Exercise "integration"

1. (mandatory)

   • Using the integrate.quad routine from integrate.cs calculate numerically the intergral (exact value is -4),

     ∫01 dx (ln(x)/√x)
     

     You can either copy/paste the content of the link into the file "integrate.cs", or wget the file,

     wget http://80.251.205.75/~fedorov/prog/matlib/integrate/integrate.cs
     

     Your main.cs might look like

     Func<double,double> f = delegate(double x){return Log(x)/Sqrt(x);};
     double result = integrate.quad(f,0,1);
     

     or, equivalently, using a lambda-expression

     Func<double,double> f = x => Log(x)/Sqrt(x);
     double result = integrate.quad(f,0,1);
     

     Your makefile might look like

     out.txt:main.exe
             mono $< > $@
     main.exe: main.cs matlib.dll
             mcs -target:exe -reference:matlib.dll -out:$@ $<
     matlib.dll: integrate.cs
             mcs -target:library -out:$@ $<
     

   • Using the integrate.quad routine from integrate.cs implement the [error-function] using its integral representation. Make a plot.

     Something like

     public static double erf(double z){
     	Func<double,double> f = x => Exp(-x*x);
     	return integrate.quad(f,0,z)*2/Sqrt(PI);
     }
     

2. (optional) Using the integrate.quad routine from integrate.cs implement the Bessel function (of the first kind) using its [integral representation]. Make plots, compare with your numpy/scipy routines.

3. (optional) Improve the effectiveness of your error-function implementation by switching the integration limits from {0,x} to {x,∞} for, say, x>1. Check the number of integrand calls for the naive and the (probably) more effective implementation. Hint: positive infinity in Csharp is double.PositiveInfinity,

   double infin = double.PositiveInfinity;