[publish] Improve precision bit estimation

README.md

@@ -155,7 +155,7 @@ Options:
    -n <int>     number of terms (default: 10)
    -d <int>     number of precision bits (default: 512)
    -q <int>     quadrature order (default: 500)
-   -m <int>     precision multiplier (default: 1.5)
+   -m <float>   precision multiplier (default, minimum: 1.5)
    -nc <int>    controls the maximum exponent (default: 4)
    -e <float>   tolerance (default: 1E-8)
    -k <string>  file name of kernel function (default: exp(-t^2/4))
@@ -164,6 +164,11 @@ Options:
    -h           print this help message
 ```
 
+The minimum required precision can be estimated by the parameter $$n$$.
+The algorithm involves the computation of $$C^{4n}_{2n}$$ and $$2^{4n}$$.
+The number of precision bits shall be at least $$\log_2(C^{4n}_{2n})+4n$$.
+In the implementation, this number will be further multiplied by the parameter $$m$$.
+
 #### Example
 
 The default kernel is `exp(-t^2/4)`. One can run the application with the following command:

src/VPMR.cpp

@@ -308,17 +308,17 @@ int main(const int argc, const char** argv) {
     bool has_digit = false;
     for(auto I = 1; I < argc; ++I) {
         if(const auto token = std::string(argv[I]); token == "-nc")
-            NC = std::stoi(argv[++I]);
+            NC = std::max(1, std::stoi(argv[++I]));
         else if(token == "-n")
-            N = std::stoi(argv[++I]);
+            N = std::max(1, std::stoi(argv[++I]));
         else if(token == "-d") {
-            DIGIT = std::stoi(argv[++I]);
+            DIGIT = std::max(1, std::stoi(argv[++I]));
             has_digit = true;
         }
         else if(token == "-q")
-            QUAD_ORDER = std::stoi(argv[++I]);
+            QUAD_ORDER = std::max(1, std::stoi(argv[++I]));
         else if(token == "-m")
-            SCALE = std::stod(argv[++I]);
+            SCALE = std::max(1.5, std::stod(argv[++I]));
         else if(token == "-e")
             TOL = mpreal(argv[++I]);
         else if(token == "-w")
@@ -411,11 +411,11 @@ int main(const int argc, const char** argv) {
 
 std::tuple<std::vector<std::complex<double>>, std::vector<std::complex<double>>> vpmr_wrapper(
     const int n, const int d, const int q, const double m, const int nc, const double e, const std::string& k) {
-    N = n;
-    DIGIT = d;
-    QUAD_ORDER = q;
-    SCALE = m;
-    NC = nc;
+    N = std::max(1, n);
+    DIGIT = std::max(1, d);
+    QUAD_ORDER = std::max(1, q);
+    SCALE = std::max(1.5, m);
+    NC = std::max(1, nc);
     TOL = mpreal(e);
     if(!k.empty()) KERNEL = k;