Mathematics Source Library
C & ASM


Basforth 9 - Moulton 8 Steps

Adams-Bashforth 9 Steps Method
Adams-Moulton 8 Steps Method

The Adams-Bashforth 9 steps method and Adams-Moulton 8 steps method form a predictor-corrector multistep procedure for approximating the solution of a differential equation given historical values.

Function List

  • int Adams_9_Steps( double (*f)(double, double), double y[ ], double x0, double h, double f_history[ ], double *y_bashforth, double tolerance, int iterations )

    This function uses the Adams-Bashforth method and Adams-Moulton method to estimate the solution of the initial value problem, y' = f(x,y); y(x0) = y[0], at x = x0 + h, where h is the step size and f_history[ ] is the array containing the values of the function f(x,y) evaluated at the starting values f(x0 - i*h,y(x0 - i*h)) for i = 1, …, 8. The Adams-Moulton method terminates either when successive estimates are within tolerance of each other or when the number of iterations exceeds iterations. Upon return, the y[1] contains the estimate of y(x0 + h), y_bashforth contains the estimate of y(x0 + h) using the Adams-Bashforth algorithm, the array f_history[ ] has been updated for a subsequent call to this function, and the function itself returns the number of iterations used in the Adams-Mouton procedure.

  • double Adams_Bashforth_9_Steps( double y, double h, double f_history[ ] )

    This function uses Adams-Bashforth 9 step method to return the estimate of the solution of the initial value problem, y' = f(x,y); y(x0) = y[0], at x = x0 + h, where h is the step size and f_history[ ] is the array containing the values of the function f(x,y) evaluated at the starting values f(x0 - i*h,y(x0 - i*h)) for i = 0, …, 8.

  • int Adams_Moulton_8_Steps( double (*f)(double, double), double y[ ], double x, double h, double f_history[ ], double tolerance, int iterations )

    This function uses the Adams-Moulton 8 step method to estimate the solution of the initial value problem, y' = f(x,y); y(x0) = y[0], at x = x0 + h, where h is the step size and f_history[ ] is the array containing the values of the function f(x,y) evaluated at the starting values f(x0 - i*h,y(x0 - i*h)) for i = 1, . . ., 8. The Adams-Moulton method terminates either when successive estimates are within tolerance of each other or when the number of iterations exceeds iterations. Upon return, the y[1] contains the estimate of y(x0 + h), and the function itself returns the number of iterations used in the Adams-Mouton procedure.

  • void Adams_9_Build_History(double (*f)(double,double), double f_history[ ], double y[ ], double x, double h)

    This function builds the f_history[ ] array using the historical values y[i] = y(x + i*h) for
    i = 0, …, 7. The ith element of the f_history[ ] array is then
    f_history[i] = f( x-(8-i)*h, y(x-(8-i)*h) ), i = 0, …, 7.

C Source Code

  • The file, adams_9_steps.c, contains the versions of Adams_9_Steps( ), Adams_Bashforth_9_Steps( ), Adams_Moulton_8_Steps( ), and Adams_9_Build_History( ) written in C.