Problems 10

1. Theory.
   1. Find the file on you box which contains the declaration of the object gsl_multiroot_function and read the declaration.
      Hint: the file is gsl_multiroots.h;
2. Practice.
   1. 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.
   2. Multidimentional root finding 'without derivatives': firing projectiles at a target.
      • Consider a cannon that fires cannonballs with muzzle velocity v at angle θ from zenith and azimuth φ.
        The coordinates of the projectile as function of elapsed time are given as (check it)

        z(t)=v cos(θ) t - gt²/2,

        x(t)=v sin(θ) sin(φ) t,

        y(t)=v sin(θ) cos(φ) t.

      • Suppose you want to hit a target with coordinates [x_target, y_target, z_target=0]
      • Build a program that calculates the aiming angles to hit the target using gsl_multiroot_fsolver_hybrids algorithm.
      • Make a plot which illustrates the progress of the algorithm.
   3. (optional) Compare with an algorithm with derivatives.
   4. (optional) Generalize to z_target≠0.
   5. (optional) Add the wind-resistance drag force to the equation of motion of the cannonball in the form F_drag=-γvv, where boldface denotes vectors, and γ is the wind-resistance parameter of the cannonball.
   6. (optional) Add the atmospheric wind.