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tests.c

/* integration/tests.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Brian Gough
 * 
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 2 of the License, or (at
 * your option) any later version.
 * 
 * This program is distributed in the hope that it will be useful, but
 * WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * General Public License for more details.
 * 
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 */

#include <config.h>
#include <math.h>
#include <gsl/gsl_math.h>

#include "tests.h"

/* These are the test functions from table 4.1 of the QUADPACK book */

/* f1(x) = x^alpha * log(1/x) */
/* integ(f1,x,0,1) = 1/(alpha + 1)^2 */

double f1 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(x,alpha) * log(1/x) ;
}

/* f2(x) = 4^-alpha / ((x-pi/4)^2 + 16^-alpha) */
/* integ(f2,x,0,1) = arctan((4-pi)4^(alpha-1)) + arctan(pi 4^(alpha-1)) */

double f2 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(4.0,-alpha) / (pow((x-M_PI/4.0),2.0) + pow(16.0,-alpha)) ;
}

/* f3(x) = cos(2^alpha * sin(x)) */
/* integ(f3,x,0,pi) = pi J_0(2^alpha) */

double f3 (double x, void * params) {
  double alpha = *(double *) params ;
  return cos(pow(2.0,alpha) * sin(x)) ;
}

/* Functions 4, 5 and 6 are duplicates of functions  1, 2 and 3 */
/* ....                                                         */

/* f7(x) = |x - 1/3|^alpha */
/* integ(f7,x,0,1) = ((2/3)^(alpha+1) + (1/3)^(alpha+1))/(alpha + 1) */

double f7 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(fabs(x - (1.0/3.0)),alpha) ;
}

/* f8(x) = |x - pi/4|^alpha */
/* integ(f8,x,0,1) = 
   ((1 - pi/4)^(alpha+1) + (pi/4)^(alpha+1))/(alpha + 1) */

double f8 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(fabs(x - (M_PI/4.0)),alpha) ;
}

/* f9(x) = sqrt(1 - x^2) / (x + 1 + 2^-alpha) */
/* integ(f9,x,-1,1) = pi/sqrt((1+2^-alpha)^2-1) */

double f9 (double x, void * params) {
  double alpha = *(double *) params ;
  return 1 / ((x + 1 + pow(2.0,-alpha)) * sqrt(1-x*x)) ;
}

/* f10(x) = sin(x)^(alpha - 1) */
/* integ(f10,x,0,pi/2) = 2^(alpha-2) ((Gamma(alpha/2))^2)/Gamma(alpha) */

double f10 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(sin(x), alpha-1) ;
}

/* f11(x) = log(1/x)^(alpha - 1) */
/* integ(f11,x,0,1) = Gamma(alpha) */

double f11 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(log(1/x), alpha-1) ;
}

/* f12(x) = exp(20*(x-1)) * sin(2^alpha * x) */
/* integ(f12,x,0,1) = 
   (20 sin(2^alpha) - 2^alpha cos(2^alpha) + 2^alpha exp(-20))
   /(400 + 4^alpha) */

double f12 (double x, void * params) {
  double alpha = *(double *) params ;
  return exp(20*(x-1)) * sin(pow(2.0,alpha) * x) ;
}

/* f13(x) = cos(2^alpha * x)/sqrt(x(1 - x)) */
/* integ(f13,x,0,1) = pi cos(2^(alpha-1)) J_0(2^(alpha-1))  */

double f13 (double x, void * params) {
  double alpha = *(double *) params ;
  return cos(pow(2.0,alpha)*x)/sqrt(x*(1-x)) ;
}

double f14 (double x, void * params) {
  double alpha = *(double *) params ;
  return exp(-pow(2.0,-alpha)*x)*cos(x)/sqrt(x) ;
}

double f15 (double x, void * params) {
  double alpha = *(double *) params ;
  return x*x * exp(-pow(2.0,-alpha)*x) ;
}

double f16 (double x, void * params) {
  double alpha = *(double *) params ;
  if (x==0 && alpha == 1) return 1 ;  /* make the function continuous in x */
  if (x==0 && alpha > 1) return 0 ;   /* avoid problems with pow(0,1) */
  return pow(x,alpha-1)/pow((1+10*x),2.0) ;
}

double f17 (double x, void * params) {
  double alpha = *(double *) params ;
  return pow(2.0,-alpha)/(((x-1)*(x-1)+pow(4.0,-alpha))*(x-2)) ;
}

/* f454(x) = x^3 log|(x^2-1)(x^2-2)| */
/* integ(f454,x,0,inf) = 61 log(2) + (77/4) log(7) - 27 */

double f454 (double x, void * params) {
  double x2 = x * x;
  double x3 = x * x2;
  params = 0 ;
  return x3 * log(fabs((x2 - 1.0) * (x2 - 2.0))) ;
}

/* f455(x) = log(x)/(1+100*x^2) */
/* integ(f455,x,0,inf) = -log(10)/20 */

double f455 (double x, void * params) {
  params = 0 ;
  return log(x) / (1.0 + 100.0 * x * x) ;
}

/* f456(x) = log(x) */
/* integ(f456*sin(10 pi x),x,0,1) = -(gamma + log(10pi) - Ci(10pi))/(10pi) */

double f456 (double x, void * params) {
  params = 0 ;
  if (x == 0.0)
    {
      return 0;
    }
  return log(x) ;
}

/* f457(x) = 1/sqrt(x) */
/* integ(f457*cos(pi x / 2),x,0,+inf) = 1 */

double f457 (double x, void * params) {
  params = 0 ;
  if (x == 0.0)
    {
      return 0;
    }
  return 1/sqrt(x) ;
}

/* f458(x) = 1/(1 + log(x)^2)^2 */
/* integ(log(x) f458(x),x,0,1) = (Ci(1) sin(1) + (pi/2 - Si(1)) cos(1))/pi 
                               = -0.1892752 */

double f458 (double x, void * params) {
  params = 0 ;

  if (x == 0.0) 
    {
      return 0;
    }
  else 
    {
      double u = log(x);
      double v = 1 + u * u;
      
      return 1.0 / (v * v) ;
    }
}

/* f459(x) = 1/(5 x^3 + 6) */
/* integ(f459/(x-0),x,-1,5) = log(125/631)/18 */

double f459 (double x, void * params) {
  params = 0 ;
  return 1.0 / (5.0 * x * x * x + 6.0) ;
}

/* myfn1(x) = exp(-x - x^2) */
/* integ(myfn1,x,-inf,inf) = sqrt(pi) exp(-1/4) */

double myfn1 (double x, void * params) {
  params = 0;
  return exp(-x - x*x) ;
}

/* myfn2(x) = exp(alpha*x) */
/* integ(myfn2,x,-inf,b) = exp(alpha*b)/alpha */

double myfn2 (double x, void * params) {
  double alpha = *(double *) params ;
  return exp(alpha*x) ;
}

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