Fehlberg's 7th and 8th Order Embedded Runge-Kutta Method
Function List
Function(s):
- int Embedded_Fehlberg_7_8( 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_fehlberg_7_8.c //
// Routines: //
// Embedded_Fehlberg_7_8 //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// //
// Description: //
// The Runge-Kutta-Fehlberg 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) //
// thirteen times per step using embedded seventh order and eight 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 * (41/840 * k1 + 34/105 * k6 + 9/35 * k7 //
// + 9/35 * k8 + 9/280 * k9 + 9/280 k10 + 41/840 k11 ) //
// where //
// k1 = f( x[i],y[i] ), //
// k2 = f( x[i]+2h/27, y[i] + 2h*k1/27), //
// k3 = f( x[i]+h/9, y[i]+h/36*( k1 + 3 k2) ), //
// k4 = f( x[i]+h/6, y[i]+h/24*( k1 + 3 k3) ), //
// k5 = f( x[i]+5h/12, y[i]+h/48*(20 k1 - 75 k3 + 75 k4)), //
// k6 = f( x[i]+h/2, y[i]+h/20*( k1 + 5 k4 + 4 k5 ) ), //
// k7 = f( x[i]+5h/6, y[i]+h/108*( -25 k1 + 125 k4 - 260 k5 + 250 k6 ) ), //
// k8 = f( x[i]+h/6, y[i]+h*( 31/300 k1 + 61/225 k5 - 2/9 k6 //
// + 13/900 K7) ) //
// k9 = f( x[i]+2h/3, y[i]+h*( 2 k1 - 53/6 k4 + 704/45 k5 - 107/9 k6 //
// + 67/90 k7 + 3 k8) ), //
// k10 = f( x[i]+h/3, y[i]+h*( -91/108 k1 + 23/108 k4 - 976/135 k5 //
// + 311/54 k6 - 19/60 k7 + 17/6 K8 - 1/12 k9) ), //
// k11 = f( x[i]+h, y[i]+h*( 2383/4100 k1 - 341/164 k4 + 4496/1025 k5 //
// - 301/82 k6 + 2133/4100 k7 + 45/82 K8 + 45/164 k9 + 18/41 k10) ) //
// k12 = f( x[i], y[i]+h*( 3/205 k1 - 6/41 k6 - 3/205 k7 - 3/41 K8 //
// + 3/41 k9 + 6/41 k10) ) //
// k13 = f( x[i]+h, y[i]+h*( -1777/4100 k1 - 341/164 k4 + 4496/1025 k5 //
// - 289/82 k6 + 2193/4100 k7 + 51/82 K8 + 33/164 k9 + //
// 12/41 k10 + k12) ) //
// x[i+1] = x[i] + h. //
// //
// The error is estimated to be //
// err = -41/840 * h * ( k1 + k11 - k12 - k13) //
// The step size h is then scaled by the scale factor //
// scale = 0.8 * | epsilon * y[i] / [err * (xmax - x[0])] | ^ 1/7 //
// The scale factor is further constrained 0.125 < scale < 4.0. //
// The new step size is h := scale * h. //
////////////////////////////////////////////////////////////////////////////////
#include < math.h >
#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_Fehlberg_7_8( 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_Fehlberg_7_8. //
// 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_Fehlberg_7_8( double (*f)(double, double), double y[], double x,
double h, double xmax, double *h_next, double tolerance ) {
static const double err_exponent = 1.0 / 7.0;
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. //
if (h > (xmax - x) ) { h = xmax - x; last_interval = 1;}
// 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 Fehlberg's embedded 7th and 8th 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 c_1_11 = 41.0 / 840.0;
static const double c6 = 34.0 / 105.0;
static const double c_7_8= 9.0 / 35.0;
static const double c_9_10 = 9.0 / 280.0;
static const double a2 = 2.0 / 27.0;
static const double a3 = 1.0 / 9.0;
static const double a4 = 1.0 / 6.0;
static const double a5 = 5.0 / 12.0;
static const double a6 = 1.0 / 2.0;
static const double a7 = 5.0 / 6.0;
static const double a8 = 1.0 / 6.0;
static const double a9 = 2.0 / 3.0;
static const double a10 = 1.0 / 3.0;
static const double b31 = 1.0 / 36.0;
static const double b32 = 3.0 / 36.0;
static const double b41 = 1.0 / 24.0;
static const double b43 = 3.0 / 24.0;
static const double b51 = 20.0 / 48.0;
static const double b53 = -75.0 / 48.0;
static const double b54 = 75.0 / 48.0;
static const double b61 = 1.0 / 20.0;
static const double b64 = 5.0 / 20.0;
static const double b65 = 4.0 / 20.0;
static const double b71 = -25.0 / 108.0;
static const double b74 = 125.0 / 108.0;
static const double b75 = -260.0 / 108.0;
static const double b76 = 250.0 / 108.0;
static const double b81 = 31.0/300.0;
static const double b85 = 61.0/225.0;
static const double b86 = -2.0/9.0;
static const double b87 = 13.0/900.0;
static const double b91 = 2.0;
static const double b94 = -53.0/6.0;
static const double b95 = 704.0 / 45.0;
static const double b96 = -107.0 / 9.0;
static const double b97 = 67.0 / 90.0;
static const double b98 = 3.0;
static const double b10_1 = -91.0 / 108.0;
static const double b10_4 = 23.0 / 108.0;
static const double b10_5 = -976.0 / 135.0;
static const double b10_6 = 311.0 / 54.0;
static const double b10_7 = -19.0 / 60.0;
static const double b10_8 = 17.0 / 6.0;
static const double b10_9 = -1.0 / 12.0;
static const double b11_1 = 2383.0 / 4100.0;
static const double b11_4 = -341.0 / 164.0;
static const double b11_5 = 4496.0 / 1025.0;
static const double b11_6 = -301.0 / 82.0;
static const double b11_7 = 2133.0 / 4100.0;
static const double b11_8 = 45.0 / 82.0;
static const double b11_9 = 45.0 / 164.0;
static const double b11_10 = 18.0 / 41.0;
static const double b12_1 = 3.0 / 205.0;
static const double b12_6 = - 6.0 / 41.0;
static const double b12_7 = - 3.0 / 205.0;
static const double b12_8 = - 3.0 / 41.0;
static const double b12_9 = 3.0 / 41.0;
static const double b12_10 = 6.0 / 41.0;
static const double b13_1 = -1777.0 / 4100.0;
static const double b13_4 = -341.0 / 164.0;
static const double b13_5 = 4496.0 / 1025.0;
static const double b13_6 = -289.0 / 82.0;
static const double b13_7 = 2193.0 / 4100.0;
static const double b13_8 = 51.0 / 82.0;
static const double b13_9 = 33.0 / 164.0;
static const double b13_10 = 12.0 / 41.0;
static const double err_factor = -41.0 / 840.0;
double k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13;
double h2_7 = a2 * h;
k1 = (*f)(x0, *y);
k2 = (*f)(x0+h2_7, *y + h2_7 * k1);
k3 = (*f)(x0+a3*h, *y + h * ( b31*k1 + b32*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 + b64*k4 + b65*k5) );
k7 = (*f)(x0+a7*h, *y + h * ( b71*k1 + b74*k4 + b75*k5 + b76*k6) );
k8 = (*f)(x0+a8*h, *y + h * ( b81*k1 + b85*k5 + b86*k6 + b87*k7) );
k9 = (*f)(x0+a9*h, *y + h * ( b91*k1 + b94*k4 + b95*k5 + b96*k6
+ b97*k7 + b98*k8) );
k10 = (*f)(x0+a10*h, *y + h * ( b10_1*k1 + b10_4*k4 + b10_5*k5 + b10_6*k6
+ b10_7*k7 + b10_8*k8 + b10_9*k9 ) );
k11 = (*f)(x0+h, *y + h * ( b11_1*k1 + b11_4*k4 + b11_5*k5 + b11_6*k6
+ b11_7*k7 + b11_8*k8 + b11_9*k9 + b11_10 * k10 ) );
k12 = (*f)(x0, *y + h * ( b12_1*k1 + b12_6*k6 + b12_7*k7 + b12_8*k8
+ b12_9*k9 + b12_10 * k10 ) );
k13 = (*f)(x0+h, *y + h * ( b13_1*k1 + b13_4*k4 + b13_5*k5 + b13_6*k6
+ b13_7*k7 + b13_8*k8 + b13_9*k9 + b13_10*k10 + k12 ) );
*(y+1) = *y + h * (c_1_11 * (k1 + k11) + c6 * k6 + c_7_8 * (k7 + k8)
+ c_9_10 * (k9 + k10) );
return err_factor * (k1 + k11 - k12 - k13);
}