Modified Midpoint (Gragg's) Method

For approximating the solution of the initial value problem y ' = f(x, y), y(x₀) = y₀, the modified midpoint method begins by applying the midpoint method:



Then at a certain step N, where N is an even integer, the oscillating error term is damped by redefining y_N as


The method then proceeds as if solving the initial value problem y ' = f(x, y), y(x_N) = y_N .
The modified midpoint method is globally a second order method for which Richardson extrapolation may be applied to increase the order and accuracy.

Function List

• double Graggs_Method( double (*f)(double, double), double y0, double x0, double x, int number_of_steps )
  
  This function uses Gragg's modified midpoint method to return the estimate of the solution of the initial value problem, y' = f(x,y);  y = y0 when x = x0, at the argument x. The step size h used is given by h = ( x - x0 ) / n where n is the number_of_steps. The number_of_steps must be a positive even integer.
  
  
• double Graggs_Method_Richardson( double (*f)(double, double), double y0, double x0, double x, int number_of_steps, int richardson_columns )
  
  This function uses Gragg's modified midpoint method together with Richardson extrapolation to return the estimate of the solution of the initial value problem,
  y' = f(x,y); y = y0 when x = x0, at the argument x. The initial step size h used is given by h = ( x - x0 ) / n where n is the number_of_steps. The number_of_steps must be a positive even integer. 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, graggs_method.c, contains versions of Graggs_Method_Method( ) and Graggs_Method_Richardson( ) written in C.