Trapezoidal Method
The trapeziodal method is an implicit method for approximating the solution of the initial value problem y' = f(x,y); y(x_{0}) = y_{0} at x for a given step size h. The trapezoidal method can be derived by expanding y(x) in a Taylor seriesy(x+h) = y(x) + h y'(x) + (h^{2}/2) y''(x) + O(h^{3})
and then subsituting (y'(x+h) - y(x))/h for y''(x) followed by substituting f(x,y) for y'(x). The approximation for y(x+h) is then given implicitly by
y(x+h) = y(x) + (h / 2) (f(x,y(x)) + f(x+h,y(x+h))).
ory(x+h) - (h / 2) f(x+h,y(x+h)) = y(x) + (h / 2) f(x,y(x)).
In order to use the trapezoidal method, the user must program a function g whose value is that value of y such that
y(x+h) - h' f(x+h,y(x+h)) = u,
where h'=h / 2 and u = y(x) + h' f(x,y(x)), i.e.
g(x+h,h',u) - h' f(x+h,g(x+h,h',u)) = u.
Locally the trapezoidal method is a third order method and therefore globally a second order method.
The trapezoidal method is a stable and convergent method with region of absolute stability
| (2 + µh) / (2 - µh) | < 1
in the complex µh plane.
As a rule, the trapezoidal method is convenient if the function g(x,h,u) is simple.
Function List
- double Trapezoidal_Method( double (*f)(double, double), double (*g)(double,double,double), double y0, double x0, double h, int number_of_steps )
This function uses the trapezoidal method to return the estimate of the solution of the initial value problem, y' = f(x,y); y = y0 when x = x0, at x0 + nh where n is the number_of_steps and h is the step size. The user supplied function g(x,h,u) returns the value of y such that y - h * f(x,y) = u (here h is the step size / 2).
- void Trapezoidal_Integral_Curve( double (*f)(double, double), double (*g)(double,double,double), double y[ ], double x0, double h, int number_of_steps_per_interval, int number_of_intervals )
This function uses the trapezoidal method to estimate the solution, y[n], of the initial value problem, y' = f(x,y); y = y[0] when x = x0, at x0+nmh where n is the interval number n = 1, ..., number_of_intervals, m is the number_of_steps_per_interval and h is the step size. Note that on input y[0] is the user-specified initial value at x = x0. The user supplied function g(x,h,u) returns the value of y such that y - h * f(x,y) = u (here h is step size / 2).
C Source Code
- The file, trapezoidal_method.c, contains versions of Trapezoidal_Method( ) and Trapezoidal_Integral_Curve( ) written in C.