FIX: Update Numba Lecture to Address Deprecation of @jit (#296)

* update a section on type inference.

* update lecture to avoid literal box warning

* check the type of the function

* Update lectures/numba.md

Co-authored-by: mmcky <[email protected]>

* reduce redundancy

* further simplifies descriptions

* fix typos

---------

Co-authored-by: mmcky <[email protected]>

lectures/numba.md

@@ -3,8 +3,10 @@ jupytext:
   text_representation:
     extension: .md
     format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.4
 kernelspec:
-  display_name: Python 3
+  display_name: Python 3 (ipykernel)
   language: python
   name: python3
 ---
@@ -26,10 +28,9 @@ kernelspec:
 
 In addition to what's in Anaconda, this lecture will need the following libraries:
 
-```{code-cell} ipython
----
-tags: [hide-output]
----
+```{code-cell} ipython3
+:tags: [hide-output]
+
 !pip install quantecon
 ```
 
@@ -38,7 +39,7 @@ versions are a {doc}`common source of errors <troubleshooting>`.
 
 Let's start with some imports:
 
-```{code-cell} ipython
+```{code-cell} ipython3
 %matplotlib inline
 import numpy as np
 import quantecon as qe
@@ -98,13 +99,13 @@ $$
 
 In what follows we set
 
-```{code-cell} python3
+```{code-cell} ipython3
 α = 4.0
 ```
 
 Here's the plot of a typical trajectory, starting from $x_0 = 0.1$, with $t$ on the x-axis
 
-```{code-cell} python3
+```{code-cell} ipython3
 def qm(x0, n):
     x = np.empty(n+1)
     x[0] = x0
@@ -122,10 +123,10 @@ plt.show()
 
 To speed the function `qm` up using Numba, our first step is
 
-```{code-cell} python3
-from numba import jit
+```{code-cell} ipython3
+from numba import njit
 
-qm_numba = jit(qm)
+qm_numba = njit(qm)
 ```
 
 The function `qm_numba` is a version of `qm` that is "targeted" for
@@ -135,7 +136,7 @@ We will explain what this means momentarily.
 
 Let's time and compare identical function calls across these two versions, starting with the original function `qm`:
 
-```{code-cell} python3
+```{code-cell} ipython3
 n = 10_000_000
 
 qe.tic()
@@ -145,7 +146,7 @@ time1 = qe.toc()
 
 Now let's try qm_numba
 
-```{code-cell} python3
+```{code-cell} ipython3
 qe.tic()
 qm_numba(0.1, int(n))
 time2 = qe.toc()
@@ -156,13 +157,14 @@ This is already a massive speed gain.
 In fact, the next time and all subsequent times it runs even faster as the function has been compiled and is in memory:
 
 (qm_numba_result)=
-```{code-cell} python3
+
+```{code-cell} ipython3
 qe.tic()
 qm_numba(0.1, int(n))
 time3 = qe.toc()
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 time1 / time3  # Calculate speed gain
 ```
 
@@ -194,12 +196,12 @@ Note that, if you make the call `qm(0.5, 10)` and then follow it with `qm(0.9, 2
 
 The compiled code is then cached and recycled as required.
 
-## Decorators and "nopython" Mode
+## Decorator Notation
 
 In the code above we created a JIT compiled version of `qm` via the call
 
-```{code-cell} python3
-qm_numba = jit(qm)
+```{code-cell} ipython3
+qm_numba = njit(qm)
 ```
 
 In practice this would typically be done using an alternative *decorator* syntax.
@@ -208,14 +210,12 @@ In practice this would typically be done using an alternative *decorator* syntax
 
 Let's see how this is done.
 
-### Decorator Notation
-
-To target a function for JIT compilation we can put `@jit` before the function definition.
+To target a function for JIT compilation we can put `@njit` before the function definition.
 
 Here's what this looks like for `qm`
 
-```{code-cell} python3
-@jit
+```{code-cell} ipython3
+@njit
 def qm(x0, n):
     x = np.empty(n+1)
     x[0] = x0
@@ -224,15 +224,21 @@ def qm(x0, n):
     return x
 ```
 
-This is equivalent to `qm = jit(qm)`.
+This is equivalent to `qm = njit(qm)`.
 
 The following now uses the jitted version:
 
-```{code-cell} python3
-qm(0.1, 10)
+```{code-cell} ipython3
+%%time 
+
+qm(0.1, 100_000)
 ```
 
-### Type Inference and "nopython" Mode
+Numba provides several arguments for decorators to accelerate computation and cache functions [here](https://numba.readthedocs.io/en/stable/user/performance-tips.html).
+
+In the [following lecture on parallelization](parallel), we will discuss how to use the `parallel` argument to achieve automatic parallelization.
+
+## Type Inference
 
 Clearly type inference is a key part of JIT compilation.
 
@@ -246,29 +252,83 @@ This allows it to generate native machine code, without having to call the Pytho
 
 In such a setting, Numba will be on par with machine code from low-level languages.
 
-When Numba cannot infer all type information, some Python objects are given generic object status and execution falls back to the Python runtime.
+When Numba cannot infer all type information, it will raise an error.
 
-When this happens, Numba provides only minor speed gains or none at all.
+For example, in the case below, Numba is unable to determine the type of function `mean` when compiling the function `bootstrap`
 
-We generally prefer to force an error when this occurs, so we know effective
-compilation is failing.
+```{code-cell} ipython3
+@njit
+def bootstrap(data, statistics, n):
+    bootstrap_stat = np.empty(n)
+    n = len(data)
+    for i in range(n_resamples):
+        resample = np.random.choice(data, size=n, replace=True)
+        bootstrap_stat[i] = statistics(resample)
+    return bootstrap_stat
 
-This is done by using either `@jit(nopython=True)` or, equivalently, `@njit` instead of `@jit`.
+def mean(data):
+    return np.mean(data)
 
-For example,
+data = np.array([2.3, 3.1, 4.3, 5.9, 2.1, 3.8, 2.2])
+n_resamples = 10
 
-```{code-cell} python3
-from numba import njit
+print('Type of function:', type(mean))
+
+#Error
+try:
+    bootstrap(data, mean, n_resamples)
+except Exception as e:
+    print(e)
+```
 
+But Numba recognizes JIT-compiled functions
+
+```{code-cell} ipython3
 @njit
-def qm(x0, n):
-    x = np.empty(n+1)
-    x[0] = x0
-    for t in range(n):
-        x[t+1] = 4 * x[t] * (1 - x[t])
-    return x
+def mean(data):
+    return np.mean(data)
+
+print('Type of function:', type(mean))
+
+%time bootstrap(data, mean, n_resamples)
+```
+
+We can check the signature of the JIT-compiled function
+
+```{code-cell} ipython3
+bootstrap.signatures
+```
+
+The function `bootstrap` takes one `float64` floating point array, one function called `mean` and an `int64` integer.
+
+Now let's see what happens when we change the inputs.
+
+Running it again with a larger integer for `n` and a different set of data does not change the signature of the function.
+
+```{code-cell} ipython3
+data = np.array([4.1, 1.1, 2.3, 1.9, 0.1, 2.8, 1.2])
+%time bootstrap(data, mean, 100)
+bootstrap.signatures
 ```
 
+As expected, the second run is much faster.
+
+Let's try to change the data again and use an integer array as data
+
+```{code-cell} ipython3
+data = np.array([1, 2, 3, 4, 5], dtype=np.int64)
+%time bootstrap(data, mean, 100)
+bootstrap.signatures
+```
+
+Note that a second signature is added.
+
+It also takes longer to run, suggesting that Numba recompiles this function as the type changes.
+
+Overall, type inference helps Numba to achieve its performance, but it also limits what Numba supports and sometimes requires careful type checks.
+
+You can refer to the list of supported Python and Numpy features [here](https://numba.pydata.org/numba-doc/dev/reference/pysupported.html).
+
 ## Compiling Classes
 
 As mentioned above, at present Numba can only compile a subset of Python.
@@ -285,7 +345,7 @@ created in {doc}`this lecture <python_oop>`.
 
 To compile this class we use the `@jitclass` decorator:
 
-```{code-cell} python3
+```{code-cell} ipython3
 from numba import float64
 from numba.experimental import jitclass
 ```
@@ -294,11 +354,11 @@ Notice that we also imported something called `float64`.
 
 This is a data type representing standard floating point numbers.
 
-We are importing it here because Numba needs a bit of extra help with types when it trys to deal with classes.
+We are importing it here because Numba needs a bit of extra help with types when it tries to deal with classes.
 
 Here's our code:
 
-```{code-cell} python3
+```{code-cell} ipython3
 solow_data = [
     ('n', float64),
     ('s', float64),
@@ -361,7 +421,7 @@ After that, targeting the class for JIT compilation only requires adding
 
 When we call the methods in the class, the methods are compiled just like functions.
 
-```{code-cell} python3
+```{code-cell} ipython3
 s1 = Solow()
 s2 = Solow(k=8.0)
 
@@ -444,25 +504,25 @@ For larger ones, or for routines using external libraries, it can easily fail.
 
 Hence, it's prudent when using Numba to focus on speeding up small, time-critical snippets of code.
 
-This will give you much better performance than blanketing your Python programs with `@jit` statements.
+This will give you much better performance than blanketing your Python programs with `@njit` statements.
 
 ### A Gotcha: Global Variables
 
 Here's another thing to be careful about when using Numba.
 
 Consider the following example
 
-```{code-cell} python3
+```{code-cell} ipython3
 a = 1
 
-@jit
+@njit
 def add_a(x):
     return a + x
 
 print(add_a(10))
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 a = 2
 
 print(add_a(10))
@@ -492,7 +552,7 @@ Compare speed with and without Numba when the sample size is large.
 
 Here is one solution:
 
-```{code-cell} python3
+```{code-cell} ipython3
 from random import uniform
 
 @njit
@@ -581,13 +641,13 @@ We let
 - 0 represent "low"
 - 1 represent "high"
 
-```{code-cell} python3
+```{code-cell} ipython3
 p, q = 0.1, 0.2  # Prob of leaving low and high state respectively
 ```
 
 Here's a pure Python version of the function
 
-```{code-cell} python3
+```{code-cell} ipython3
 def compute_series(n):
     x = np.empty(n, dtype=np.int_)
     x[0] = 1  # Start in state 1
@@ -604,7 +664,7 @@ def compute_series(n):
 Let's run this code and check that the fraction of time spent in the low
 state is about 0.666
 
-```{code-cell} python3
+```{code-cell} ipython3
 n = 1_000_000
 x = compute_series(n)
 print(np.mean(x == 0))  # Fraction of time x is in state 0
@@ -614,30 +674,28 @@ This is (approximately) the right output.
 
 Now let's time it:
 
-```{code-cell} python3
+```{code-cell} ipython3
 qe.tic()
 compute_series(n)
 qe.toc()
 ```
 
 Next let's implement a Numba version, which is easy
 
-```{code-cell} python3
-from numba import jit
-
-compute_series_numba = jit(compute_series)
+```{code-cell} ipython3
+compute_series_numba = njit(compute_series)
 ```
 
 Let's check we still get the right numbers
 
-```{code-cell} python3
+```{code-cell} ipython3
 x = compute_series_numba(n)
 print(np.mean(x == 0))
 ```
 
 Let's see the time
 
-```{code-cell} python3
+```{code-cell} ipython3
 qe.tic()
 compute_series_numba(n)
 qe.toc()