Verner's 6th and 7th Order Embedded Runge-Kutta Method
Function List
Function(s):
- int Embedded_Verner_6_7( double (*f)(double, double), double y[ ], double x0, double h, double xmax, double *h_next, double tolerance )
Solve the differential equation y' = f(x,y) from x0 to xmax with initial condition y(x0) = y[0] using the initial step size h. The result at x = xmax is returned in y[1]. Upon returning h_next contains the estimated step size so that the final answer is within tolerance of the actual solution at x = xmax, this value of h_next can be used as the initial step size h in the subsequent call to this function. This function returns a 0 if a solution was found, -1 if a solution could not be found, and -2 if xmax < x0 or if h < = 0.
C Source
////////////////////////////////////////////////////////////////////////////////
// File: embedded_verner_6_7.c //
// Routines: //
// Embedded_Verner_6_7 //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// //
// Description: //
// The Runge-Kutta-Verner method is an adaptive procedure for approxi- //
// mating the solution of the differential equation y'(x) = f(x,y) with //
// initial condition y(x0) = c. This implementation evaluates f(x,y) //
// ten times per step using embedded sixth order and seventh order //
// Runge-Kutta estimates to estimate the not only the solution but also //
// the error. //
// The next step size is then calculated using the preassigned tolerance //
// and error estimate. //
// For step i+1, //
// y[i+1] = y[i] + h / 90 * ( 7 * k1 + 32 * k4 + 32 * k5 //
// + 12 * k7 + 7 * k8 ) //
// where //
// k1 = f( x[i],y[i] ), //
// k2 = f( x[i]+h/12, y[i] + h*k1/12), //
// k3 = f( x[i]+h/6, y[i]+h/6*( k2) ), //
// k4 = f( x[i]+h/4, y[i]+h/16*( k1 + 3 k3) ), //
// k5 = f( x[i]+3h/4, y[i]+h/16*(21 k1 - 81 k3 + 72 k4)), //
// k6 = f( x[i]+16h/17, y[i]+h/250563*(1344688 k1 - 5127552 k3 //
// + 4096896 k4 - 78208 k5 ) ), //
// k7 = f( x[i]+h/2, y[i]+h/234624*( -341549 k1 + 1407744 k3 - 1018368 k4 //
// + 84224 k5 - 14739 k6 ) ), //
// k8 = f( x[i]+h, y[i]+h/136864*( -381875 k1 + 1642368 k3 - 1327872 k4 //
// + 72192 k5 + 14739 k6 + 117312 K7) ) //
// k9 = f( x[i]+2h/3, y[i]+h/16755336*( -2070757 k1 + 9929088 k3 //
// + 584064 k4 + 3023488 k5 - 447083 k6 + 151424 k7) ), //
// k10 = f( x[i]+h, y[i]+h/10743824*( 130521209 k1 - 499279872 k3 //
// - 391267968 k4 + 13012608 k5 - 3522621 k6 //
// + 9033024 K7 - 30288492 k9) ) //
// x[i+1] = x[i] + h. //
// //
// The error is estimated to be //
// err = h*( - 1090635 k1 + 9504768 k4 - 171816960 k5 + 72412707 k6 //
// - 55840512 k7 - 13412672 k8 + 181730952 k9 //
// - 21487648 k10) / 172448640 //
// The step size h is then scaled by the scale factor //
// scale = 0.8 * | epsilon * y[i] / [err * (xmax - x[0])] | ^ 1/6 //
// The scale factor is further constrained 0.125 < scale < 4.0. //
// The new step size is h := scale * h. //
////////////////////////////////////////////////////////////////////////////////
#include
#define max(x,y) ( (x) < (y) ? (y) : (x) )
#define min(x,y) ( (x) < (y) ? (x) : (y) )
#define ATTEMPTS 12
#define MIN_SCALE_FACTOR 0.125
#define MAX_SCALE_FACTOR 4.0
static double Runge_Kutta(double (*f)(double,double), double *y, double x,
double h);
////////////////////////////////////////////////////////////////////////////////
// int Embedded_Verner_6_7( double (*f)(double, double), double y[], //
// double x, double h, double xmax, double *h_next, double tolerance ) //
// //
// Description: //
// This function solves the differential equation y'=f(x,y) with the //
// initial condition y(x) = y[0]. The value at xmax is returned in y[1]. //
// The function returns 0 if successful or -1 if it fails. //
// //
// Arguments: //
// double *f Pointer to the function which returns the slope at (x,y) of //
// integral curve of the differential equation y' = f(x,y) //
// which passes through the point (x0,y0) corresponding to the //
// initial condition y(x0) = y0. //
// double y[] On input y[0] is the initial value of y at x, on output //
// y[1] is the solution at xmax. //
// double x The initial value of x. //
// double h Initial step size. //
// double xmax The endpoint of x. //
// double *h_next A pointer to the estimated step size for successive //
// calls to Embedded_Verner_6_7. //
// double tolerance The tolerance of y(xmax), i.e. a solution is sought //
// so that the relative error < tolerance. //
// //
// Return Values: //
// 0 The solution of y' = f(x,y) from x to xmax is stored y[1] and //
// h_next has the value to the next size to try. //
// -1 The solution of y' = f(x,y) from x to xmax failed. //
// -2 Failed because either xmax < x or the step size h <= 0. //
// //
////////////////////////////////////////////////////////////////////////////////
// //
int Embedded_Verner_6_7( double (*f)(double, double), double y[], double x,
double h, double xmax, double *h_next, double tolerance ) {
static const double err_exponent = 1.0 / 6.0;
double scale;
double temp_y[2];
double err;
double yy;
int i;
int last_interval = 0;
// Verify that the step size is positive and that the upper endpoint //
// of integration is greater than the initial enpoint. //
if (xmax < x || h <= 0.0) return -2;
// If the upper endpoint of the independent variable agrees with the //
// initial value of the independent variable. Set the value of the //
// dependent variable and return success. //
*h_next = h;
y[1] = y[0];
if (xmax == x) return 0;
// Insure that the step size h is not larger than the length of the //
// integration interval. //
h = min(h, xmax - x);
// Redefine the error tolerance to an error tolerance per unit //
// length of the integration interval. //
tolerance /= (xmax - x);
// Integrate the diff eq y'=f(x,y) from x=x to x=xmax trying to //
// maintain an error less than tolerance * (xmax-x) using an //
// initial step size of h and initial value: y = y[0] //
temp_y[0] = y[0];
while ( x < xmax ) {
scale = 1.0;
for (i = 0; i < ATTEMPTS; i++) {
err = fabs( Runge_Kutta(f, temp_y, x, h) );
if (err == 0.0) { scale = MAX_SCALE_FACTOR; break; }
yy = (temp_y[0] == 0.0) ? tolerance : fabs(temp_y[0]);
scale = 0.8 * pow( tolerance * yy / err , err_exponent );
scale = min( max(scale,MIN_SCALE_FACTOR), MAX_SCALE_FACTOR);
if ( err < ( tolerance * yy ) ) break;
h *= scale;
if ( x + h > xmax ) h = xmax - x;
else if ( x + h + 0.5 * h > xmax ) h = 0.5 * h;
}
if ( i >= ATTEMPTS ) { *h_next = h * scale; return -1; };
temp_y[0] = temp_y[1];
x += h;
h *= scale;
*h_next = h;
if ( last_interval ) break;
if ( x + h > xmax ) { last_interval = 1; h = xmax - x; }
else if ( x + h + 0.5 * h > xmax ) h = 0.5 * h;
}
y[1] = temp_y[1];
return 0;
}
////////////////////////////////////////////////////////////////////////////////
// static double Runge_Kutta(double (*f)(double,double), double *y, //
// double x0, double h) //
// //
// Description: //
// This routine uses Verner's embedded 6th and 7th order methods to //
// approximate the solution of the differential equation y'=f(x,y) with //
// the initial condition y = y[0] at x = x0. The value at x + h is //
// returned in y[1]. The function returns err / h ( the absolute error //
// per step size ). //
// //
// Arguments: //
// double *f Pointer to the function which returns the slope at (x,y) of //
// integral curve of the differential equation y' = f(x,y) //
// which passes through the point (x0,y[0]). //
// double y[] On input y[0] is the initial value of y at x, on output //
// y[1] is the solution at x + h. //
// double x Initial value of x. //
// double h Step size //
// //
// Return Values: //
// This routine returns the err / h. The solution of y(x) at x + h is //
// returned in y[1]. //
// //
////////////////////////////////////////////////////////////////////////////////
static double Runge_Kutta(double (*f)(double,double), double *y, double x0,
double h) {
static const double c1 = 7.0 / 90.0;
static const double c4 = 32.0 / 90.0;
static const double c5 = 32.0 / 90.0;
static const double c7 = 12.0 / 90.0;
static const double c8 = 7.0 / 90.0;
static const double a2 = 1.0 / 12.0;
static const double a3 = 1.0 / 6.0;
static const double a4 = 1.0 / 4.0;
static const double a5 = 3.0 / 4.0;
static const double a6 = 16.0 / 17.0;
static const double a7 = 1.0 / 2.0;
static const double a9 = 2.0 / 3.0;
static const double b41 = 1.0 / 16.0;
static const double b43 = 3.0 / 16.0;
static const double b51 = 21.0 / 16.0;
static const double b53 = -81.0 / 16.0;
static const double b54 = 72.0 / 16.0;
static const double b61 = 1344688.0 / 250563.0;
static const double b63 = -5127552.0 / 250563.0;
static const double b64 = 4096896.0 / 250563.0;
static const double b65 = -78208.0 / 250563.0;
static const double b71 = -341549.0 / 234624.0;
static const double b73 = 1407744.0 / 234624.0;
static const double b74 = -1018368.0 / 234624.0;
static const double b75 = 84224.0 / 234624.0;
static const double b76 = -14739.0 / 234624.0;
static const double b81 = -381875.0 / 136864.0;
static const double b83 = 1642368.0 / 136864.0;
static const double b84 = -1327872.0 / 136864.0;
static const double b85 = 72192.0 / 136864.0;
static const double b86 = 14739.0 / 136864.0;
static const double b87 = 117312.0 / 136864.0;
static const double b91 = -2070757.0 / 16755336.0;
static const double b93 = 9929088.0 / 16755336.0;
static const double b94 = 584064.0 / 16755336.0;
static const double b95 = 3023488.0 / 16755336.0;
static const double b96 = -447083.0 / 16755336.0;
static const double b97 = 151424.0 / 16755336.0;
static const double b10_1 = 130521209.0 / 10743824.0;
static const double b10_3 = -499279872.0 / 10743824.0;
static const double b10_4 = 391267968.0 / 10743824.0;
static const double b10_5 = 13012608.0 / 10743824.0;
static const double b10_6 = -3522621.0 / 10743824.0;
static const double b10_7 = 9033024.0 / 10743824.0;
static const double b10_9 = -30288492.0 / 10743824.0;
static const double e1 = -1090635.0 / 172448640.0;
static const double e4 = 9504768.0 / 172448640.0;
static const double e5 = - 171816960.0 / 172448640.0;
static const double e6 = 72412707.0 / 172448640.0;
static const double e7 = - 55840512.0 / 172448640.0;
static const double e8 = - 13412672.0 / 172448640.0;
static const double e9 = 181730952.0 / 172448640.0;
static const double e10 = - 21487648.0 / 172448640.0;
double k1, k2, k3, k4, k5, k6, k7, k8, k9, k10;
double h12 = a2 * h;
double h6 = a3 * h;
k1 = (*f)(x0, *y);
k2 = (*f)(x0+h12, *y + h12 * k1);
k3 = (*f)(x0+h6, *y + h6 * k2 );
k4 = (*f)(x0+a4*h, *y + h * ( b41*k1 + b43*k3) );
k5 = (*f)(x0+a5*h, *y + h * ( b51*k1 + b53*k3 + b54*k4) );
k6 = (*f)(x0+a6*h, *y + h * ( b61*k1 + b63*k3 + b64*k4 + b65*k5) );
k7 = (*f)(x0+a7*h, *y + h * ( b71*k1 + b73*k3 + b74*k4 + b75*k5 + b76*k6) );
k8 = (*f)(x0+h, *y + h * ( b81*k1 + b83*k3 + b84*k4 + b85*k5 + b86*k6
+ b87*k7) );
k9 = (*f)(x0+a9*h, *y + h * ( b91*k1 + b93*k3 + b94*k4 + b95*k5 + b96*k6
+ b97*k7) );
k10 = (*f)(x0+h, *y + h * ( b10_1*k1 + b10_3*k3 + b10_4*k4 + b10_5*k5
+ b10_6*k6 + b10_7*k7 + b10_9*k9 ) );
*(y+1) = *y + h * (c1 * k1 + c4 * k4 + c5 * k5 + c7 * k7 + c8 * k8);
return e1*k1 + e4*k4 + e5*k5 + e6*k6 + e7*k7 + e8*k8 + e9*k9 + e10*k10;
}