/* From: "COMPUTATIONAL PHYSICS, 2nd Ed" 
   by RH Landau, MJ Paez, and CC Bordeianu 
   Copyright Wiley-VCH, 2007.
   Electronic Materials copyright: R Landau, Oregon State Univ, 2007;
   MJ Paez, Univ Antioquia, 2007; & CC Bordeianu, Univ Bucharest, 2007
   Support by National Science Foundation
*/
/* rk45.c Ordinary Differential Equations Solver (ODES).
  Solve the differential equation using Runge-Kutta-Fehlberg Method
  with variable step size.                                        */ 

#include<stdio.h>
#include<stdlib.h>
#include<math.h>

main()  {
	
  FILE *out1;              // open  file rk45.dat for output data
  out1 = fopen("rk45.dat","w");
	
  double h,t,s,s1,hmin,hmax;
  double y[2], fReturn[2], ydumb[2];
  double k1[2], k2[2], k3[2];
  double k4[2], k5[2], k6[2];
  double err[2]; 
  double Tol = 1.0E-8;                      //error control tolerance
  double a = 0.0;                                         //endpoints
  double b = 10.0;
  int i,j,n = 20;
  void f(double t, double y[], double fReturn[]);
	
  y[0] = 1.0;  y[1]= 0.0; 
  h = (b-a)/n;                            //tentative number of steps
  hmin = h/64;
  hmax = h*64;                        //minimum and maximum step size
  t = a;
  j = 0; 
  fprintf(out1,"%f\t%f\t%f\n",t,y[0],y[1]);   //output answer to file
 
	while (t<b)  {
    if ( (t + h) > b ) h= b - t;                     // the last step
    f(t, y, fReturn);    // evaluate both RHS's and return in fReturn
    k1[0] = h*fReturn[0];               //compute the function values
    k1[1] = h*fReturn[1];
   
		for (i = 0;i<= 1;i++) ydumb[i]= y[i] + k1[i]/4;
    f(t + h/4, ydumb, fReturn);
    k2[0] = h*fReturn[0];
    k2[1] = h*fReturn[1];
    
		for (i = 0;i<= 1;i++) ydumb[i]= y[i] + 3*k1[i]/32 + 9*k2[i]/32;
    f(t + 3*h/8, ydumb, fReturn);
    k3[0]= h*fReturn[0]; 
    k3[1]= h*fReturn[1];
   
		for (i = 0;i<= 1;i++) ydumb[i]= y[i] + 1932*k1[i]/2197
           -7200*k2[i]/2197. +7296*k3[i]/2197;
    f(t+ 12*h/13, ydumb, fReturn);
    k4[0]= h*fReturn[0];  
    k4[1]= h*fReturn[1];  
    
		for (i = 0;i<= 1;i++) ydumb[i]= y[i] +439*k1[i]/216 -8*k2[i]
                                  +3680*k3[i]/513 -845*k4[i]/4104;
    f(t+h, ydumb, fReturn);
    k5[0]= h*fReturn[0];  
    k5[1]= h*fReturn[1];  
   
		for (i = 0;i<= 1;i++) ydumb[i]= y[i] -8*k1[i]/27 +2*k2[i]
                  -3544*k3[i]/2565 +1859*k4[i]/4104 -11*k5[i]/40;
    f(t+ h/2, ydumb, fReturn);
    k6[0]= h*fReturn[0];  
    k6[1]= h*fReturn[1];
   
		for (i = 0;i<= 1;i++) err[i] = abs( k1[i]/360 - 128*k3[i]/4275 
      - 2197*k4[i]/75240 + k5[i]/50.0 + 2*k6[i]/55 );      
    if ((err[0]<Tol)||(err[1]<Tol)||(h<= 2*hmin))  {         //accept
      for (i = 0;i<= 1;i++)   y[i]= y[i] + 25*k1[i]/216. 
				                                          + 1408*k3[i]/2565.
        + 2197*k4[i]/4104. - k5[i]/5.;
      t= t + h;
      j++;
    }
    if (( err[0] == 0)||(err[1] == 0)) s = 0;    //trap division by 0
    else s = 0.84*pow(Tol*h/err[0],0.25);          //step size scalar
    if ( (s < 0.75) && (h > 2*hmin) )   h /=  2.;       //reduce step
    else if ( (s > 1.5) && (2* h < hmax) ) h *=  2.;  //increase step
    fprintf(out1,"%f\t%f\t%f\n",t,y[0],y[1]); //output answer to file
 }                                               // End of while loop
} 

void f(double t, double y[], double fReturn[])  {
	                                                    // RHS function
    fReturn[0] = y[1];                       // RHS of first equation
    fReturn[1] = -6.0*pow(y[0],5.0);        // RHS of second equation
    return;
}