is the hallmark of the iterative approach to obtaining eigenvalues:",
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-    "text": "Wait, isn’t T arbitrary?\nUnfortunately—and unlike the spectral decomposition2—there is no canonical choice of T. Indeed, as T is a family with n - 1 degrees of freedom and v \\in \\mathbb{R}^n was chosen arbitrarily, there are infinitely many essentially distinct such decompositions.\nNot all hope is lost though, as it turns out that T is actually fully characterized by v. To see this, notice that since Q is an orthogonal matrix, we have:\n Q Q^T = I_n = [e_1, e_2, \\dots, e_n] \nBy extension, given an initial pair (A, q_1) satisfying \\lVert q_1 \\rVert = 1, the following holds:\n\nK_n(A, q_1) = Q Q^T K_n(A, q_1) = Q[ \\, e_1 \\mid T e_1 \\mid T^2 e_1 \\mid \\dots \\mid T^{n-1} e_1 \\, ]\n\n…this is actually a QR factorization, which is essentially unique! Indeed, the Implicit Q Theorem asserts that if an upper Hessenburg matrix T \\in \\mathbb{R}^{n \\times n} has only positive elements on its first subdiagonal and there exists an orthogonal matrix Q such that Q^T A Q = T, then Q and T are uniquely determined3 by (A, q_1).",
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-    "text": "primate supports a variety of matrix-types of the box, including numpy ndarray’s, compressed sparse matrices (a lá SciPy), and LinearOperators—the latter enables the use of matrix free operators.\nOutside of the natively types above, the basic requirements for any operator A to be used with e.g. the Lanczos method in primate are:\n\nA method A.matvec(input: ndarray) -&gt; ndarray implementing v \\mapsto Av\nAn attribute A.shape -&gt; tuple[int, int] giving the output/input dimensions of A\n\n\n\nHere’s an example of a simple operator representing a Diagonal matrix, which inherits a .matvec() method by following the subclassing rules of SciPy’s LinearOperator:\nimport numpy as np \nfrom numpy.typing import ArrayLike\nfrom scipy.sparse.linalg import LinearOperator \n\nclass DiagonalOp(LinearOperator):\n  diag: np.ndarray = None\n  \n  def __init__(self, d: ArrayLike, dtype = None):\n    self.diag = np.array(d)\n    self.shape = (len(d), len(d))\n    self.dtype = np.dtype('float32') if dtype is None else dtype\n\n  def _matvec(self, x: ArrayLike) -&gt; np.ndarray:\n    out = self.diag * np.ravel(x)\n    return out.reshape(x.shape)",
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-    "text": "C++ usage\nSimilarly, to get started calling any matrix-free function provided by primate on the C++ side, such hutch or lanczos, simply pass any type with .shape() and .matvec() member functions:\nclass LinOp {\n  int nrow, ncol;\n  \n  LinOp(int nr, int nc) : nrow(nr), ncol(nc) {}\n  \n  void matvec(const float* input, float* output) const {\n    ... // implementation details \n  }\n\n  void shape() const { return std::make_pair(nrow, ncol); }\n}\nIt’s up to you to ensure shape() yields the correct size; primate will supply vectors to input of size .shape().second (number of columns) and guarantees the pointer to the output will be at least shape().first (number of rows), no more.\nTo read more about how semantics extend to the C++ side as well via C++20 concepts—see the C++ integration guide. If you’re using pybind11 and you want to extend primate’s Python API to work natively with linear operator implemented in C++, see the pybind11 integration guide.",
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-    "text": "Below is a quick introduction to primate. For more introductory material, theor\nTo do trace estimation, use functions in the trace module:\n\nimport primate.trace as TR\nfrom primate.random import symmetric\nA = symmetric(150)  ## random positive-definite matrix \n\nprint(f\"Actual trace: {A.trace():6f}\")        ## Actual trace\nprint(f\"Girard-Hutch: {TR.hutch(A):6f}\")      ## Monte-carlo tyoe estimator\nprint(f\"XTrace:       {TR.xtrace(A):6f}\")     ## Epperly's algorithm\n\nActual trace: 75.697397\nGirard-Hutch: 75.981329\nXTrace:       75.697398\n\n\nFor matrix functions, you can either construct a LinearOperator directly via the matrix_function API, or supply a string to the parameter fun describing the spectral function to apply. For example, one might compute the log-determinant as follows:\n\nfrom primate.operator import matrix_function\nM = matrix_function(A, fun=\"log\")\n\new = np.linalg.eigvalsh(A)\nprint(f\"logdet(A):  {np.sum(np.log(ew)):6f}\")\nprint(f\"GR approx:  {TR.hutch(M):6f}\")\nprint(f\"XTrace:     {TR.xtrace(M):6f}\")\n\n## Equivalently could've used: \n## M = matrix_function(A, fun=np.log)\n\nlogdet(A):  -148.321844\nGR approx:  -147.250366\nXTrace:     -148.315937\n\n\nNote in the above example you can supply to fun either string describing a built-in spectral function or an arbitrary Callable. The former is preferred when possible, as function evaluations will generally be faster and hutch can also be parallelized. Multi-threaded execution of e.g. hutch with arbitrary functions is not currently allowed due to the GIL, though there are options available, see the integration docs for more details.\nFor ‘plain’ operators, XTrace should recover the exact trace (up to roundoff error). For matrix functions f(A), there will be some inherent inaccuracy as the underlying matrix-vector multiplication is approximated with the Lanczos method.\nIn general, the amount of accuracy depends both on the Lanczos parameters and the type of matrix function. Spectral functions that are difficult or impossible to approximate via low-degree polynomials, for example, may suffer more from inaccuracy issues than otherwise. For example, consider the example below that computes that rank:\n\n## Make a rank-deficient operator\new = np.sort(ew)\new[:30] = 0.0\nA = symmetric(150, ew = ew, pd = False)\nM = matrix_function(A, fun=np.sign)\n\nprint(f\"Rank:       {np.linalg.matrix_rank(A)}\")\nprint(f\"GR approx:  {TR.hutch(M)}\")\nprint(f\"XTrace:     {TR.xtrace(M)}\")\n\nRank:       120\nGR approx:  144.8700408935547\nXTrace:     143.97018151807674\n\n\nThis is not so much a fault of hutch or xtrace as much as it is the choice of approximation and Lanczos parameters. The sign function has a discontinuity at 0, is not smooth, and is difficult to approximate with low-degree polynomials. One workaround to handle this issue is relax the sign function with a low-degree “soft-sign” function:  \\mathcal{S}_\\lambda(x) = \\sum\\limits_{i=0}^q \\left( x(1 - x^2)^i \\prod_{j=1}^i \\frac{2j - 1}{2j} \\right)\nVisually, the soft-sign function looks like this:\n\nfrom primate.special import soft_sign, figure_fun\nshow(figure_fun(\"smoothstep\"))\n\n\n  \n\n\n\n\n\nIt’s been shown that there is a low degree polynomial p^\\ast that uniformly approximates \\mathcal{S}_\\lambda up to a small error on the interval [-1,1]. Since matrix_function uses a low degree Krylov subspace to approximate the action v \\mapsto f(A)v, one way to improve the accuracy of rank estimation is to replace \\mathrm{sign} \\mapsto \\mathcal{S}_{\\lambda} for some choice of q \\in \\mathbb{Z}_+ (this function is available in primate under the name soft_sign):\n\nfrom primate.special import soft_sign\nfor q in range(0, 50, 5):\n  M = matrix_function(A, fun=soft_sign(q=q))\n  print(f\"XTrace S(A) for q={q}: {TR.xtrace(M):6f}\")\n\nXTrace S(A) for q=0: 72.776775\nXTrace S(A) for q=5: 109.325087\nXTrace S(A) for q=10: 114.843157\nXTrace S(A) for q=15: 116.952146\nXTrace S(A) for q=20: 118.027760\nXTrace S(A) for q=25: 118.657212\nXTrace S(A) for q=30: 119.055340\nXTrace S(A) for q=35: 119.319911\nXTrace S(A) for q=40: 119.501839\nXTrace S(A) for q=45: 119.630120\n\n\nIf the type of operator A is known to typically have a large spectral gap, another option is to compute the numerical rank by thresholding values above some fixed value \\lambda_{\\text{min}}. This is equivalent to applying the following spectral function:\n S_{\\lambda_{\\text{min}}}(x) =\n\\begin{cases}\n1 & \\text{ if } x \\geq \\lambda_{\\text{min}} \\\\\n0 & \\text{ otherwise }\n\\end{cases}\n\nIn the above example, the optimal cutoff \\lambda_{\\text{min}} is given by the smallest non-zero eigenvalue. Since the trace estimators all stochastic to some degree, we set the cutoff to slightly less than this value:\n\nlambda_min = min(ew[ew != 0.0])\nprint(f\"Smallest non-zero eigenvalue: {lambda_min:.6f}\")\n\nstep = lambda x: 1 if x &gt; (lambda_min * 0.90) else 0\nM = matrix_function(A, fun=step, deg=50)\nprint(f\"XTrace S_t(A) for t={lambda_min*0.90:.4f}: {TR.xtrace(M):6f}\")\n\nSmallest non-zero eigenvalue: 0.191916\nXTrace S_t(A) for t=0.1727: 120.000000\n\n\nIndeed, this works! Of course, here we’ve used the fact that we know the optimal cutoff value, but this can also be estimated with the lanczos method itself.\n\nfrom primate.diagonalize import lanczos\nfrom scipy.linalg import eigvalsh_tridiagonal\na,b = lanczos(A)\nrr = eigvalsh_tridiagonal(a,b) # Rayleigh-Ritz values\ntol = 10 * np.finfo(A.dtype).resolution\nprint(f\"Approx. cutoff: {np.min(rr[rr &gt; tol]):.6f}\")\n\nApprox. cutoff: 0.191916"
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-    "text": "primate contains a variety of methods for estimating quantities from matrix functions. One popular such quantity is the trace of f(A), for some generic f: [a,b] \\to \\mathbb{R} defined on the spectrum of A:  \\mathrm{tr}(f(A)) \\triangleq \\mathrm{tr}(U f(\\Lambda) U^{\\intercal}), \\quad \\quad f : [a,b] \\to \\mathbb{R}\nMany quantities of common interest can be expressed in as traces of matrix functions with the right choice of f. A few such function specializations include:\n\\begin{align*}\nf &= \\mathrm{id} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{tr}(A) \\\\\nf &= f^{-1} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{tr}(A^{-1}) \\\\\nf &= \\log \\quad &\\Longleftrightarrow& \\quad  &\\mathrm{logdet}(A) \\\\\nf &= \\mathrm{exp} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{exp}(A) \\\\\nf &= \\mathrm{sign} \\quad &\\Longleftrightarrow& \\quad &\\mathrm{rank}(A)  \\\\\n&\\vdots \\quad &\\hphantom{\\Longleftrightarrow}& \\quad & \\vdots &\n\\end{align*}\nIn this introduction, the basics of randomized trace estimation are introduced, with a focus on how to use matrix-free algorithms estimate traces of matrix functions.",
+    "text": "Diagonalization methods\n\n\n\ndiagonalize.lanczos\nLanczos method for matrix tridiagonalization.",
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-    "text": "Basics: Trace estimation\n\nSuppose we wanted to compute the trace of some matrix A. By the very definition of the trace, we have a variety of means to do it: \\mathrm{tr}(A) \\triangleq \\sum\\limits_{i=1}^n A_{ii} = \\sum\\limits_{i=1}^n \\lambda_i = \\sum\\limits_{i=1}^n e_i^T A e_i \nwhere, in the right-most sum, we’re using e_i to represent the ith identity vector:  A_{ii} = e_i^T A e_i, \\quad e_i = [0, \\dots, 0, \\underbrace{1}_{i}, 0, \\dots, 0] \nLet’s see some code. First, we start with a simple (random) positive-definite matrix A \\in \\mathbb{R}^{n \\times n}.\n\nfrom primate.random import symmetric\nA = symmetric(150, pd = True)\n\nBy default, symmetric normalizes A such that the eigenvalues are uniformly distributed in the interval [0, 1]. For reference, the spectrum of A looks as follows:\n\n\n\n  \n\n\n\n\n\nNow, using numpy, we can verify all of these trace definitions are indeed equivalent:\n\n\nimport numpy as np\neye = lambda i: np.ravel(np.eye(1, 150, k=i))\nprint(f\"Direct: {np.sum(A.diagonal()):.8f}\")\nprint(f\"Eigen:  {np.sum(np.linalg.eigvalsh(A)):.8f}\")\nprint(f\"Matvec: {sum([eye(i) @ A @ eye(i) for i in range(150)]):.8f}\")\n\nDirect: 75.69739746\nEigen:  75.69739746\nMatvec: 75.69739746\n\n\nWhile (1) trivially yields \\mathrm{tr}(A) in (optimal) O(n) time, there exist situations where the diagonal entries of A are not available—particularly in the large-scale regime. In contrast, though both (2) and (3) are less efficient, they are nonetheless viable altenrnatives to obtain \\mathrm{tr}(A).",
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-    "text": "The implicit trace estimation problem\nThe implicit trace estimation problem can be stated as follows:\n\nGiven access to a square matrix A \\in \\mathbb{R}^{n \\times n} via its matrix-vector product operator x \\mapsto Ax, estimate its trace \\mathrm{tr}(A) = \\sum\\limits_{i=1}^n A_{ii}\n\nThough method (3) may be used to solve the implicit trace estimation problem exactly, its pretty inefficient from a computational standpoint—after all, most of the entries of e_i are zero, thus most inner product computations do not contribute to the trace estimate at all.\nOne idea, attributed to A. Girard, is to replace e_i with random zero-mean vectors, e.g. random sign vectors v \\in \\{-1, +1\\}^{n} or random normal vectors v \\sim \\mathcal{N}(\\mu=0, \\sigma=1):\n\\mathtt{tr}(A) \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nIt was shown more formally later by M.F. Huchinson that if these random vectors v \\in \\mathbb{R}^n satisfy \\mathbb{E}[vv^T] = I then this approach indeed forms an unbiased estimator of \\mathrm{tr}(A). To see this, its sufficient to combine the linearity of expectation, the cyclic-property of the trace function, and the aforementioned isotropy conditions: \\mathtt{tr}(A) = \\mathtt{tr}(A I) = \\mathtt{tr}(A \\mathbb{E}[v v^T]) = \\mathbb{E}[\\mathtt{tr}(Avv^T)] = \\mathbb{E}[\\mathtt{tr}(v^T A v)] = \\mathbb{E}[v^T A v]\nNaturally we expect the approximation to gradually improve as more vectors are sampled, converging as n_v \\to \\infty. Let’s see if we can check this: we start by collecting a few sample quadratic forms\n\ndef isotropic(n):\n  yield from [np.random.choice([-1,+1], size=A.shape[1]) for i in range(n)]\nestimates = np.array([v @ A @ v for v in isotropic(500)])\n\nPlotting both the estimator formed by averaging the first k samples along with its absolute error, we get the following figures for k = 50 and k = 500, respectively:\n\n\n\n  \n\n\n\n\n\nSure enough, the estimator quickly gets within an error rate of about 1-5% away from the actual trace, getting generally closer as more samples are collected. On the other hand, improvement is not gaurenteed, and securing additional precision much less than 1% becomes increasingly difficult due to the randomness and variance of the sample estimates.",
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+    "text": "random.normal(size, rng='splitmix64', seed=-1, dtype=np.float32)\nGenerates random vectors from the standard normal distribution.\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nsize\nint or tuple\nOutput shape to generate.\nrequired\n\n\nrng\nstr = “splitmix64”\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint = -1\nSeed for the generator. Use -1 to for random (non-deterministic) behavior.\n-1\n\n\ndtype\ndtype = float32\nFloating point dtype for the output. Must be float32 or float64.\nnp.float32\n\n\n\n\n\n\n: Randomly generated matrix of shape size with entries in { -1, 1 }.",
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-    "text": "The statistical side\nThis forms an unbiased estimator of \\mathrm{tr}(A); to see this, its sufficient to combine the linearity of expectation, the cyclic-property of the trace function, and the aforementioned isotropy conditions \\mathtt{tr}(A) = \\mathtt{tr}(A I) = \\mathtt{tr}(A \\mathbb{E}[v v^T]) = \\mathbb{E}[\\mathtt{tr}(Avv^T)] = \\mathbb{E}[\\mathtt{tr}(v^T A v)] = \\mathbb{E}[v^T A v]",
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-    "text": "Ex: Logdet computation\nThe basic way to approximate the trace of a matrix function is to simply set fun to the name the spectral function. For example, to approximate the log-determinant:\n\nfrom primate.trace import hutch\n\n## Log-determinant\nlogdet_test = hutch(A, fun=\"log\", maxiter=25)\nlogdet_true = np.sum(np.log(np.linalg.eigvalsh(A)))\n\nprint(f\"Logdet (exact):  {logdet_true}\")\nprint(f\"Logdet (approx): {logdet_test}\")\n\nLogdet (exact):  -148.3218440234385\nLogdet (approx): -148.3090874345191\n\n\nEven using n / 6 matvecs, we get a decent approximation. But how good is this estimate?\nTo get a slightly better idea, you can set verbose=True:\n\nest = hutch(A, fun=\"log\", maxiter=25, seed=5, verbose=True)\n\nGirard-Hutchinson estimator (fun=log, deg=20, quad=fttr)\nEst: -146.830 +/- 5.59 (95% CI), CV: 2%, Evals: 25 [R] (seed: 5)\n\n\nThe first line of the statement contains fixed about the estimator, including the type of estimator (Girard-Hutchinson), the function applied to the spectrum (log), the degree of the Krylov expansion (20), and the quadrature method used (fttr).\nThe second line prints the runtime information about the samples, such as the final trace estimate, its margin of error and coefficient of variation (aka the relative std. deviation), and the number of samples used + their distribution prefix (‘R’ for rademacher).\nAnother way to get an idea of how well the estimator is converging is to pass the plot=True option. Here, we pass the same seed for reproducibility with the above call.\n\nest = hutch(A, fun=\"log\", maxiter=100, seed=5, plot=True)\n\n\n  \n\n\n\n\n\nSimilar information reported by the verbose=True flag is plotted, now cumulatively applied to every sample. The left plot shows the sample point estimates alongside yellow bands denoting the margin of error associated with the 95% confidence interval (CI) at every sample index. The right plot show convergence information, e.g. the running coefficient of variation is shown on the top-right and an upper bound on the absolute error is derived from the CI is shown on the bottom right.\nWhile the margin of error is typically a valid, conservative estimate of estimators error, in some situations these bounds may not be valid as they are based solely on the CLT implications for normally distributed random variables. In particular, there is an inherent amount of error associated with the Lanczos-based quadrature approximations that is not taken into account in the error bound, which may invalidate the corresponding CI’s.",
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-    "text": "operator.matrix_function(A, fun='identity', deg=20, rtol=None, orth=0, **kwargs)\nConstructs an operator approximating the action v \\mapsto f(A)v\nThis function constructs an operator that uses the Lanczos quadrature method to approximate the action of matrix-vector multiplication with the matrix function f(A) for any choice of fun.\n\n\nA : ndarray, sparray, or LinearOperator real symmetric operator. fun : str or Callable, default=“identity” real-valued function defined on the spectrum of A. deg : int, default=20 Degree of the Krylov expansion. rtol : float, default=1e-8 Relative tolerance to consider two Lanczos vectors are numerically orthogonal. orth: int, default=0 Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis. kwargs : dict, optional additional key-values to parameterize the chosen function ‘fun’.\n\n\n\noperator : LinearOperator Operator approximating the action of fun on the spectrum of A\n\n\n\nThe matrix-function approximation is implemented via Lanczos quadrature, which combines the Lanczos method with either: 1. The Golub-Welsch algorithm (GW), or 2. The Forward Three Term Recurrence algorithm (FTTR) for computing the deg-point Gaussian quadrature rule of a symmetric tridiagonal / Jacobi matrix T.\nThe GW computation uses implicit symmetric QR steps with Wilkinson shift to compute the full eigen-decomposition of T, while the FTTR algorithm uses the explicit expression for orthogonal polynomials to compute only the weights of the quadrature. The former uses O(\\mathrm{deg}^2) time and space and is highly accurate, while the latter uses O(\\mathrm{deg}^2) time and O(1) space at the cost of some accuracy. If deg is large, the fttr method should be preferred."
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-    "text": "The matrix-function approximation is implemented via Lanczos quadrature, which combines the Lanczos method with either: 1. The Golub-Welsch algorithm (GW), or 2. The Forward Three Term Recurrence algorithm (FTTR) for computing the deg-point Gaussian quadrature rule of a symmetric tridiagonal / Jacobi matrix T.\nThe GW computation uses implicit symmetric QR steps with Wilkinson shift to compute the full eigen-decomposition of T, while the FTTR algorithm uses the explicit expression for orthogonal polynomials to compute only the weights of the quadrature. The former uses O(\\mathrm{deg}^2) time and space and is highly accurate, while the latter uses O(\\mathrm{deg}^2) time and O(1) space at the cost of some accuracy. If deg is large, the fttr method should be preferred."
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-    "text": "operator.matrix_function(A, fun='identity', deg=20, rtol=None, orth=0, **kwargs)\nConstructs a LinearOperator approximating the action of the matrix function fun(A).\nThis function uses the Lanczos quadrature method to approximate the action of matrix function:  v \\mapsto f(A)v, \\quad f(A) = U f(\\lambda) U^T  The resulting operator may be used in conjunction with other trace and vector-based estimation methods, such as hutch or xtrace.\nThe weights of the quadrature may be computed using either the Golub-Welsch (GW) or Forward Three Term Recurrence algorithms (FTTR) (see the quad parameter). For a description of the other parameters, see the Lanczos function.\n\n\n\n\n\n\nNote\n\n\n\nTo compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require O(\\mathrm{deg}^2) time to execute, the former requires O(\\mathrm{deg}^2) space but is highly accurate, while the latter uses only O(1) space at the cost of stability. If deg is large, fttr is preferred.\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\nscipy.sparse.linalg.LinearOperator\na LinearOperator approximating the action of fun on the spectrum of A",
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-    "text": "The matrix-function approximation is implemented via Lanczos quadrature, which combines the Lanczos method with either: 1. The Golub-Welsch algorithm (GW), or 2. The Forward Three Term Recurrence algorithm (FTTR) for computing the deg-point Gaussian quadrature rule of a symmetric tridiagonal / Jacobi matrix T.\nThe GW computation uses implicit symmetric QR steps with Wilkinson shift to compute the full eigen-decomposition of T, while the FTTR algorithm uses the explicit expression for orthogonal polynomials to compute only the weights of the quadrature. The former uses O(\\mathrm{deg}^2) time and space and is highly accurate, while the latter uses O(\\mathrm{deg}^2) time and O(1) space at the cost of some accuracy. If deg is large, the fttr method should be preferred."
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-    "text": "trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='fttr', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)\nEstimates the trace of a symmetric A or matrix function f(A) via a Girard-Hutchinson estimator.\nThis function uses up to maxiter random isotropic vectors to estimate of the trace of f(A), where: \\mathrm{tr}(f(A)) = \\mathrm{tr}(U f(\\Lambda) U^T) = \\sum\\limits_{i=1}^n f(\\lambda_i)  The estimator is obtained by averaging quadratic forms v \\mapsto v^T f(A)v, rescaling as necessary. This estimator may be used to quickly approximate of a variety of quantities, such as the trace inverse, the log-determinant, the numerical rank, etc. See the online documentation for more details.\n\n\n\n\n\n\nNote\n\n\n\nConvergence behavior is controlled by the stop parameter: “confidence” uses the central limit theorem to generate confidence intervals on the fly, which may be used in conjunction with atol and rtol to upper-bound the error of the approximation. Alternatively, when stop = “change”, the estimator is considered converged when the error between the last two iterates is less than atol (or rtol, respectively), similar to the behavior of scipy.integrate.quadrature.\n\n\n\n\n\n\n\n\n\n\n\n\n\nName\nType\nDescription\nDefault\n\n\n\n\nA\nndarray, sparray, or LinearOperator\nreal symmetric operator.\nrequired\n\n\nfun\nstr or typing.Callable\nreal-valued function defined on the spectrum of A.\n\"identity\"\n\n\nmaxiter\nint\nMaximum number of random vectors to sample for the trace estimate.\n10\n\n\ndeg\nint\nDegree of the quadrature approximation. Must be at least 1.\n20\n\n\natol\nfloat\nAbsolute tolerance to signal convergence for early-stopping. See notes.\nNone\n\n\nrtol\nfloat\nRelative tolerance to signal convergence for early-stopping. See notes.\n1e-2\n\n\nstop\nstr\nEarly-stopping criteria to test estimator convergence. See details.\n\"confidence\"\n\n\nncv\nint\nNumber of Lanczos vectors to allocate. Must be at least 2.\n2\n\n\north\nint\nNumber of additional Lanczos vectors to orthogonalize against. Must be less than ncv.\n0\n\n\nquad\nstr\nMethod used to obtain the weights of the Gaussian quadrature. See notes.\n'fttr'\n\n\nconfidence\nfloat\nConfidence level to consider estimator as converged. Only used when stop = “confidence”.\n0.95\n\n\npdf\n‘rademacher’, ‘normal’\nChoice of zero-centered distribution to sample random vectors from.\n'rademacher'\n\n\nrng\n‘splitmix64’, ’xoshiro256**‘, ’pcg64’, ‘mt64’\nRandom number generator to use.\n'splitmix64'\n\n\nseed\nint\nSeed to initialize the rng entropy source. Set seed &gt; -1 for reproducibility.\n-1\n\n\nnum_threads\nint\nNumber of threads to use to parallelize the computation. Set to &lt;= 0 to let OpenMP decide.\n0\n\n\nplot\nbool\nIf true, plots the samples of the trace estimate along with their convergence characteristics.\nFalse\n\n\ninfo\nbool\nIf True, returns a dictionary containing all relevant information about the computation.\nFalse\n\n\nkwargs\ndict\nadditional key-values to parameterize the chosen function ‘fun’.\n{}\n\n\n\n\n\n\n\n\n\n\n\n\n\nType\nDescription\n\n\n\n\ntyping.Union[float, tuple]\nEstimate the trace of f(A). If ‘info = True’, additional information about the computation is also returned.\n\n\n\n\n\n\nlanczos : the lanczos tridiagonalization algorithm.\n\n\n\n\nUbaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.",
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-    "text": "For matrix functions (when fun is not None), the Lanczos method is used to approximate the action of f(A), and this function yields estimates of the quantity: \\mathrm{tr}(f(A)) = \\mathrm{tr}(U f(\\Lambda) U^T) This allows for quick approximating of a variety of quantities, such as the trace inverse, the log-determinant, the numerical rank, etc. See the online documentation for more details.",
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-    "text": "primate provides a variety of efficient algorithms for estimating quantities derived from matrix functions. These algorithms are largely implemented in C++ to minimize overhead, and for some computational problems primate can out-perform the standard algorithms for estimating spectral quantities by several orders of magnitude. Nonetheless, there are some performance-related caveats to be aware of.\n\n\n\nfrom scipy.linalg import toeplitz\nfrom primate.trace import hutch \n\nc = np.random.uniform(size=100, low=0, high=1)\nT = toeplitz(c)\n\n# np.sum(np.reciprocal(np.linalg.eigvalsh(T)))\nnp.sum(np.abs(np.linalg.eigvalsh(T)))\nest, info = hutch(T, fun=np.abs, maxiter=200, pdf='rademacher', rng=\"mt\", deg=20, seed=-1, plot=True)\n\nest, info = hutch(T, fun=np.abs, maxiter=800, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=True)\nest, info = hutch(C, fun=np.abs, maxiter=800, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=True)\n\n\n\nnp.sum(np.linalg.svd(T)[1])\n\nZ = np.zeros(T.shape)\nC = np.block([[Z, T], [T.T, Z]])\nhutch(C, fun=np.abs, maxiter=200, pdf='rademacher', rng=\"mt\", quad=\"golub_welsch\", deg=200, seed=-1, plot=False)\n\n\nhutch(T, fun=np.abs, maxiter=200, pdf='rademacher', deg=50, seed=-1, plot=False)\n\np = info['figure']\n\nfrom primate.trace import xtrace\nfrom primate.operator import matrix_function\nM = matrix_function(T, \"abs\")\nxtrace(M)\n\ns = info['samples']\nv = np.random.choice([-1, 1], size=T.shape[0])\nT @ v\n\ns[np.abs(s) &lt;= np.linalg.norm(T)].mean()\n\n\nfrom primate.diagonalize import lanczos\nfrom scipy.linalg import eigvalsh_tridiagonal\na, b = lanczos(T, deg=499, orth=150)\nnp.sum(np.abs(eigvalsh_tridiagonal(a,b)))\n\n\nimport timeit \ntimeit.timeit(lambda: hutch(A, maxiter=20, deg=5, fun=\"log\", quad=\"fttr\"), number = 1000)\ntimeit.timeit(lambda: np.sum(np.log(np.linalg.eigvalsh(A))), number = 1000)"
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-    "text": "Footnotes\n\n\nTechnically, we could solve a system of equations via either Gaussian elimination or, when A is rank-deficient, via least squares: (A + \\epsilon I)^{-1}v \\simeq u \\Leftrightarrow \\mathrm{argmin}_{u \\in \\mathbb{R}^n} \\lVert v - (A + \\epsilon I)u \\rVert^2. Then again, this could be just as expensive—if not more so—than just executing the Lanczos iteration!↩︎",
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-    "text": "… but how do I code it up?\nTo clarify that that means, here’s an abstract presentation of the generic SLQ procedure:\n\n\n\\begin{algorithm} \\caption{Stochastic Lanczos Quadrature} \\begin{algorithmic} \\Input Symmetric operator ($A \\in \\mathbb{R}^{n \\times n}$) \\Require Number of queries ($n_v$), Degree of quadrature ($k$) \\Function{SLQ}{$A$, $n_v$, $k$} \\State $\\Gamma \\gets 0$ \\For{$j = 1, 2, \\dots, n_v$} \\State $v_i \\sim \\mathcal{D}$ where $\\mathcal{D}$ satisfies $\\mathbb{E}(v v^\\top) = I$ \\State $T^{(j)}(\\alpha, \\beta)$ $\\gets$ $\\mathrm{Lanczos}(A,v_j,k+1)$ \\State $[\\Theta, Y] \\gets \\mathrm{eigh\\_tridiag}(T^{(j)}(\\alpha, \\beta))$ \\State $\\tau_i \\gets \\langle e_1, y_i \\rangle$ \\State &lt; Do something with the node/weight pairs $(\\theta_i, \\tau_i^2)$ &gt; \\EndFor \\EndFunction \\end{algorithmic} \\end{algorithm}",
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-    "text": "Beating the bound\nElegant and as theoretically founded as the Lanczos method may be, is it efficient in practice?\nLet’s start by establishing a baseline on its complexity:\n\nTheorem 1 (Parlett 1994) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos method computes \\Lambda(A) in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic\n\nAs its clear from the theorem, if we specialize it such that r = n and \\eta = \\nu = n, then the Lanczos method requires just O(n^2) time and O(n) space to execute. In other words, the Lanczos method drops both the time and space complexity4 of obtaining spectral information by order of magnitude over similar eigen-algorithms that decompose A directly.\nTo see why this is true, note that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then carrying out the recurrence clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nNow, if we need to store all of Y or Q explicitly, we clearly need O(n^2) space to do so. However, if we only need the eigen-values \\Lambda(A) (and not their eigen-vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!",
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-    "text": "Basics: Trace computation\n\nSuppose we wanted to compute the trace of some matrix A. By the very definition of the trace, we have a variety of means to do it: \\mathrm{tr}(A) \\triangleq \\sum\\limits_{i=1}^n A_{ii} = \\sum\\limits_{i=1}^n \\lambda_i = \\sum\\limits_{i=1}^n e_i^T A e_i \nwhere, in the right-most sum, we’re using e_i to represent the ith identity vector:  A_{ii} = e_i^T A e_i, \\quad e_i = [0, \\dots, 0, \\underbrace{1}_{i}, 0, \\dots, 0] \nLet’s see some code. First, we start with a simple (random) positive-definite matrix A \\in \\mathbb{R}^{n \\times n}.\n\nfrom primate.random import symmetric\nA = symmetric(150, pd = True)\n\nBy default, symmetric normalizes A such that the eigenvalues are uniformly distributed in the interval [0, 1]. For reference, the spectrum of A looks as follows:\n\n\n\n  \n\n\n\n\n\nNow, using numpy, we can verify all of these trace definitions are indeed equivalent:\n\n\nimport numpy as np\neye = lambda i: np.ravel(np.eye(1, 150, k=i))\nprint(f\"Direct: {np.sum(A.diagonal()):.8f}\")\nprint(f\"Eigen:  {np.sum(np.linalg.eigvalsh(A)):.8f}\")\nprint(f\"Matvec: {sum([eye(i) @ A @ eye(i) for i in range(150)]):.8f}\")\n\nDirect: 75.69739746\nEigen:  75.69739746\nMatvec: 75.69739746\n\n\nWhile (1) trivially yields \\mathrm{tr}(A) in (optimal) O(n) time, there exist situations where the diagonal entries of A are not available—particularly in the large-scale regime. In contrast, though both (2) and (3) are less efficient, they are nonetheless viable alternatives to obtain \\mathrm{tr}(A).",
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-    "text": "Implicit trace estimation\nThe implicit trace estimation problem can be stated as follows:\n\nGiven access to a square matrix A \\in \\mathbb{R}^{n \\times n} via its matrix-vector product operator x \\mapsto Ax, estimate its trace \\mathrm{tr}(A) = \\sum\\limits_{i=1}^n A_{ii}\n\nNote we no longer assume we have direct access to the diagonal here, leaving us with the Eigen or Matvec methods. The Eigen method is expensive, and though the Matvec method solves the problem exactly, its pretty inefficient from a computational standpoint—most of the entries of e_i are zero, thus most inner product computations do not contribute to the trace estimate at all.\nOne idea, attributed to A. Girard, is to replace e_i with random zero-mean vectors, e.g. random sign vectors v \\in \\{-1, +1\\}^{n} or random normal vectors v \\sim \\mathcal{N}(\\mu=0, \\sigma=1):\n\\mathtt{tr}(A) \\approx \\frac{1}{n_v}\\sum\\limits_{i=1}^{n_v} v_i^\\top A v_i \nIt was shown more formally later by M.F. Huchinson that if these random vectors v \\in \\mathbb{R}^n satisfy \\mathbb{E}[vv^T] = I then this approach indeed forms an unbiased estimator of \\mathrm{tr}(A). To see this, its sufficient to combine the linearity of expectation, the cyclic-property of the trace function, and the aforementioned isotropy conditions: \\mathtt{tr}(A) = \\mathtt{tr}(A I) = \\mathtt{tr}(A \\mathbb{E}[v v^T]) = \\mathbb{E}[\\mathtt{tr}(Avv^T)] = \\mathbb{E}[\\mathtt{tr}(v^T A v)] = \\mathbb{E}[v^T A v]\nNaturally we expect the approximation to gradually improve as more vectors are sampled, converging as n_v \\to \\infty. Let’s see if we can check this: we start by collecting a few sample quadratic forms\n\ndef isotropic(n):\n  yield from [np.random.choice([-1,+1], size=A.shape[1]) for i in range(n)]\nestimates = np.array([v @ A @ v for v in isotropic(500)])\n\nPlotting both the estimator formed by averaging the first k samples along with its absolute error, we get the following figures for k = 50 and k = 500, respectively:\n\n\n\n  \n\n\n\n\n\nSure enough, the estimator quickly gets within an error rate of about 1-5% away from the actual trace, getting generally closer as more samples are collected. On the other hand, improvement is not gaurenteed, and securing additional precision much less than 1% becomes increasingly difficult due to the randomness and variance of the sample estimates.",
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-    "text": "Beating the complexity bounds\nElegant and as theoretically founded as the Lanczos method may be, is it efficient in practice?\nLet’s start by establishing a baseline on its complexity:\n\nTheorem 1 (Parlett 1994) Given a symmetric rank-r matrix A \\in \\mathbb{R}^{n \\times n} whose operator x \\mapsto A x requires O(\\eta) time and O(\\nu) space, the Lanczos method computes \\Lambda(A) in O(\\max\\{\\eta, n\\}\\cdot r) time and O(\\max\\{\\nu, n\\}) space, when computation is done in exact arithmetic\n\nAs its clear from the theorem, if we specialize it such that r = n and \\eta = \\nu = n, then the Lanczos method requires just O(n^2) time and O(n) space to execute. In other words, the Lanczos method drops both the time and space complexity4 of obtaining spectral information by order of magnitude over similar eigen-algorithms that decompose A directly.\nTo see why this is true, note that a symmetric tridiagonal matrix is fully characterized by its diagonal and subdiagonal terms, which requires just O(n) space. If we assume that v \\mapsto Av \\sim O(n), then carrying out the recurrence clearly takes at most O(n^2) time, since there are most n such vectors \\{q_i\\}_{i=1}^n to generate!\nNow, if we need to store all of Y or Q explicitly, we clearly need O(n^2) space to do so. However, if we only need the eigen-values \\Lambda(A) (and not their eigen-vectors U), then we may execute the recurrence keeping at most three vectors \\{q_{j-1}, q_{j}, q_{j+1}\\} in memory at any given time. Since each of these is O(n) is size, the claim of O(n) space is justified!",
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-    "text": "Example: Logdet computation\nPutting all of the theory above together, a simple way to estimate the trace of a matrix function is to average Lanczos quadrature estimates:\n\\mathrm{tr}(f(A)) \\approx \\frac{n}{n_v} \\sum\\limits_{i=1}^{n_v} e_1^T f(T_m) e_1  = \\frac{n}{n_v} \\sum\\limits_{j=1}^{n_v} \\left ( \\sum\\limits_{i=1}^m \\tau_i^{(j)} f(\\theta_i)^{(j)} \\right )\nTo enable this kind of estimator in hutch, simply set the fun argument to either the name of a built-in function or an arbitrary Callable. For example, to approximate the log-determinant:\n\n## Log-determinant\nlogdet_test = hutch(A, fun=\"log\", maxiter=25)\nlogdet_true = np.sum(np.log(np.linalg.eigvalsh(A)))\n\nprint(f\"Logdet (exact):  {logdet_true}\")\nprint(f\"Logdet (approx): {logdet_test}\")\n\nLogdet (exact):  -148.3218440234385\nLogdet (approx): -148.71526016046562\n\n\nHere we see that using n / 6 evaluatons of e_1^T f(T_m) e_1, we get a decent approximation of logdet(A). To get a better idea of the quality of this estimate, you can set verbose=True:\n\nest = hutch(A, fun=\"log\", maxiter=25, seed=5, verbose=True)\n\nGirard-Hutchinson estimator (fun=log, deg=20, quad=fttr)\nEst: -143.207 +/- 5.40 (95% CI), CV: 2%, Evals: 25 [R] (seed: 5)\n\n\nThe first line of the statement contains fixed about the estimator, including the type of estimator (Girard-Hutchinson), the function applied to the spectrum (log), the degree of the Lanczos quadrature approximation (20), and the quadrature method used (fttr).\nThe second line prints the runtime information about the samples, such as the final trace estimate, its margin of error and coefficient of variation (aka the relative std. deviation), and the number of samples used + their distribution prefix (‘R’ for rademacher).\nAnother way to get an idea of how well the estimator is converging is to pass the plot=True option. Here, we pass the same seed for reproducibility with the above call.\n\nest = hutch(A, fun=\"log\", maxiter=100, seed=5, plot=True)\n\n\n  \n\n\n\n\n\nSimilar information as reported by the verbose=True flag is shown cumulatively applied to every sample. The left plot shows the sample point estimates alongside yellow bands denoting the margin of error associated with the 95% confidence interval (CI) at every sample index. The right plot show convergence information, e.g. the running coefficient of variation is shown on the top-right and an upper bound on the absolute error is derived from the CI is shown on the bottom right.\nWhile the margin of error is typically a valid, conservative estimate of estimators error, in some situations these bounds may not be valid as they are based solely on the CLT implications for normally distributed random variables. In particular, there is an inherent amount of error associated with the Lanczos-based quadrature approximations that is not taken into account in the error bound, which may invalidate the corresponding CI’s.",
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-    "text": "Implementation\nRather than manually averaging samples, trace estimates can be obtained via the hutch function:\n\nfrom primate.trace import hutch\nprint(f\"Trace estimate: {hutch(A, maxiter=50)}\")\n\nTrace estimate: 76.03071528930039\n\n\nThough its estimation is identical to averaging quadratic forms v^T A v of isotropic vectors as above, hutch comes with a few extra features to simplify its usage. In particular, all v \\mapsto v^T A v evaluations are done in parallel, there are a variety of isotropic distributions and random number generators (RNGs) to choose from, and the estimator may be suppleid various convergence criteria to allow for early-stopping; see the hutch documentation page for more details.",
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-    "text": "Estimator Implementation\nRather than manually averaging samples, trace estimates can be obtained via the hutch function:\n\nfrom primate.trace import hutch\nprint(f\"Trace estimate: {hutch(A, maxiter=50)}\")\n\nTrace estimate: 76.32787158595767\n\n\nThough its estimation is identical to averaging quadratic forms v^T A v of isotropic vectors as above, hutch comes with a few extra features to simplify its usage. In particular, all v \\mapsto v^T A v evaluations are done in parallel, there are a variety of isotropic distributions and random number generators (RNGs) to choose from, and the estimator may be suppleid various convergence criteria to allow for early-stopping; see the hutch documentation page for more details.",
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-    "text": "Stochastic Lanczos Quadrature\nWhile there are some real-world applications for estimating \\mathrm{tr}(A), in many setting the trace of a given matrix is not a very informative quantity. On the other hand, as shown in the beginning, there are many situations where we might be interested in estimating the trace of a matrix function f(A). It is natural to consider extending the Hutchinson estimator above with something like:\n \\mathrm{tr}(f(A)) \\approx \\frac{n}{n_v} \\sum\\limits_{i=1}^{n_v} v_i^T f(A) v_i \nOf course, the remaining difficult lies in computing quadratic forms v \\mapsto v^T f(A) v efficiently. The approach taken by Ubaru, Saad, and Seghouane (2017) is to notice that sums of these quantities can transformed into a Riemann-Stieltjes integral:\n v^T f(A) v = v^T U f(\\Lambda) U^T v = \\sum\\limits_{i}^n f(\\lambda_i) \\mu_i^2 = \\int\\limits_{a}^b f(t) \\mu(t)\nwhere scalars \\mu_i \\in \\mathbb{R} constitute a cumulative, piecewise constant function \\mu : \\mathbb{R} \\to \\mathbb{R}_+:\n\n\\mu_i = (U^T v)_i, \\quad \\mu(t) = \\begin{cases}\n0, & \\text{if } t &lt; a = \\lambda_1 \\\\\n\\sum_{j=1}^{i-1} \\mu_j^2, & \\text{if } \\lambda_{i-1} \\leq t &lt; \\lambda_i, i = 2, \\dots, n \\\\\n\\sum_{j=1}^n \\mu_j^2, & \\text{if } b = \\lambda_n \\leq t\n\\end{cases}\n\nAs with any definite integral, one would ideally like to approximate its value via some quadrature rule. An exemplary such approximation is the m-point Gaussian quadrature rule:\n \\int\\limits_{a}^b f(t) \\mu(t) \\approx \\sum\\limits_{k=1}^m \\omega_k f(\\eta_k) \nwith weights \\{\\omega_k\\} and nodes \\{ \\eta_k\\}. On one hand, Gaussian Quadrature (GQ) is just one of many quadrature rules that could be used to approximate v^T A v. On the other hand, GQ is often preferred over other rules due to its exactness on polynomials up to degree 2m - 1.\nOf course, both the weights and nodes of GQ rule are unknown for the integral above. This is in strong contrast to the typical setting, wherein \\omega is assumed uniform or otherwise known ahead-of-time. A surprising and non-trivial fact is that both the weights \\{\\omega_k\\} and nodes \\{ \\eta_k\\} of the m-point GQ above are directly computable the degree m matrix T_m produced by the Lanczos method:\n Q_m^T A Q_m = T_m = Y \\Theta Y^T \\quad \\Leftrightarrow \\quad (\\eta_i, \\omega_i) = (\\theta_i, \\tau_i), \\; \\tau_i = (e_1^T y_i)^2 \nIn other words, the Ritz values \\{\\theta_i\\} of T_m are exactly the nodes of GQ quadrature rule applied to \\mu(t), while the first components of the Ritz vectors \\tau_i (squared) are exactly the weights. For more information about this non-trivial fact, see Golub and Meurant (2009).",
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-    "text": "To clarify that that means, here’s an abstract presentation of the generic SLQ procedure:\n\n\n\\begin{algorithm} \\caption{Stochastic Lanczos Quadrature} \\begin{algorithmic} \\Input Symmetric operator ($A \\in \\mathbb{R}^{n \\times n}$) \\Require Number of queries ($n_v$), Degree of quadrature ($k$) \\Function{SLQ}{$A$, $n_v$, $k$} \\State $\\Gamma \\gets 0$ \\For{$j = 1, 2, \\dots, n_v$} \\State $v_i \\sim \\mathcal{D}$ where $\\mathcal{D}$ satisfies $\\mathbb{E}(v v^\\top) = I$ \\State $T^{(j)}(\\alpha, \\beta)$ $\\gets$ $\\mathrm{Lanczos}(A,v_j,k+1)$ \\State $[\\Theta, Y] \\gets \\mathrm{eigh\\_tridiag}(T^{(j)}(\\alpha, \\beta))$ \\State $\\tau_i \\gets \\langle e_1, y_i \\rangle$ \\State &lt; Do something with the node/weight pairs $(\\theta_i, \\tau_i^2)$ &gt; \\EndFor \\EndFunction \\end{algorithmic} \\end{algorithm}"
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-    "text": "In the introduction, the basics of implicit trace estimation were introduced, wherein it shown both in theory and with code using primate how to estimate the trace of matrix functions:\nf(A) = U f(\\Lambda) U^T, \\quad f: [a,b] \\to \\mathbb{R}\nIn particular, the introduction briefly covered how the Lanczos method is intimately connected to Gaussian Quadrature, and how this connection enables fast randomized trace estimation of f(A).\nIn this post, I’ll cover how to approximate the action v \\mapsto f(A)v for any function f with primate with its matrix_function API, and how to compose this functionality with other trace estimators.",
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-    "text": "# a, b = 0.8, 2\n# x = np.random.uniform(low=0, high=10, size=40)\n# eps = np.random.normal(loc=0, scale=1.0, size=40)\n# y = (a * x + b) + eps\n\n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# show(p)\n\n# X = np.c_[x,y]\n# from scipy.linalg import lstsq\n# from scipy.optimize import least_squares, leastsq, minimize, minimize_scalar\n# def L(beta: tuple):\n#   b, a = beta\n#   return np.linalg.norm(y - a * x + b)**2\n\n# def L(beta: tuple):\n#   b, a = beta\n#   return (y - (a * x + b))\n\n\n\n# res = minimize_scalar(L, x0=(0,1))\n# b_opt, a_opt = res.x\n# L([a_opt,b_opt])\n\n# res = least_squares(L, x0=(0,1))\n# b_opt, a_opt = res.x\n# f_opt = lambda x: x * a_opt + b_opt \n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# p.line(x=[0, 10], y=[b_opt, f_opt(10)])\n# show(p)\n\n# ## Normal equations...\n# XI = np.c_[np.ones(X.shape[0]), X]\n\n# c, b_opt, a_opt = (np.linalg.inv((XI.T @ XI)) @ XI.T) @ y\n# f_opt = lambda x: x * a_opt + b_opt \n# p = figure(width=350, height=200)\n# p.scatter(x, y)\n# p.line(x=[0, 10], y=[b_opt, f_opt(10)])\n# show(p)"
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-    "text": "Establishing a baseline\nprimate allows the user to construct a LinearOperator directly from an existing matrix-like object A and callable fun through its matrix_function API.\n\nAs a baseline example, consider the action that adds an \\epsilon amount of mass to the diagonal of A:\nv \\mapsto (A + \\epsilon I)v\nFor any fixed \\epsilon, imitating this matrix action can be done in three different ways:\n\nObtain u = Av and then add u \\gets u + \\epsilon \\cdot v\nForm (A + \\epsilon I) and explicitly carry out the multiplication\nMultiply v by f(A) induced by f(\\lambda) = \\lambda + \\epsilon\n\nLets see what the code to accomplish this using (3) looks like:\n\nfrom primate.random import symmetric\nfrom primate.operator import matrix_function\n\n## Random matrix + input vector\nA = symmetric(150, pd = True)\nv = np.random.uniform(size=A.shape[1])\n\n## Ground truth v |-&gt; f(A)v\nLambda, U = np.linalg.eigh(A)\nv_truth = (U @ np.diag(Lambda + 0.10) @ U.T) @ v\n\n## Lanczos approximation\nM = matrix_function(A, fun = lambda x: x + 0.10)\nv_test = np.ravel(M @ v)\n\nvn, vn_approx = np.linalg.norm(v_truth), np.linalg.norm(v_test)\nprint(f\"f(A)v norm:  {vn:.6f}\")\nprint(f\"Approx norm: {vn_approx:.6f}\")\nprint(f\"Max diff:    {np.max(np.abs(v_truth - np.ravel(v_test))):.6e}\")\nprint(f\"cosine sim:  {np.dot(v_test, v_truth) / (vn * vn_approx):6e}\")\n\nf(A)v norm:  4.392660\nApprox norm: 4.392660\nMax diff:    5.218048e-15\ncosine sim:  1.000000e+00\n\n\nObserve M matches the ground truth v \\mapsto (A + \\epsilon I)v. In this way, one benefit of using matrix_function is that it allows one to approximate f(A) by thinking only about what is happening at the spectral level (as opposed to the matrix level). Of course, this example is a bit non-convincing as there are simpler ways of achieving the same result as shown by (1) and (2), for example:\n\nnp.allclose(A @ v + 0.10 * v, v_truth)\n\nTrue\n\n\nBaseline established.",
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-    "section": "When v \\mapsto f(A)v is not known",
-    "text": "When v \\mapsto f(A)v is not known\nOn the other hand, there are plenty of situations where one doesn’t have access to these simpler expressions. For example, consider the map:\nv \\mapsto (A + \\epsilon I)^{-1} v\nSuch expressions pop up in a variety of settings, such as in Tikhonov regularization, in Schatten-norm estimation Ubaru, Saad, and Seghouane (2017), in the Cholesky factorization of PSD matrices, and so on. Unlike the previous setting, we cannot readily access v \\mapsto f(A)v unless we explicitly compute the full spectral decomposition of A or the inverse of A, both of which are expensive to obtain.\n\n## Alternative: v_truth = np.linalg.inv((A + 0.10 * np.eye(A.shape[0]))) @ v\nv_truth = (U @ np.diag(np.reciprocal(Lambda + 0.10)) @ U.T) @ v\n\n## Lanczos approximation\nM = matrix_function(A, fun = lambda x: 1.0 / (x + 0.10))\nv_test = np.ravel(M @ v)\n\nvn, vn_approx = np.linalg.norm(v_truth), np.linalg.norm(v_test)\nprint(f\"f(A)v norm:  {vn:.6f}\")\nprint(f\"Approx norm: {vn_approx:.6f}\")\nprint(f\"Max diff:    {np.max(np.abs(v_truth - np.ravel(v_test))):.6e}\")\nprint(f\"cosine sim:  {np.dot(v_test, v_truth) / (vn * vn_approx):6e}\")\n\nf(A)v norm:  32.982583\nApprox norm: 32.982583\nMax diff:    1.626085e-05\ncosine sim:  1.000000e+00\n\n\nThere is a larger degree of error compared to the base as evidenced by the \\lVert \\cdot \\rVert_\\infty-normed difference between v_truth and v_test, however this is to be expected, as in general approximating the action v \\mapsto A^{-1} v will always be more difficult that v \\mapsto A v, even if A is well-conditioned.",
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-    "text": "Unfortunately, by its very definition, if A \\in \\mathbb{R}^{n \\times n} then obtaining f(A) \\in \\mathbb{R}^{n \\times n} explicitly can be very expensive. One well-known approach to sidestep this difficulty is to use the degree-k Lanczos expansion to iteratively approximate the action x \\mapsto f(A)x:\n Q^T A Q = T \\quad \\Leftrightarrow \\quad f(A)x \\approx \\lVert x \\rVert \\cdot Q f(T) e_1 \nDespite its rich theory, most results about the Lanczos method only hold in exact arithmetic; historically, practitioners have been hesitant about using the Lanczos method for eigen-estimation application due to its well-known instability with respect to roundoff and cancellation errors. Such errors typically manifest as loss of orthogonality between Lanczos vectors, which can cause a host of issues related to eigen- estimations, such as muddled convergence and spurious Ritz values. While there exist fixes to these issues, such fixes typically involve difficult to implement convergence heuristics or expensive re-orthogonalization techniques.\nOn the other hand, there is mounting empirical evidence (e.g. Ghorbani, Krishnan, and Xiao (2019)) that suggests using the (plain) Lanczos method in randomized setting is often more than sufficient for several eigenvalue-related estimation tasks. In fact, it’s been recently shown[^2] that the using the Lanczos method to approximate the action x \\mapsto f(A)x has the following guarantee:\n\\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| \\leq 2\\|\\mathbf{x}\\| \\cdot \\min _{\\substack{\\text { polynomial } p \\\\ \\text { with degree }&lt;k}}\\left(\\max _{x \\in\\left[\\lambda_{\\min }(\\mathbf{A}), \\lambda_{\\max }(\\mathbf{A})\\right]}|f(x)-p(x)|\\right)\nIn other words, up to a factor of 2, the error \\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| is bounded by the uniform error of the best polynomial approximation to f with degree &lt; k. For general matrix functions, this implies that finite-precision Lanczos essentially matches strongest known exact arithmetic bounds.",
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-    "text": "Matrix function approximation\nIf A \\in \\mathbb{R}^{n \\times n} and n is large, obtaining f(A) \\in \\mathbb{R}^{n \\times n} explicitly can be very expensive. One way to sidestep this difficulty is to approximate v \\mapsto f(A)v using the degree-k Lanczos expansion:\n Q^T A Q = T \\quad \\Leftrightarrow \\quad f(A)v \\approx \\lVert x \\rVert \\cdot Q f(T) e_1 \nIt’s been shown by (Musco, Musco, and Sidford 2018) that this approximation has the following guarantee:\n\\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| \\leq 2\\|\\mathbf{x}\\| \\cdot \\min _{\\substack{\\text { polynomial } p \\\\ \\text { with degree }&lt;k}}\\left(\\max _{x \\in\\left[\\lambda_{\\min }(\\mathbf{A}), \\lambda_{\\max }(\\mathbf{A})\\right]}|f(x)-p(x)|\\right)\nIn other words, up to a factor of 2, the error \\|f(\\mathbf{A}) \\mathbf{x}-\\mathbf{y}\\| is bounded by the uniform error of the best polynomial approximation to f with degree &lt; k. For general matrix functions, this implies that finite-precision Lanczos essentially matches strongest known exact arithmetic bounds.\nThis suggests an idea: can we convert any matrix-like object into a LinearOperator that transparently converts matrix-vector products Av into products with matrix-functions f(A)v?",
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-    "text": "Back to trace estimation\nThere are several use-cases wherein one might be interested in the output f(A)v itself—see Musco, Musco, and Sidford (2018) and references within for some examples. One use-case is the extension of existing matvec-dependent implicit trace estimators to matrix-function trace estimators.\n\nfrom primate.trace import hutch, xtrace\nM = matrix_function(A, fun=\"log\")\nprint(f\"Logdet exact:  {np.sum(np.log(np.linalg.eigvalsh(A))):6e}\")\nprint(f\"Logdet Hutch:  {hutch(A, fun='log'):6e}\")\nprint(f\"Logdet XTrace: {xtrace(M):6e}\")\n\nLogdet exact:  -1.483218e+02\nLogdet Hutch:  -1.477567e+02\nLogdet XTrace: -1.483155e+02\n\n\nAs with the hutch estimators applied to matrix functions, note that the action v \\mapto f(A)v is subject to the approximation errors studied by Musco, Musco, and Sidford (2018), making such extensions limited functions that are well-approximated by Lanczos.",
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diff --git a/docs/src/basic/install.qmd b/docs/src/basic/install.qmd
index 637823f..a2cb885 100644
--- a/docs/src/basic/install.qmd
+++ b/docs/src/basic/install.qmd
@@ -14,14 +14,24 @@ Currently the package must be built from source via cloning the repository. PYPI
 
 ### Platform support
 
-Platform-specific wheels are currently built with cibuildwheel and uploaded to PyPI. These enable `primate` to be installed on supported architecture / CPython implementations without compilation. The currently supported platforms include: 
+Platform-specific wheels are currently built with cibuildwheel and uploaded to PyPI. These enable `primate` to be installed on supported architecture / CPython implementations without compilation. As of 12/29/23, native `primate` wheels are buitl for the following platforms:
 
+| Platform                          | 3.8 | 3.9 | 3.10 | 3.11 | 3.12 |
+|-----------------------------------|:---:|:---:|:----:|:----:|:----:|
+| **Linux (manylinux x86_64)**      | ✅  | ✅  | ✅   | ✅   | ✅   |
+| **MacOS (x86_64)**                | ✅  | ✅  | ✅   | ✅   | ✅   |
+| **MacOS ARM (arm64) [^1]**        | ✅  | ✅  | ✅   | ✅   | ✅   |
+| **Windows (AMD64) [^2]**          | ✅  | ✅  | ✅   | ✅   | ✅   |
 
-If your platform isn't on this list, feel free to make an issue requesting support for your platform. 
+Wheels are currently built with [cibuildwheel](https://cibuildwheel.readthedocs.io/en/stable/).
+Currently no support is offered for PyPy, 32-bit runtimes, or [unsupported versions of CPython](https://devguide.python.org/versions/#supported-versions). If your platform isn't on this list, feel free to make an issue requesting support. 
+
+[^1]: Single-thread execution only; ARM-based OSX runners come stocked with Apple's clang, which doesn't natively ship with `libomp.dylib`, though this [may be fixable](https://mac.r-project.org/openmp/). Feel free to file an PR if you can get this working.  
+[^2]: Single-thread execution only; `primate` depends on OpenMP 4.5+, which isn't supported on any Windows compiler I'm aware of. 
 
 ### Compiling from source
 
-To install the package from its source distribution, a C++20 compiler is required; the current builds are all built with some variant of [clang](). For platform- and compiler-specific settings, consult the build scripts and CI configuration files.
+To install the package from its source distribution, a C++20 compiler is required; the current builds are all built with some variant of [clang](https://clang.llvm.org/). For platform- and compiler-specific settings, consult the build scripts and CI configuration files.
 
 ### C++ Installation
 
diff --git a/docs/src/reference/diagonalize.lanczos.qmd b/docs/src/reference/diagonalize.lanczos.qmd
index 63ba0a7..004c37e 100644
--- a/docs/src/reference/diagonalize.lanczos.qmd
+++ b/docs/src/reference/diagonalize.lanczos.qmd
@@ -12,17 +12,17 @@ modifications to support varying degrees of re-orthogonalization. In particular,
 
 ## Notes
 
-The Lanczos method iteratively builds a tridiagonal matrix `T` of a symmetric `A` via an orthogonal change-of-basis `Q`:
+The Lanczos method builds a tridiagonal `T` from a symmetric `A` via an orthogonal change-of-basis `Q`:
 $$ Q^T A Q  = T $$
 Unlike other Lanczos implementations (e.g. SciPy's `eigsh`), which includes e.g. sophisticated restarting, 
 deflation, and selective-reorthogonalization steps, this method simply executes `deg` steps of the Lanczos method with 
-the supplied `v0` and returns the resulting tridiagonal matrix.
+the supplied `v0` and returns the resulting tridiagonal matrix `T`.
 
-Diagonalizing `T` via e.g. `scipy.linalg.eigh_tridiagonal` yields Rayleigh-Ritz approximations of the eigenvalues of `A`,
-though note no checking is performed for 'ghost' or already converged eigenvalues. To increase the accuracy of these eigenvalue 
-approximation, try increasing `orth` and `deg`. Supplying either negative values or values larger than `deg` for `orth` will 
-result in full re-orthogonalization, though note the number of matvecs scales linearly with `deg` and the number of inner-products 
-scales quadratically with `orth`.
+Rayleigh-Ritz approximations of the eigenvalues of `A` can be further obtained by diagonalizing `T` via any 
+symmetric tridiagonal eigenvalue solver, `scipy.linalg.eigh_tridiagonal` though note unlike `eigsh` no checking is performed 
+for 'ghost' or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing `orth` 
+and `deg`. Supplying either negative values or values larger than `deg` for `orth` will result in full re-orthogonalization, 
+though note the number of matvecs scales linearly with `deg` and the number of inner-products scales quadratically with `orth`.
 
 ## Parameters
 
diff --git a/docs/src/reference/operator.matrix_function.qmd b/docs/src/reference/operator.matrix_function.qmd
index ce46746..ac53877 100644
--- a/docs/src/reference/operator.matrix_function.qmd
+++ b/docs/src/reference/operator.matrix_function.qmd
@@ -4,10 +4,9 @@
 
 Constructs a `LinearOperator` approximating the action of the matrix function `fun(A)`. 
 
-This function uses the Lanczos quadrature method to approximate the action of matrix function:
+This function uses the Lanczos method to approximate the action of matrix function:
 $$ v \mapsto f(A)v, \quad f(A) = U f(\lambda) U^T $$
-The resulting operator may be used in conjunction with other trace and vector-based estimation methods, 
-such as `hutch` or `xtrace`.
+The resulting operator may be used in conjunction with other estimation methods, such as `hutch` or `xtrace`.
 
 The weights of the quadrature may be computed using either the Golub-Welsch (GW) or 
 Forward Three Term Recurrence algorithms (FTTR) (see the `quad` parameter). For a description 
@@ -20,6 +19,17 @@ the former requires $O(\mathrm{deg}^2)$ space but is highly accurate, while the
 If `deg` is large, `fttr` is preferred. 
 :::
 
+## Parameters
+
+| Name     | Type                                                     | Description                                                                                   | Default      |
+|----------|----------------------------------------------------------|-----------------------------------------------------------------------------------------------|--------------|
+| `A`      | scipy.sparse.linalg.LinearOperator or ndarray or sparray | real symmetric operator.                                                                      | _required_   |
+| `fun`    | str or typing.Callable                                   | real-valued function defined on the spectrum of `A`.                                          | `"identity"` |
+| `deg`    | int                                                      | Degree of the Krylov expansion.                                                               | `20`         |
+| `rtol`   | float                                                    | Relative tolerance to consider two Lanczos vectors are numerically orthogonal.                | `1e-8`       |
+| `orth`   | int                                                      | Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis. | `0`          |
+| `kwargs` | dict                                                     | additional key-values to parameterize the chosen function 'fun'.                              | `{}`         |
+
 ## Returns
 
 | Type                               | Description                                                                 |
diff --git a/docs/src/reference/trace.hutch.qmd b/docs/src/reference/trace.hutch.qmd
index d1438bb..fe4c137 100644
--- a/docs/src/reference/trace.hutch.qmd
+++ b/docs/src/reference/trace.hutch.qmd
@@ -1,6 +1,6 @@
 # trace.hutch { #primate.trace.hutch }
 
-`trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='fttr', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)`
+`trace.hutch(A, fun=None, maxiter=200, deg=20, atol=None, rtol=None, stop=['confidence', 'change'], ncv=2, orth=0, quad='golub_welsch', confidence=0.95, pdf='rademacher', rng='pcg64', seed=-1, num_threads=0, verbose=False, info=False, plot=False, **kwargs)`
 
 Estimates the trace of a symmetric $A$ or matrix function $f(A)$ via a Girard-Hutchinson estimator.
 
@@ -19,32 +19,41 @@ Alternatively, when `stop` = "change", the estimator is considered converged whe
 
 ## Parameters
 
-| Name          | Type                                          | Description                                                                                    | Default         |
-|---------------|-----------------------------------------------|------------------------------------------------------------------------------------------------|-----------------|
-| `A`           | ndarray, sparray, or LinearOperator           | real symmetric operator.                                                                       | _required_      |
-| `fun`         | str or typing.Callable                        | real-valued function defined on the spectrum of `A`.                                           | `"identity"`    |
-| `maxiter`     | int                                           | Maximum number of random vectors to sample for the trace estimate.                             | `10`            |
-| `deg`         | int                                           | Degree of the quadrature approximation. Must be at least 1.                                    | `20`            |
-| `atol`        | float                                         | Absolute tolerance to signal convergence for early-stopping. See notes.                        | `None`          |
-| `rtol`        | float                                         | Relative tolerance to signal convergence for early-stopping. See notes.                        | `1e-2`          |
-| `stop`        | str                                           | Early-stopping criteria to test estimator convergence. See details.                            | `"confidence"`  |
-| `ncv`         | int                                           | Number of Lanczos vectors to allocate. Must be at least 2.                                     | `2`             |
-| `orth`        | int                                           | Number of additional Lanczos vectors to orthogonalize against. Must be less than `ncv`.        | `0`             |
-| `quad`        | str                                           | Method used to obtain the weights of the Gaussian quadrature. See notes.                       | `'fttr'`        |
-| `confidence`  | float                                         | Confidence level to consider estimator as converged. Only used when `stop` = "confidence".     | `0.95`          |
-| `pdf`         | 'rademacher', 'normal'                        | Choice of zero-centered distribution to sample random vectors from.                            | ` 'rademacher'` |
-| `rng`         | 'splitmix64', 'xoshiro256**', 'pcg64', 'mt64' | Random number generator to use.                                                                | ` 'splitmix64'` |
-| `seed`        | int                                           | Seed to initialize the `rng` entropy source. Set `seed` > -1 for reproducibility.              | `-1`            |
-| `num_threads` | int                                           | Number of threads to use to parallelize the computation. Set to <= 0 to let OpenMP decide.     | `0`             |
-| `plot`        | bool                                          | If true, plots the samples of the trace estimate along with their convergence characteristics. | `False`         |
-| `info`        | bool                                          | If True, returns a dictionary containing all relevant information about the computation.       | `False`         |
-| `kwargs`      | dict                                          | additional key-values to parameterize the chosen function 'fun'.                               | `{}`            |
+| Name          | Type                                          | Description                                                                                    | Default          |
+|---------------|-----------------------------------------------|------------------------------------------------------------------------------------------------|------------------|
+| `A`           | ndarray, sparray, or LinearOperator           | real symmetric operator.                                                                       | _required_       |
+| `fun`         | str or typing.Callable                        | real-valued function defined on the spectrum of `A`.                                           | `"identity"`     |
+| `maxiter`     | int                                           | Maximum number of random vectors to sample for the trace estimate.                             | `10`             |
+| `deg`         | int                                           | Degree of the quadrature approximation. Must be at least 1.                                    | `20`             |
+| `atol`        | float                                         | Absolute tolerance to signal convergence for early-stopping. See notes.                        | `None`           |
+| `rtol`        | float                                         | Relative tolerance to signal convergence for early-stopping. See notes.                        | `1e-2`           |
+| `stop`        | str                                           | Early-stopping criteria to test estimator convergence. See details.                            | `"confidence"`   |
+| `ncv`         | int                                           | Number of Lanczos vectors to allocate. Must be at least 2.                                     | `2`              |
+| `orth`        | int                                           | Number of additional Lanczos vectors to orthogonalize against. Must be less than `ncv`.        | `0`              |
+| `quad`        | str                                           | Method used to obtain the weights of the Gaussian quadrature. See notes.                       | `'golub_welsch'` |
+| `confidence`  | float                                         | Confidence level to consider estimator as converged. Only used when `stop` = "confidence".     | `0.95`           |
+| `pdf`         | 'rademacher', 'normal'                        | Choice of zero-centered distribution to sample random vectors from.                            | ` 'rademacher'`  |
+| `rng`         | 'splitmix64', 'xoshiro256**', 'pcg64', 'mt64' | Random number generator to use.                                                                | ` 'splitmix64'`  |
+| `seed`        | int                                           | Seed to initialize the `rng` entropy source. Set `seed` > -1 for reproducibility.              | `-1`             |
+| `num_threads` | int                                           | Number of threads to use to parallelize the computation. Set to <= 0 to let OpenMP decide.     | `0`              |
+| `plot`        | bool                                          | If true, plots the samples of the trace estimate along with their convergence characteristics. | `False`          |
+| `info`        | bool                                          | If True, returns a dictionary containing all relevant information about the computation.       | `False`          |
+| `kwargs`      | dict                                          | additional key-values to parameterize the chosen function 'fun'.                               | `{}`             |
 
 ## Returns
 
 | Type                         | Description                                                                                                    |
 |------------------------------|----------------------------------------------------------------------------------------------------------------|
-| typing.Union\[float, tuple\] | Estimate the trace of $f(A)$. If 'info = True', additional information about the computation is also returned. |
+| typing.Union\[float, tuple\] | Estimate the trace of $f(A)$. If `info = True`, additional information about the computation is also returned. |
+
+## Notes:
+
+To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, 
+while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require $O(\mathrm{deg}^2)$ time to execute, 
+the former requires $O(\mathrm{deg}^2)$ space but is highly accurate, while the latter uses only $O(1)$ space at the cost of stability. 
+If `deg` is large, `fttr` is preferred. 
+
+
 
 ## See Also
 
@@ -54,4 +63,5 @@ lanczos : the lanczos tridiagonalization algorithm.
 
 ## Reference
 
-1. Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.
\ No newline at end of file
+1. Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.
+2. Hutchinson, Michael F. "A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines." Communications in Statistics-Simulation and Computation 18.3 (1989): 1059-1076.
\ No newline at end of file
diff --git a/docs/theory/intro.html b/docs/theory/intro.html
index 1c2bd6d..f9738f6 100644
--- a/docs/theory/intro.html
+++ b/docs/theory/intro.html
@@ -84,7 +84,7 @@
 <script src="../site_libs/quarto-search/fuse.min.js"></script>
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
-<link href="../theory/lanczos.html" rel="next">
+<link href="../theory/matrix_functions.html" rel="next">
 <link href="../imate_compare.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
@@ -251,14 +251,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -404,7 +404,7 @@ <h1 class="title">Introduction to trace estimation with <code>primate</code></h1
 </span></p>
 <p>In this introduction, the basics of randomized trace estimation are introduced, with a focus on how to use matrix-free algorithms estimate traces of <em>matrix functions</em>.</p>
 <!-- One such method, often used in trace estimation, is the "stochastic Lanczos quadrature" (SLQ) method. Unfortunately, SLQ refer not to one but a host of methods in the literature; though each is typically related, they are often used in different contexts.  -->
-<div id="dfc9be7f" class="cell" data-execution_count="1">
+<div id="654b7c6d" class="cell" data-execution_count="1">
 <div class="cell-output cell-output-display">
 <script type="application/javascript">
 (function(root) {
@@ -693,22 +693,22 @@ <h2 class="anchored" data-anchor-id="basics-trace-computation">Basics: Trace com
 <p>Suppose we wanted to compute the trace of some matrix <span class="math inline">A</span>. By the very definition of the trace, we have a variety of means to do it: <span class="math display">\mathrm{tr}(A) \triangleq \sum\limits_{i=1}^n A_{ii} = \sum\limits_{i=1}^n \lambda_i = \sum\limits_{i=1}^n e_i^T A e_i </span></p>
 <p>where, in the right-most sum, we’re using <span class="math inline">e_i</span> to represent the <span class="math inline">i</span>th <a href="https://en.wikipedia.org/wiki/Indicator_vector">identity vector</a>: <span class="math display"> A_{ii} = e_i^T A e_i, \quad e_i = [0, \dots, 0, \underbrace{1}_{i}, 0, \dots, 0] </span></p>
 <p>Let’s see some code. First, we start with a simple (random) positive-definite matrix <span class="math inline">A \in \mathbb{R}^{n \times n}</span>.</p>
-<div id="b0925e4a" class="cell" data-execution_count="2">
+<div id="1b0165f3" class="cell" data-execution_count="2">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.random <span class="im">import</span> symmetric</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>A <span class="op">=</span> symmetric(<span class="dv">150</span>, pd <span class="op">=</span> <span class="va">True</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>By default, <code>symmetric</code> normalizes <span class="math inline">A</span> such that the eigenvalues are uniformly distributed in the interval <span class="math inline">[0, 1]</span>. For reference, the spectrum of <span class="math inline">A</span> looks as follows:</p>
-<div id="bd4c3a24" class="cell" data-execution_count="3">
+<div id="9efee939" class="cell" data-execution_count="3">
 <div class="cell-output cell-output-display">
 
-  <div id="c8dcce5f-0f17-4904-8000-ae4a3c8c5b22" data-root-id="p1001" style="display: contents;"></div>
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+  const render_items = [{"docid":"c1fc4bcf-44c0-471a-b144-c522ee334407","roots":{"p1001":"f0743771-b798-4e69-b7d7-9e42be68a71d"},"root_ids":["p1001"]}];
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@@ -736,7 +736,7 @@ <h2 class="anchored" data-anchor-id="basics-trace-computation">Basics: Trace com
 <!-- 1. Sum the diagonal entries (or use numpy's built-in `.trace()`)
   2. Sum the eigenvalues
   3. Sum $\mathrm{e}_i^T A \mathrm{e}_i$ for all $i \in [1, n]$ --></p>
-<div id="7af142b7" class="cell" data-execution_count="4">
+<div id="992ee3e9" class="cell" data-execution_count="4">
 <div class="sourceCode cell-code" id="cb2"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
 <span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a>eye <span class="op">=</span> <span class="kw">lambda</span> i: np.ravel(np.eye(<span class="dv">1</span>, <span class="dv">150</span>, k<span class="op">=</span>i))</span>
 <span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Direct: </span><span class="sc">{</span>np<span class="sc">.</span><span class="bu">sum</span>(A.diagonal())<span class="sc">:.8f}</span><span class="ss">"</span>)</span>
@@ -761,23 +761,23 @@ <h2 class="anchored" data-anchor-id="implicit-trace-estimation"><em>Implicit</em
 <p><span class="math display">\mathtt{tr}(A) \approx \frac{1}{n_v}\sum\limits_{i=1}^{n_v} v_i^\top A v_i </span></p>
 <p>It was shown more formally later by <a href="https://www.tandfonline.com/doi/abs/10.1080/03610918908812806">M.F. Huchinson</a> that if these random vectors <span class="math inline">v \in \mathbb{R}^n</span> satisfy <span class="math inline">\mathbb{E}[vv^T] = I</span> then this approach indeed forms an unbiased estimator of <span class="math inline">\mathrm{tr}(A)</span>. To see this, its sufficient to combine the linearity of expectation, the cyclic-property of the trace function, and the aforementioned <a href="https://en.wikipedia.org/wiki/Isotropic_position">isotropy conditions</a>: <span class="math display">\mathtt{tr}(A) = \mathtt{tr}(A I) = \mathtt{tr}(A \mathbb{E}[v v^T]) = \mathbb{E}[\mathtt{tr}(Avv^T)] = \mathbb{E}[\mathtt{tr}(v^T A v)] = \mathbb{E}[v^T A v]</span></p>
 <p>Naturally we expect the approximation to gradually improve as more vectors are sampled, converging as <span class="math inline">n_v \to \infty</span>. Let’s see if we can check this: we start by collecting a few sample quadratic forms</p>
-<div id="189f75a1" class="cell" data-execution_count="5">
+<div id="1f7d3071" class="cell" data-execution_count="5">
 <div class="sourceCode cell-code" id="cb4"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> isotropic(n):</span>
 <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>  <span class="cf">yield</span> <span class="cf">from</span> [np.random.choice([<span class="op">-</span><span class="dv">1</span>,<span class="op">+</span><span class="dv">1</span>], size<span class="op">=</span>A.shape[<span class="dv">1</span>]) <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(n)]</span>
 <span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>estimates <span class="op">=</span> np.array([v <span class="op">@</span> A <span class="op">@</span> v <span class="cf">for</span> v <span class="kw">in</span> isotropic(<span class="dv">500</span>)])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 </div>
 <p>Plotting both the estimator formed by averaging the first <span class="math inline">k</span> samples along with its absolute error, we get the following figures for <span class="math inline">k = 50</span> and <span class="math inline">k = 500</span>, respectively:</p>
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@@ -806,18 +806,18 @@ <h2 class="anchored" data-anchor-id="implicit-trace-estimation"><em>Implicit</em
 <section id="estimator-implementation" class="level2">
 <h2 class="anchored" data-anchor-id="estimator-implementation">Estimator Implementation</h2>
 <p>Rather than manually averaging samples, trace estimates can be obtained via the <code>hutch</code> function:</p>
-<div id="b3c4e7cf" class="cell" data-execution_count="7">
+<div id="7542c58f" class="cell" data-execution_count="7">
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.trace <span class="im">import</span> hutch</span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Trace estimate: </span><span class="sc">{</span>hutch(A, maxiter<span class="op">=</span><span class="dv">50</span>)<span class="sc">}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>Trace estimate: 76.32787158595767</code></pre>
+<pre><code>Trace estimate: 76.84235846113064</code></pre>
 </div>
 </div>
 <p>Though its estimation is identical to averaging quadratic forms <span class="math inline">v^T A v</span> of isotropic vectors as above, <code>hutch</code> comes with a few extra features to simplify its usage. In particular, all <span class="math inline">v \mapsto v^T A v</span> evaluations are done in parallel, there are a variety of isotropic distributions and random number generators (RNGs) to choose from, and the estimator may be suppleid various convergence criteria to allow for early-stopping; see the <a href="../reference/trace.hutch.html">hutch documentation page</a> for more details.</p>
 <!-- 
 Let's see how this fares using, let's say, $\frac{1}{4}$ the number of _matvecs_: 
 
-::: {#62474ab4 .cell execution_count=8}
+::: {#df4f20e7 .cell execution_count=8}
 ``` {.python .cell-code}
 n: int = A.shape[0]
 trace_estimate = 0.0
@@ -863,7 +863,7 @@ <h2 class="anchored" data-anchor-id="example-logdet-computation">Example: Logdet
 <p>Putting all of the theory above together, a simple way to estimate the trace of a matrix function is to average Lanczos quadrature estimates:</p>
 <p><span class="math display">\mathrm{tr}(f(A)) \approx \frac{n}{n_v} \sum\limits_{i=1}^{n_v} e_1^T f(T_m) e_1  = \frac{n}{n_v} \sum\limits_{j=1}^{n_v} \left ( \sum\limits_{i=1}^m \tau_i^{(j)} f(\theta_i)^{(j)} \right )</span></p>
 <p>To enable this kind of estimator in <code>hutch</code>, simply set the <code>fun</code> argument to either the name of a built-in function or an arbitrary Callable. For example, to approximate the log-determinant:</p>
-<div id="6ec04cdd" class="cell" data-execution_count="9">
+<div id="9b0d91b4" class="cell" data-execution_count="9">
 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="co">## Log-determinant</span></span>
 <span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>logdet_test <span class="op">=</span> hutch(A, fun<span class="op">=</span><span class="st">"log"</span>, maxiter<span class="op">=</span><span class="dv">25</span>)</span>
 <span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a>logdet_true <span class="op">=</span> np.<span class="bu">sum</span>(np.log(np.linalg.eigvalsh(A)))</span>
@@ -872,32 +872,32 @@ <h2 class="anchored" data-anchor-id="example-logdet-computation">Example: Logdet
 <span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Logdet (approx): </span><span class="sc">{</span>logdet_test<span class="sc">}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>Logdet (exact):  -148.3218440234385
-Logdet (approx): -148.71526016046562</code></pre>
+Logdet (approx): -148.15107274186403</code></pre>
 </div>
 </div>
 <p>Here we see that using <span class="math inline">n / 6</span> evaluatons of <span class="math inline">e_1^T f(T_m) e_1</span>, we get a decent approximation of <code>logdet(A)</code>. To get a better idea of the quality of this estimate, you can set <code>verbose=True</code>:</p>
-<div id="84c98af3" class="cell" data-execution_count="10">
+<div id="a87954c9" class="cell" data-execution_count="10">
 <div class="sourceCode cell-code" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>est <span class="op">=</span> hutch(A, fun<span class="op">=</span><span class="st">"log"</span>, maxiter<span class="op">=</span><span class="dv">25</span>, seed<span class="op">=</span><span class="dv">5</span>, verbose<span class="op">=</span><span class="va">True</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
-<pre><code>Girard-Hutchinson estimator (fun=log, deg=20, quad=fttr)
-Est: -143.207 +/- 5.40 (95% CI), CV: 2%, Evals: 25 [R] (seed: 5)</code></pre>
+<pre><code>Girard-Hutchinson estimator (fun=log, deg=20, quad=golub_welsch)
+Est: -144.400 +/- 6.05 (95% CI), CV: 2%, Evals: 25 [R] (seed: 5)</code></pre>
 </div>
 </div>
 <p>The first line of the statement contains fixed about the estimator, including the type of estimator (<code>Girard-Hutchinson</code>), the function applied to the spectrum (<code>log</code>), the degree of the Lanczos quadrature approximation (<code>20</code>), and the quadrature method used (<code>fttr</code>).</p>
 <p>The second line prints the runtime information about the samples, such as the final trace estimate, its <a href="https://en.wikipedia.org/wiki/Margin_of_error">margin of error</a> and <a href="https://en.wikipedia.org/wiki/Coefficient_of_variation">coefficient of variation</a> (aka the relative std. deviation), and the number of samples used + their distribution prefix (‘R’ for rademacher).</p>
 <p>Another way to get an idea of how well the estimator is converging is to pass the <code>plot=True</code> option. Here, we pass the same <code>seed</code> for reproducibility with the above call.</p>
-<div id="810e1204" class="cell" data-execution_count="11">
+<div id="1a49a953" class="cell" data-execution_count="11">
 <div class="sourceCode cell-code" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>est <span class="op">=</span> hutch(A, fun<span class="op">=</span><span class="st">"log"</span>, maxiter<span class="op">=</span><span class="dv">100</span>, seed<span class="op">=</span><span class="dv">5</span>, plot<span class="op">=</span><span class="va">True</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
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@@ -1485,8 +1485,8 @@ <h2 class="anchored" data-anchor-id="example-logdet-computation">Example: Logdet
       </a>          
   </div>
   <div class="nav-page nav-page-next">
-      <a href="../theory/lanczos.html" class="pagination-link" aria-label="The Lanczos Method">
-        <span class="nav-page-text">The Lanczos Method</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../theory/matrix_functions.html" class="pagination-link" aria-label="Matrix functions">
+        <span class="nav-page-text">Matrix functions</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/theory/lanczos.html b/docs/theory/lanczos.html
index b677252..7bf19df 100644
--- a/docs/theory/lanczos.html
+++ b/docs/theory/lanczos.html
@@ -30,8 +30,8 @@
 <script src="../site_libs/quarto-search/fuse.min.js"></script>
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
-<link href="../theory/matrix_functions.html" rel="next">
-<link href="../theory/intro.html" rel="prev">
+<link href="../advanced/cpp_integration.html" rel="next">
+<link href="../theory/matrix_functions.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
@@ -194,14 +194,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link active">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link active">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -1042,13 +1042,13 @@ <h2 class="anchored" data-anchor-id="beating-the-complexity-bounds">Beating the
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../theory/intro.html" class="pagination-link  aria-label=" introduction"="">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Introduction</span>
+      <a href="../theory/matrix_functions.html" class="pagination-link  aria-label=" matrix="" functions"="">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Matrix functions</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
-      <a href="../theory/matrix_functions.html" class="pagination-link" aria-label="Matrix functions">
-        <span class="nav-page-text">Matrix functions</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../advanced/cpp_integration.html" class="pagination-link" aria-label="Usage from C++">
+        <span class="nav-page-text">Usage from C++</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/theory/matrix_functions.html b/docs/theory/matrix_functions.html
index 94a2d72..07fd46e 100644
--- a/docs/theory/matrix_functions.html
+++ b/docs/theory/matrix_functions.html
@@ -84,8 +84,8 @@
 <script src="../site_libs/quarto-search/fuse.min.js"></script>
 <script src="../site_libs/quarto-search/quarto-search.js"></script>
 <meta name="quarto:offset" content="../">
-<link href="../advanced/cpp_integration.html" rel="next">
-<link href="../theory/lanczos.html" rel="prev">
+<link href="../theory/lanczos.html" rel="next">
+<link href="../theory/intro.html" rel="prev">
 <script src="../site_libs/quarto-html/quarto.js"></script>
 <script src="../site_libs/quarto-html/popper.min.js"></script>
 <script src="../site_libs/quarto-html/tippy.umd.min.js"></script>
@@ -251,14 +251,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link active">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link active">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -389,7 +389,7 @@ <h1 class="title">Matrix function estimation with <code>primate</code></h1>
 </header>
 
 
-<div id="ca2897fb" class="cell" data-execution_count="1">
+<div id="e899bc6c" class="cell" data-execution_count="1">
 <div class="cell-output cell-output-display">
 <script type="application/javascript">
 (function(root) {
@@ -700,7 +700,7 @@ <h2 class="anchored" data-anchor-id="establishing-a-baseline">Establishing a bas
 <li>Multiply <span class="math inline">v</span> by <span class="math inline">f(A)</span> induced by <span class="math inline">f(\lambda) = \lambda + \epsilon</span></li>
 </ol>
 <p>Lets see what the code to accomplish this using (3) looks like:</p>
-<div id="ee45b900" class="cell" data-execution_count="2">
+<div id="bfb75a66" class="cell" data-execution_count="2">
 <div class="sourceCode cell-code" id="cb1"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.random <span class="im">import</span> symmetric</span>
 <span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.operator <span class="im">import</span> matrix_function</span>
 <span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -729,7 +729,7 @@ <h2 class="anchored" data-anchor-id="establishing-a-baseline">Establishing a bas
 </div>
 </div>
 <p>Observe <span class="math inline">M</span> matches the ground truth <span class="math inline">v \mapsto (A + \epsilon I)v</span>. In this way, one benefit of using <code>matrix_function</code> is that it allows one to approximate <span class="math inline">f(A)</span> by thinking only about what is happening at the spectral level (as opposed to the matrix level). Of course, this example is a bit non-convincing as there are simpler ways of achieving the same result as shown by (1) and (2), for example:</p>
-<div id="9b23852d" class="cell" data-execution_count="3">
+<div id="7deb6c3f" class="cell" data-execution_count="3">
 <div class="sourceCode cell-code" id="cb3"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>np.allclose(A <span class="op">@</span> v <span class="op">+</span> <span class="fl">0.10</span> <span class="op">*</span> v, v_truth)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-display" data-execution_count="3">
 <pre><code>True</code></pre>
@@ -742,7 +742,7 @@ <h2 class="anchored" data-anchor-id="when-v-mapsto-fav-is-not-known">When <span
 <p>On the other hand, there are plenty of situations where one <em>doesn’t have access to these simpler expressions</em>. For example, consider the map:</p>
 <p><span class="math display">v \mapsto (A + \epsilon I)^{-1} v</span></p>
 <p>Such expressions pop up in a variety of settings, such as in <a href="https://en.wikipedia.org/wiki/Ridge_regression">Tikhonov regularization</a>, in Schatten-norm estimation <span class="citation" data-cites="ubaru2017fast">Ubaru, Saad, and Seghouane (<a href="#ref-ubaru2017fast" role="doc-biblioref">2017</a>)</span>, in the <a href="https://scikit-sparse.readthedocs.io/en/latest/cholmod.html#sksparse.cholmod.Factor.cholesky_AAt_inplace">Cholesky factorization of PSD matrices</a>, and so on. Unlike the previous setting, we cannot readily access <span class="math inline">v \mapsto f(A)v</span> unless we explicitly compute the full spectral decomposition of <span class="math inline">A</span> or the inverse of <span class="math inline">A</span>, both of which are expensive to obtain.</p>
-<div id="ad6f392e" class="cell" data-execution_count="4">
+<div id="eff05621" class="cell" data-execution_count="4">
 <div class="sourceCode cell-code" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="co">## Alternative: v_truth = np.linalg.inv((A + 0.10 * np.eye(A.shape[0]))) @ v</span></span>
 <span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>v_truth <span class="op">=</span> (U <span class="op">@</span> np.diag(np.reciprocal(Lambda <span class="op">+</span> <span class="fl">0.10</span>)) <span class="op">@</span> U.T) <span class="op">@</span> v</span>
 <span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a></span>
@@ -767,7 +767,7 @@ <h2 class="anchored" data-anchor-id="when-v-mapsto-fav-is-not-known">When <span
 <section id="back-to-trace-estimation" class="level2">
 <h2 class="anchored" data-anchor-id="back-to-trace-estimation">Back to trace estimation</h2>
 <p>There are several use-cases wherein one might be interested in the output <span class="math inline">f(A)v</span> itself—see <span class="citation" data-cites="musco2018stability">Musco, Musco, and Sidford (<a href="#ref-musco2018stability" role="doc-biblioref">2018</a>)</span> and references within for some examples. One use-case is the extension of existing <code>matvec</code>-dependent implicit trace estimators to matrix-function trace estimators.</p>
-<div id="ebae49f4" class="cell" data-execution_count="5">
+<div id="8c78498e" class="cell" data-execution_count="5">
 <div class="sourceCode cell-code" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> primate.trace <span class="im">import</span> hutch, xtrace</span>
 <span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a>M <span class="op">=</span> matrix_function(A, fun<span class="op">=</span><span class="st">"log"</span>)</span>
 <span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Logdet exact:  </span><span class="sc">{</span>np<span class="sc">.</span><span class="bu">sum</span>(np.log(np.linalg.eigvalsh(A)))<span class="sc">:6e}</span><span class="ss">"</span>)</span>
@@ -775,7 +775,7 @@ <h2 class="anchored" data-anchor-id="back-to-trace-estimation">Back to trace est
 <span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Logdet XTrace: </span><span class="sc">{</span>xtrace(M)<span class="sc">:6e}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
 <div class="cell-output cell-output-stdout">
 <pre><code>Logdet exact:  -1.483218e+02
-Logdet Hutch:  -1.477567e+02
+Logdet Hutch:  -1.475913e+02
 Logdet XTrace: -1.483155e+02</code></pre>
 </div>
 </div>
@@ -1336,13 +1336,13 @@ <h2 class="anchored" data-anchor-id="back-to-trace-estimation">Back to trace est
 </script>
 <nav class="page-navigation">
   <div class="nav-page nav-page-previous">
-      <a href="../theory/lanczos.html" class="pagination-link  aria-label=" the="" lanczos="" method"="">
-        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">The Lanczos Method</span>
+      <a href="../theory/intro.html" class="pagination-link  aria-label=" introduction"="">
+        <i class="bi bi-arrow-left-short"></i> <span class="nav-page-text">Introduction</span>
       </a>          
   </div>
   <div class="nav-page nav-page-next">
-      <a href="../advanced/cpp_integration.html" class="pagination-link" aria-label="Usage from C++">
-        <span class="nav-page-text">Usage from C++</span> <i class="bi bi-arrow-right-short"></i>
+      <a href="../theory/lanczos.html" class="pagination-link" aria-label="The Lanczos Method">
+        <span class="nav-page-text">The Lanczos Method</span> <i class="bi bi-arrow-right-short"></i>
       </a>
   </div>
 </nav>
diff --git a/docs/theory/slq_pseudo.html b/docs/theory/slq_pseudo.html
index dcef9a2..2cb5eed 100644
--- a/docs/theory/slq_pseudo.html
+++ b/docs/theory/slq_pseudo.html
@@ -194,14 +194,14 @@
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">The Lanczos Method</span></a>
+  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">Matrix functions</span></a>
   </div>
 </li>
           <li class="sidebar-item">
   <div class="sidebar-item-container"> 
-  <a href="../theory/matrix_functions.html" class="sidebar-item-text sidebar-link">
- <span class="menu-text">Matrix functions</span></a>
+  <a href="../theory/lanczos.html" class="sidebar-item-text sidebar-link">
+ <span class="menu-text">The Lanczos Method</span></a>
   </div>
 </li>
       </ul>
@@ -324,7 +324,7 @@ <h1 class="title">SLQ Trace guide</h1>
 
 
 <p>To clarify that that means, here’s an abstract presentation of the generic SLQ procedure:</p>
-<div id="alg-slq" class="pseudocode-container" data-comment-delimiter="//" data-line-number="true" data-alg-title="Algorithm" data-no-end="true" data-indent-size="1.1em" data-line-number-punc=":" data-pseudocode-index="1">
+<div id="alg-slq" class="pseudocode-container" data-line-number-punc=":" data-line-number="true" data-pseudocode-index="1" data-no-end="true" data-alg-title="Algorithm" data-comment-delimiter="//" data-indent-size="1.1em">
 <div class="pseudocode">
 \begin{algorithm} \caption{Stochastic Lanczos Quadrature} \begin{algorithmic} \Input Symmetric operator ($A \in \mathbb{R}^{n \times n}$) \Require Number of queries ($n_v$), Degree of quadrature ($k$) \Function{SLQ}{$A$, $n_v$, $k$} \State $\Gamma \gets 0$ \For{$j = 1, 2, \dots, n_v$} \State $v_i \sim \mathcal{D}$ where $\mathcal{D}$ satisfies $\mathbb{E}(v v^\top) = I$ \State $T^{(j)}(\alpha, \beta)$ $\gets$ $\mathrm{Lanczos}(A,v_j,k+1)$ \State $[\Theta, Y] \gets \mathrm{eigh\_tridiag}(T^{(j)}(\alpha, \beta))$ \State $\tau_i \gets \langle e_1, y_i \rangle$ \State &lt; Do something with the node/weight pairs $(\theta_i, \tau_i^2)$ &gt; \EndFor \EndFunction \end{algorithmic} \end{algorithm}
 </div>
diff --git a/pyproject.toml b/pyproject.toml
index 3bb8a91..f918f1e 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -4,7 +4,7 @@ requires = ['meson-python', 'wheel', 'ninja', 'pybind11', 'numpy'] # 'pythran-op
 
 [project]
 name = "scikit-primate"
-version = "0.3.2"
+version = "0.3.3"
 readme = "README.md"
 classifiers = [
 	"Intended Audience :: Science/Research",
diff --git a/src/primate/diagonalize.py b/src/primate/diagonalize.py
index 33854da..08e7b36 100644
--- a/src/primate/diagonalize.py
+++ b/src/primate/diagonalize.py
@@ -24,17 +24,17 @@ def lanczos(
 	
 	Notes
 	-----
-	The Lanczos method iteratively builds a tridiagonal matrix `T` of a symmetric `A` via an orthogonal change-of-basis `Q`:
+	The Lanczos method builds a tridiagonal `T` from a symmetric `A` via an orthogonal change-of-basis `Q`:
 	$$ Q^T A Q  = T $$
 	Unlike other Lanczos implementations (e.g. SciPy's `eigsh`), which includes e.g. sophisticated restarting, 
 	deflation, and selective-reorthogonalization steps, this method simply executes `deg` steps of the Lanczos method with 
-	the supplied `v0` and returns the resulting tridiagonal matrix.
+	the supplied `v0` and returns the resulting tridiagonal matrix `T`.
 	
-	Diagonalizing `T` via e.g. `scipy.linalg.eigh_tridiagonal` yields Rayleigh-Ritz approximations of the eigenvalues of `A`,
-	though note no checking is performed for 'ghost' or already converged eigenvalues. To increase the accuracy of these eigenvalue 
-	approximation, try increasing `orth` and `deg`. Supplying either negative values or values larger than `deg` for `orth` will 
-	result in full re-orthogonalization, though note the number of matvecs scales linearly with `deg` and the number of inner-products 
-	scales quadratically with `orth`.
+	Rayleigh-Ritz approximations of the eigenvalues of `A` can be further obtained by diagonalizing `T` via any 
+	symmetric tridiagonal eigenvalue solver, `scipy.linalg.eigh_tridiagonal` though note unlike `eigsh` no checking is performed 
+	for 'ghost' or already converged eigenvalues. To increase the accuracy of these eigenvalue approximation, try increasing `orth` 
+	and `deg`. Supplying either negative values or values larger than `deg` for `orth` will result in full re-orthogonalization, 
+	though note the number of matvecs scales linearly with `deg` and the number of inner-products scales quadratically with `orth`.
 
 	Parameters
 	----------
diff --git a/src/primate/operator.py b/src/primate/operator.py
index 9182d99..7972702 100644
--- a/src/primate/operator.py
+++ b/src/primate/operator.py
@@ -17,7 +17,7 @@ def matrix_function(
 ) -> LinearOperator:
 	"""Constructs a `LinearOperator` approximating the action of the matrix function `fun(A)`. 
 
-	This function uses the Lanczos quadrature method to approximate the action of matrix function:
+	This function uses the Lanczos method to approximate the action of matrix function:
 	$$ v \\mapsto f(A)v, \\quad f(A) = U f(\\lambda) U^T $$
 	The resulting operator may be used in conjunction with other estimation methods, such as `hutch` or `xtrace`.
 
@@ -33,18 +33,18 @@ def matrix_function(
 	:::
 
 	Parameters
-	-----------
-		A : ndarray, sparray, or LinearOperator
-				real symmetric operator.
-		fun : str or Callable, default="identity"
-				real-valued function defined on the spectrum of `A`.
-		deg : int, default=20
-				Degree of the Krylov expansion.
-		rtol : float, default=1e-8
-				Relative tolerance to consider two Lanczos vectors are numerically orthogonal.
-		orth: int, default=0
-				Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis.
-		kwargs : dict, optional
+	----------
+	A : LinearOperator or ndarray or sparray
+			real symmetric operator.
+	fun : str or Callable, default="identity"
+			real-valued function defined on the spectrum of `A`.
+	deg : int, default=20
+			Degree of the Krylov expansion.
+	rtol : float, default=1e-8
+			Relative tolerance to consider two Lanczos vectors are numerically orthogonal.
+	orth: int, default=0
+			Number of additional Lanczos vectors to orthogonalize against when building the Krylov basis.
+	kwargs : dict, optional
 			additional key-values to parameterize the chosen function 'fun'.
 
 	Returns
diff --git a/src/primate/trace.py b/src/primate/trace.py
index e7f940e..7d93d29 100644
--- a/src/primate/trace.py
+++ b/src/primate/trace.py
@@ -21,7 +21,7 @@ def hutch(
 	stop: str = ["confidence", "change"],
 	ncv: int = 2,
 	orth: int = 0,
-	quad: str = "fttr",
+	quad: str = "golub_welsch",
 	confidence: float = 0.95,
 	pdf: str = "rademacher",
 	rng: str = "pcg64",
@@ -89,7 +89,14 @@ def hutch(
 	Returns
 	-------
 	:
-			Estimate the trace of $f(A)$. If 'info = True', additional information about the computation is also returned.
+			Estimate the trace of $f(A)$. If `info = True`, additional information about the computation is also returned.
+
+	Notes:
+	------
+	To compute the weights of the quadrature, the GW computation uses implicit symmetric QR steps with Wilkinson shifts, 
+	while the FTTR algorithm uses the explicit expression for orthogonal polynomials. While both require $O(\\mathrm{deg}^2)$ time to execute, 
+	the former requires $O(\\mathrm{deg}^2)$ space but is highly accurate, while the latter uses only $O(1)$ space at the cost of stability. 
+	If `deg` is large, `fttr` is preferred. 
 
 	See Also
 	--------
@@ -98,6 +105,7 @@ def hutch(
 	Reference
 	---------
 	1. Ubaru, S., Chen, J., & Saad, Y. (2017). Fast estimation of tr(f(A)) via stochastic Lanczos quadrature. SIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.
+	2. Hutchinson, Michael F. "A stochastic estimator of the trace of the influence matrix for Laplacian smoothing splines." Communications in Statistics-Simulation and Computation 18.3 (1989): 1059-1076.
 	"""
 	attr_checks = [hasattr(A, "__matmul__"), hasattr(A, "matmul"), hasattr(A, "dot"), hasattr(A, "matvec")]
 	assert any(attr_checks), "Invalid operator; must have an overloaded 'matvec' or 'matmul' method"