Problems 9

1. Theory.
2. Practice.
   1. Numerical integration with Gauss-Kronrod algorithm.
      • Look up the definition of the error functions erf and erfc in Wikipedia.
      • Make a subroutine that calculates the error function erf by direct numerical integration.
      • Make a plot where you compare you subroutine with the erf function from math.h .
      • When |x|<2 use erf and gsl_integration_qags, otherwise use erfc and gsl_integration_qagil gsl_integration_qagiu for negative and positive x correspondingly.
   2. Multidimensional minimization with Nelder-Mead algorithm (gsl_multimin_fminimizer_nmsimplex2rand).
      • Suppose that an experiment on measuring the activity of a radioactive substance as function of time gives the following result

        time ti	activity yi
        0	6.76804
        1.11111	5.66591
        2.22222	5.04163
        3.33333	4.55816
        4.44444	4.24362
        5.55556	3.86711
        6.66667	3.73269
        7.77778	3.70706
        8.88889	3.53118
        10	3.53251

        where the time t_i and the activity y_i is given in some rescaled units.
      • I hate to say it, but we shall disregard the experimental errors for this problem.
      • Fit the function f(t)=A*exp(-λt)+B (where A, λ, and B are fitting parameters) to the data and determine the half-life of the substance.
      • You should solve the problem by doing the multidimensional minimization using the gsl_multimin_fminimizer_nmsimplex2rand algorithm.
      • The function to minimize is F(A,λ,B)=∑_i (f(t_i)-y_i)², that is, you have got a three-dimensional minimization problem in the space of the parameters A, λ, and B.
      • Plot the experimental data and the fitting function.