Re: Some (probably) ridiculous questions

[Brian Gough <bjg at network-theory dot co dot uk>]

Jean-Max Redonnet writes:
 > I think I have defined a surface function type as you proposed.
 > This part seems to work as expected.
 > but I don't see how to define a function to create the appropriate 
 > gsl_function...
 > 

The gsl_function has to be a parametric function.  The example below
defines a gsl_function 'line.f' for |F|^2 on a path P(t),

  (u(t),v(t)) = (0.5,0.5) + 0.1*(cos(t),sin(t))

and then finds the minimum (which is at t=pi) using the standard gsl
minimization routines.  The relevant functions are line_eval() and
make_line_function().

#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_diff.h>
#include <gsl/gsl_min.h>
#include <gsl/gsl_errno.h>

#define RAD2DEG(X) (X*180/M_PI)
#define DEG2RAD(X) (X*M_PI/180)

struct surf_function_struct {
  double (* fx)(double u, double v);
  double (* fy)(double u, double v);
  double (* fz)(double u, double v);;
};

typedef struct surf_function_struct surf_function ;

double C0_fx(double u)
{
  return 80*u;
}

double C0_fy(double u)
{
  return 80*tan(DEG2RAD(22.5))*(u-0.5);
}

double C0_fz(double u)
{
  return 40;
}

double C1_fx(double u)
{
  return 80*u;
}

double C1_fy(double u)
{
  return -80*tan(DEG2RAD(22.5))*(u-0.5);
}

double C1_fz(double u)
{
  return -40;
}

double S_fx(double u, double v)
{
  return (1-v)*C0_fx(u)+v*C1_fx(u);
}

double S_fy(double u, double v)
{
  return (1-v)*C0_fy(u)+v*C1_fy(u);
}

double S_fz(double u, double v)
{
  return (1-v)*C0_fz(u)+v*C1_fz(u);
}


struct path_function_struct {
  void (* uv)(double t, double *u, double *v);
};

typedef struct path_function_struct path_function ;

void P_uv(double t, double *u, double *v)
{
  *u = 0.5 + 0.1 * cos(t);
  *v = 0.5 + 0.1 * sin(t);
}

struct line_function_struct
{
  path_function p;
  surf_function s;
  gsl_function f;
};

typedef struct line_function_struct line_function ;

double 
line_eval (double t, void * params)
{
  line_function * line = (line_function *) params;

  double u, v;

  line->p.uv(t, &u, &v);
  
  {
    double a = line->s.fx(u,v);
    double b = line->s.fy(u,v);
    double c = line->s.fz(u,v);

    /* compute |F|^2 */

    return a*a + b*b + c*c;
  }
}

void
make_line_function (path_function path, surf_function surf, 
		    line_function * line)
{
  line->p = path;
  line->s = surf;
  line->f.function = &line_eval;
  line->f.params = line;
}


int main (void)
{
  double u,v,t;
  surf_function surf;
  path_function path;

  surf.fx = &S_fx;
  surf.fy = &S_fy;
  surf.fz = &S_fz;
  
  path.uv = &P_uv;

  for (t = 0 ; t < 2*M_PI; t+= 0.5)
    {
      path.uv(t, &u, &v);

      printf("                  | %f \n", S_fx(u,v));
      printf(" S(%.3f,%.3f) = | %f \n", u, v, S_fy(u,v));
      printf("                  | %f \n", S_fz(u,v));
      
      printf("                  | %f \n", surf.fx(u,v));
      printf(" S(%.3f,%.3f) = | %f \n", u, v, surf.fy(u,v));
      printf("                  | %f \n", surf.fz(u,v));
    }

  {
    line_function line;

    make_line_function(path, surf, &line);

    {
      int status, iterations = 0;
      double m = 1.23;
      gsl_interval x = {0.0, 2*M_PI};

      gsl_min_fminimizer * s 
	= gsl_min_fminimizer_alloc (gsl_min_fminimizer_brent);

      gsl_min_fminimizer_set (s, &line.f, m, x);

      do
	{
	iterations++;
	status = gsl_min_fminimizer_iterate (s);
	
	m = gsl_min_fminimizer_minimum (s);
	x = gsl_min_fminimizer_interval (s);
	
	status = gsl_min_test_interval (x, 0.001, 0.0);
	
	if (status == GSL_SUCCESS)
	  printf ("Converged:\n");
	
	printf ("%5d [%.7f, %.7f] %.7f %.7f %+.7f\n",
		iterations, x.lower, x.upper, 
		m, s->f_minimum, x.upper - x.lower);
	}
      while (status == GSL_CONTINUE && iterations < 100);
    }
  }

      
  return 0;
}