Nystrom's Fifth Order Method
Description
Nystrom's fifth order method is a Runge-Kutta method for approximating the solution of the initial value problem y'(x) = f(x,y); y(x0) = y0 which evaluates the integrand,f(x,y), six times per step. See the comments in the source code for the algorithm.
This method is a fifth order procedure for which Richardson extrapolation can be used.
Function List
Functions:
- double Runge_Kutta_Nystrom( double (*f)(double, double), double y0, double x0, double h, int number_of_steps )
This function uses Nystrom's fifth order method to return the estimate of the solution of the initial value problem, y' = f(x,y); y(x0) = y0, at x = x0 + h * n, where h is the step size and n is number_of_steps.
- double Runge_Kutta_Nystrom_Richardson( double (*f)(double, double), double y0,
double x0, double h, int number_of_steps, int richardson_columns )
This function uses Nystrom's fifth order method together with Richardson extrapolation to the limit to return the estimate of the solution of the initial value problem, y' = f(x,y); y(x0) = y0, at x = x0 + h * n, where h is the step size and n is number_of_steps. The argument richardson_columns is
the number of step size halving + 1 used in Richardson extrapolation so that if richardson_columns = 1 then no extrapolation to the limit is performed.
- void Runge_Kutta_Nystrom_Integral_Curve( double (*f)(double, double), double y[ ], double x0, double h, int number_of_steps_per_interval, int number_of_intervals )
This function uses Nystrom's fifth order method to estimate the solution of the initial value problem, y' = f(x,y); y(x0) = y0, at x = x0 + h * n * m, where h is the step size and n is the interval number 0 <= n <= number_of_intervals, and m is the number_of_steps_per_interval. The values are return in the array y[ ] i.e. y[n] = y(x0 + h * m * n), where m, n are as above.
- void Runge_Kutta_Nystrom_Richardson_Integral_Curve( double (*f)(double, double), double y[ ], double x0, double h, int number_of_steps_per_interval, int number_of_intervals, int richardson_columns )
This function uses Nystrom's fifth order method together with Richardson extrapolation to the limit to estimate the solution of the initial value problem, y' = f(x,y); y(x0) = y0, at x = x0 + h * n * m, where h is the step size and n is the interval number 0 <= n <= number_of_intervals, and m is the number_of_steps_per_interval. The values are return in the array y[ ] i.e. y[n] = y(x0 + h * m * n), where m, n are as above. The argument richardson_columns is
the number of step size halving + 1 used in Richardson extrapolation so that if richardson_columns = 1 then no extrapolation to the limit is performed.
C Source
////////////////////////////////////////////////////////////////////////////////
// File: runge_kutta_nystrom.c //
// Routines: //
// Runge_Kutta_Nystrom //
// Runge_Kutta_Nystrom_Richardson //
// Runge_Kutta_Nystrom_Integral_Curve //
// Runge_Kutta_Nystrom_Richardson_Integral_Curve //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// //
// Description: //
// The 6 stage 5th order Runge-Kutta method for solving a differential //
// equation y'(x) = f(x,y) with initial condition y = c when x = x0 //
// evaluates f(x,y) six times per step. For step i+1, //
// y[i+1] = y[i] + ( 23 * k1 + 125 * k3 - 81 * k5 + 125 * k6 ) / 192 //
// where, //
// k1 = h * f( x[i], y[i] ), //
// k2 = h * f( x[i]+h/3, y[i]+k1/3 ), //
// k3 = h * f( x[i]+2h/5, y[i] + ( 4 k1 + 6 k2 ) / 25 ), //
// k4 = h * f( x[i]+h, y[i] + ( k1 - 12 k2 + 15 k3 ) / 4), //
// k5 = h * f( x[i]+2h/3, y[i] + (6 k1 + 90 k2 - 50 k3 + 8 k4) / 81), //
// k6 = h * f( x[i]+4h/5, y[i] + (6 k1 + 36 k2 + 10 k3 + 8 k4) / 75), //
// and x[i+1] = x[i] + h. //
// //
////////////////////////////////////////////////////////////////////////////////
#include
static const double one_twentififth = 1.0 / 25.0;
static const double one_fourth = 1.0 / 4.0;
static const double one_eightyfirst = 1.0 / 81.0;
static const double one_seventyfifth = 1.0 / 75.0;
static const double one_third = 1.0 / 3.0;
static const double two_fifths = 2.0 / 5.0;
static const double two_thirds = 2.0 / 3.0;
static const double four_fifths = 4.0 / 5.0;
static const double one_oneninetytwo = 1.0 / 192.0;
////////////////////////////////////////////////////////////////////////////////
// double Runge_Kutta_Nystrom( double (*f)(double, double), double y0, //
// double x0, double h, int number_of_steps ); //
// //
// Description: //
// This routine uses the Runge-Kutta-Nystrom method described above to //
// approximate the solution at x = x0 + h * number_of_steps of the initial//
// value problem y'=f(x,y), y(x0) = y0. //
// //
// Arguments: //
// double *f //
// Pointer to the function which returns the slope at (x,y) of the //
// integral curve of the differential equation y' = f(x,y) which //
// passes through the point (x0,y0). //
// double y0 //
// The initial value of y at x = x0. //
// double x0 //
// The initial value of x. //
// double h //
// The step size. //
// int number_of_steps //
// The number of steps. Must be a nonnegative integer. //
// //
// Return Values: //
// The solution of the initial value problem y' = f(x,y), y(x0) = y0 at //
// x = x0 + number_of_steps * h. //
// //
////////////////////////////////////////////////////////////////////////////////
// //
double Runge_Kutta_Nystrom( double (*f)(double, double), double y0, double x0,
double h, int number_of_steps ) {
double k1, k2, k3, k4, k5, k6;
double h3 = one_third * h;
double h4 = one_fourth * h;
double h25 = one_twentififth * h;
double h81 = one_eightyfirst * h;
double h2_5 = two_fifths * h;
double h2_3 = two_thirds * h;
double h4_5 = four_fifths * h;
double h75 = one_seventyfifth * h;
double h192 = one_oneninetytwo * h;
while ( --number_of_steps >= 0 ) {
k1 = (*f)(x0,y0);
k2 = (*f)(x0 + h3, y0 + h3 * k1);
k3 = (*f)(x0 + h2_5, y0
+ h25 * ( 4.0 * k1 + 6.0 * k2 ) );
k4 = (*f)(x0 + h, y0 + h4 * ( k1 - 12.0 * k2 + 15.0 * k3 ) );
k5 = (*f)(x0 + h2_3, y0
+ h81 * ( 6.0 * k1 + 90.0 * k2 - 50.0 * k3 + 8.0 * k4 ) );
k6 = (*f)(x0 + h4_5, y0 + h75 * ( 6.0 * k1 + 36.0 * k2 + 10.0 * k3
+ 8.0 * k4 ) );
y0 += h192 * ( 23.0 * k1 + 125.0 * k3 - 81.0 * k5 + 125.0 * k6 );
x0 += h;
}
return y0;
}
////////////////////////////////////////////////////////////////////////////////
// double Runge_Kutta_Nystrom_Richardson( double (*f)(double, double), //
// double y0, double x0, double h, int number_of_steps, //
// int richarson_columns); //
// //
// Description: //
// This routine uses the Runge-Kutta-Nystrom method described above //
// together with Richardson extrapolation to approximate the solution at //
// x = x0 + h * number_of_steps of the initial value problem y'=f(x,y), //
// y(x0) = y0. //
// //
// Arguments: //
// double *f //
// Pointer to the function which returns the slope at (x,y) of the //
// integral curve of the differential equation y' = f(x,y) which //
// passes through the point (x0,y0). //
// double y0 //
// The initial value of y at x = x0. //
// double x0 //
// The initial value of x. //
// double h //
// The step size. //
// int number_of_steps //
// The number of steps. Must be nonnegative. //
// int richardson_columns //
// The maximum number of columns to use in the Richardson //
// extrapolation to the limit. //
// //
// Return Values: //
// The solution of the initial value problem y' = f(x,y), y(x0) = y0 at //
// x = x0 + number_of_steps * h. //
// //
////////////////////////////////////////////////////////////////////////////////
// //
static const double richardson[] = {
1.0 / 31.0, 1.0 / 63.0, 1.0 / 127.0, 1.0 / 255.0, 1.0 / 511.0, 1.0 / 1023.0
};
#define MAX_COLUMNS 1+sizeof(richardson)/sizeof(richardson[0])
#define max(x,y) ( (x) < (y) ? (y) : (x) )
#define min(x,y) ( (x) < (y) ? (x) : (y) )
// //
////////////////////////////////////////////////////////////////////////////////
// //
double Runge_Kutta_Nystrom_Richardson( double (*f)(double, double), double y0,
double x0, double h, int number_of_steps, int richardson_columns ) {
double dt[MAX_COLUMNS]; // dt[i] is the last element in column i.
double integral, delta, h_used;
int i,j,k, number_sub_intervals;
richardson_columns = max(1, min(MAX_COLUMNS, richardson_columns));
while ( --number_of_steps >= 0 ) {
h_used = h;
number_sub_intervals = 1;
for (j = 0; j < richardson_columns; j++) {
integral = Runge_Kutta_Nystrom( f, y0, x0, h_used,
number_sub_intervals);
for ( k = 0; k < j; k++) {
delta = integral - dt[k];
dt[k] = integral;
integral += richardson[k] * delta;
}
dt[j] = integral;
h_used *= 0.5;
number_sub_intervals += number_sub_intervals;
}
y0 = integral;
x0 += h;
}
return y0;
}
////////////////////////////////////////////////////////////////////////////////
// void Runge_Kutta_Nystrom_Integral_Curve( double (*f)(double, double), //
// double y[], double x0, double h, int number_of_steps_per_interval, //
// int number_of_intervals ); //
// //
// Description: //
// This routine uses the Runge-Kutta-Nystrom method described above to //
// approximate the solution of the differential equation y'=f(x,y) with //
// the initial condition y = y[0] at x = x0. The values are returned in //
// y[n] which is the value of y evaluated at x = x0 + n * m * h, where m //
// is the number of steps per interval and n is the interval number, //
// 0 <= n <= number_of_intervals. //
// //
// Arguments: //
// double *f //
// Pointer to the function which returns the slope at (x,y) of the //
// integral curve of the differential equation y' = f(x,y) which //
// which passes through the point (x0,y[0]). //
// double y[] //
// On input y[0] is the initial value of y at x = x0. On output //
// for n >= 1, y[n] is the appproximation of the solution y(x) of //
// the initial value problem y' = f(x,y), y(x0) = y[0], at //
// x = x0 + nmh, where m is the number of steps per interval and n //
// is the interval number. //
// double x0 //
// Initial value of x. //
// double h //
// Step size //
// int number_of_steps_per_interval //
// The number of steps of length h used to calculate y[i+1] //
// starting from y[i]. //
// int number_of_intervals //
// The number of intervals, y[] should be dimensioned at least //
// number_of_intervals + 1. //
// //
// Return Values: //
// This routine is of type void and hence does not return a value. //
// The solution of y' = f(x,y) from x = x0 to x = x0 + n * m * h, //
// where n is the number of intervals and m is the number of steps per //
// interval, is stored in the input array y[]. //
// //
////////////////////////////////////////////////////////////////////////////////
// //
void Runge_Kutta_Nystrom_Integral_Curve( double (*f)(double, double),
double y[], double x0, double h, int number_of_steps_per_interval,
int number_of_intervals ) {
double k1, k2, k3, k4, k5, k6;
double h3 = one_third * h;
double h4 = one_fourth * h;
double h25 = one_twentififth * h;
double h81 = one_eightyfirst * h;
double h2_5 = two_fifths * h;
double h2_3 = two_thirds * h;
double h4_5 = four_fifths * h;
double h75 = one_seventyfifth * h;
double h192 = one_oneninetytwo * h;
int i;
while ( --number_of_intervals >= 0 ) {
*(y+1) = *y;
y++;
for (i = 0; i < number_of_steps_per_interval; i++) {
k1 = (*f)(x0,*y);
k2 = (*f)(x0 + h3, *y + h3 * k1);
k3 = (*f)(x0 + h2_5, *y
+ h25 * ( 4.0 * k1 + 6.0 * k2 ) );
k4 = (*f)(x0 + h, *y + h4 * ( k1 -12.0 * k2 + 15.0 * k3 ) );
k5 = (*f)(x0 + h2_3, *y
+ h81 * ( 6.0 * k1 + 90.0 * k2 - 50.0 * k3 + 8.0 * k4 ) );
k6 = (*f)(x0 + h4_5, *y + h75 * ( 6.0 * k1 + 36.0 * k2 + 10.0 * k3
+ 8.0 * k4 ) );
*y += h192 * ( 23.0 * k1 + 125.0 * k3 - 81.0 * k5 + 125.0 * k6 );
x0 += h;
}
}
}
////////////////////////////////////////////////////////////////////////////////
// void Runge_Kutta_Nystrom_Richardson_Integral_Curve( //
// double (*f)(double, double), double y[], double x0, double h, //
// int number_of_steps_per_interval, int number_of_intervals, //
// int richardson_columns ) //
// //
// Description: //
// This routine uses the Runge-Kutta-Nystrom method described above //
// together with Richardson extrapolation to the limit (as h -> 0) to //
// approximate the solution of the differential equation y'=f(x,y) with //
// the initial condition y = y[0] at x = x0. //
// //
// Arguments: //
// double *f //
// Pointer to the function which returns the slope at (x,y) of the //
// integral curve of the differential equation y' = f(x,y) which //
// which passes through the point (x0,y[0]). //
// double y[] //
// On input y[0] is the initial value of y at x = x0. On output //
// for n >= 1, y[n] is the appproximation of the solution y(x) of //
// the initial value problem y' = f(x,y), y(x0) = y[0], at //
// x = x0 + nmh, where m is the number of steps per interval and n //
// is the interval number. //
// double x0 //
// Initial value of x. //
// double h //
// Step size //
// int number_of_steps_per_interval //
// The number of steps of length h used to calculate y[i+1] //
// starting from y[i]. //
// int number_of_intervals //
// The number of intervals, y[] should be dimensioned at least //
// number_of_intervals + 1. //
// int richardson_columns //
// The maximum number of columns to use in the Richardson //
// extrapolation to the limit. //
// //
// Return Values: //
// This routine is of type void and hence does not return a value. //
// The solution of y' = f(x,y) from x = x0 to x = x0 + n * m * h, //
// where n is the number of intervals and m is the number of steps per //
// interval, is stored in the input array y[]. //
// //
////////////////////////////////////////////////////////////////////////////////
// //
void Runge_Kutta_Nystrom_Richardson_Integral_Curve( double (*f)(double, double),
double y[], double x0, double h, int number_of_steps_per_interval,
int number_of_intervals, int richardson_columns ) {
double mh = (double) number_of_steps_per_interval * h;
while ( --number_of_intervals >= 0 ) {
*(++y) = Runge_Kutta_Nystrom_Richardson( f, *y, x0, h,
number_of_steps_per_interval, richardson_columns );
x0 += mh;
}
return;
}