Improved Euler's Method
Description
The improved Euler 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), twice for step. For step i+1, yi+1 = yi + k2 , where k1 = h * f(xi, yi),k2 = h * f(xi + h / 2, yi + k1 / 2 ), and xi = x0 + i h.
The improved Euler method is a second order procedure for which Richardson extrapolation can be used.
Function List
Functions:
- double Improved_Euler_Method( double (*f)(double, double), double y0, double x0, double h, int number_of_steps )
This function uses the improved Euler 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 Improved_Euler_Method_Richardson( double (*f)(double, double), double y0,
double x0, double h, int number_of_steps, int richardson_columns )
This function uses the improved Euler 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 Improved_Euler_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 the improved Euler 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 Improved_Euler_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 the improved Euler 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: improved_euler_method.c //
// Routines: //
// Improved_Euler_Method //
// Improved_Euler_Method_Richardson //
// Improved_Euler_Integral_Curve //
// Improved_Euler_Richardson_Integral_Curve //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// //
// Description: //
// The improved Euler method is a Runge-Kutta method for approximating //
// the solution of the differential equation y'(x) = f(x,y) with initial //
// condition y(x0) = c. The improved Euler method evaluates f(x,y) twice //
// for each step. For step i+1, y[i+1] = y[i] + k2, where //
// k1 = h * f(x[i],y[i]), //
// k2 = h * f(x[i]+h/2,y[i]+k1/2), and x[i] = x0 + i * h. //
// //
// The improved Euler method is a second order method for which //
// Richardson extrapolation can be used. //
// //
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// double Improved_Euler_Method( double (*f)(double, double), double y0, //
// double x0, double h, int number_of_steps ); //
// //
// Description: //
// This routine uses the improved Euler method 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 Improved_Euler_Method( double (*f)(double, double), double y0,
double x0, double h, int number_of_steps ) {
double k1;
double h2 = 0.5 * h;
while ( --number_of_steps >= 0 ) {
k1 = h2 * (*f)(x0,y0);
y0 += h * (*f)(x0+h2, y0 + k1);
x0 += h;
}
return y0;
}
////////////////////////////////////////////////////////////////////////////////
// double Improved_Euler_Method_Richardson( double (*f)(double, double), //
// double y0, double x0, double h, int number_of_steps, //
// int richardson_columns) //
// //
// Description: //
// This routine uses the improved Euler method 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 / 3.0, 1.0 / 7.0, 1.0 / 15.0, 1.0 / 31.0, 1.0 / 63.0
};
#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 Improved_Euler_Method_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 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 = Improved_Euler_Method( 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 Improved_Euler_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 improved Euler method 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 Improved_Euler_Integral_Curve( double (*f)(double, double), double y[],
double x0, double h, int number_of_steps_per_interval,
int number_of_intervals ) {
double k1;
double h2 = 0.5 * h;
int i;
while ( --number_of_intervals >= 0 ) {
*(y+1) = *y;
y++;
for (i = 0; i < number_of_steps_per_interval; i++) {
k1 = h2 * (*f)(x0,*y);
*y += h * (*f)(x0+h2, *y + k1);
x0 += h;
}
}
}
////////////////////////////////////////////////////////////////////////////////
// void Improved_Euler_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 improved Euler method 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. //
// The values are returned in y[], in which y[n] 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. //
// 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 Improved_Euler_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) = Improved_Euler_Method_Richardson( f, *y, x0, h,
number_of_steps_per_interval, richardson_columns );
x0 += mh;
}
return;
}