Exercise "math"

Objectives

• Learn how to do simple math with C# [C++].
• Learn how to separately compile a function into a library [object code] and then link it to your Main [main] function.

Tasks

1. Calculate using C# System.Math class [C++ <cmath> header] √2, 2^1/5, e^π, π^e.

   Something like

   double sqrt2=System.Math.Sqrt(2.0);
   System.Console.WriteLine($"Sqrt(2) = {sqrt2}");
   

   double sqrt2=std::sqrt(2.0);
   std::cout << "sqrt(2) = " << sqrt2 << std::endl;
   

2. Using the following Stirling approximation for the gamma-function Γ(x),

   using static System.Math;
   public static class sfuns{
   public static double fgamma(double x){ ///single precision gamma function
   	if(x<0)return PI/Sin(PI*x)/fgamma(1-x); // Euler's reflection formula
   	if(x<9)return fgamma(x+1)/x; // Recurrence relation
   	double lnfgamma=x*Log(x+1/(12*x-1/x/10))-x+Log(2*PI/x)/2;
   	return Exp(lnfgamma);
   	}
   }

   #include<cmath>
   #include"sfuns.h"
   namespace sfuns{
   double fgamma(double x){
   	if(x<0)return M_PI/std::sin(M_PI*x)/fgamma(1-x);
   	if(x<9)return fgamma(x+1)/x;
   	double lnfgamma=x*std::log(x+1/(12*x-1/x/10))-x+std::log(2*PI/x)/2;
   	return std::exp(lnfgamma);
   	}
   }

   // "sfuns.h" header file
   #ifndef HAVE_SFUNS_H
   #define HAVE_SFUNS_H
   namespace sfuns{
   double fgamma(double x);
   }
   #endif
   

   calculate Γ(1), Γ(2), Γ(3), …, Γ(10). Check that the results are correct (within single precision: about 6 decimal digits).

   You should put your gamma function in a static class "sfuns" [namespace "sfuns"] in a file "sfuns.cs" ["sfuns.cc"], compile it separately into a library "sfuns.dll" [object code "sfuns.o"], and then link to your Main [main] function.

3. The gamma-function overflows very easily, so the logarithm of the gamma function, lngamma, is often a more useful function. Figure out how to modify the above formula to calculate lngamma. For simplicity you should only allow positive arguments for your lngamma:

   if(x <= 0) return double.NaN;
   if(x < 9) return lngamma(x+1) - Log(x);
   

   if(x <= 0) return NAN;
   if(x < 9) return lngamma(x+1) - std::log(x);
   

Hints

• You should redirect the output of your program to an output file for me to see it,

  Out.txt: main.exe
  	mono main.exe > Out.txt
  

  Out.txt: main
  	./main > Out.txt
  

• You compile a piece of source code, say "sfuns.cs" ["sfuns.cc"], into a library [object code], say "sfuns.dll" ["sfuns.o"], like this,

  sfuns.dll : sfuns.cs
  	mcs -target:library -out:sfuns.dll sfuns.cs 
  

  sfuns.o : sfuns.cc sfuns.h
  	$(CXX) $(CXXFLAGS) -c -o sfuns.o sfuns.cc # -c stands for compile only
  

• You link your library [object code] to your "main.cs" ["main.cc"] like this,

  main.exe : main.cs sfuns.dll
  	mcs -target:exe -out:main.exe -reference:sfuns.dll main.cs
  

  CXXFLAGS += -Wall -std=c++23 -O2
  LDLIBS += -lstdc++ -lm
  main : main.o sfuns.o
  	$(CXX) $(LDLIBS) -o main main.o sfuns.o
  main.o : main.cc sfuns.h
  	$(CXX) $(CXXFLAGS) -c -o main.o main.cc