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## Improved Euler's Method

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 each 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

• 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 Code

• The file, improved_euler_method.c, contains versions of Improved_Euler_Method( ), Improved_Euler_Method_Richardson( ), Improved_Euler_Integral_Curve( ), and Improved_Euler_Richardson_Integral_Curve( ) written in C.