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/*
---
Algo9-4.c C program for implementing Algorithm 9.4
Algorithm translated to C by: Dr. Norman Fahrer
IBM and Macintosh verification by: Daniel Mathews

NUMERICAL METHODS: C Programs, (c) John H. Mathews 1995
To accompany the text:
NUMERICAL METHODS for Mathematics, Science and Engineering, 2nd Ed, 1992
Prentice Hall, Englewood Cliffs, New Jersey, 07632, U.S.A.
Prentice Hall, Inc.; USA, Canada, Mexico ISBN 0-13-624990-6
Prentice Hall, International Editions: ISBN 0-13-625047-5
This free software is compliments of the author.
E-mail address: in%"mathews@fullerton.edu"

Algorithm 9.4 (Runge-Kutta Method of Order 4).
Section 9.5, Runge-Kutta Methods, Page 460
---
• /
/*
---

Algorithm 9.4 (Runge-Kutta Method of Order 4).

To approximate the solution of the initial value problem y' = f(t,y)
with y(a) = y_0 over [a,b] by using the formula

h
y_(k+1) = y_k + --- [K_1 + 2*K_2 + 2*K_3 + K_4]
6

User has to supply functions named : ffunction
An example is included in this program.

---
• /

5. include<stdio.h>
6. include<stdlib.h>
7. include<math.h>

8. define MAX 500

/* define prototype for USER-SUPPLIED function f(x) */

double ffunction(double t, double y);

/* EXAMPLE for "f1function" */

double ffunction(double t, double y)

{
return ( (t - y) / 2.0 );
}

/* --- */

/* Main program for algorithm 9.4 */

void main(void)

{
int J; /* Loop counter */
int M; /* INPUT: Number of steps */
double A, B, Y[MAX]; /* Endpoints and inital value */
double H; /* Compute the step size */
double T[MAX];
double K1, K2, K3, K4; /* Function values */
double t, y;

printf("--- Taylor's Method ---n");
printf("--- Example 9.10 on page 454 ---n");
printf("---n");
printf("Please enter endpoints of the interval [A,B]:n");
printf("For used Example type : 0, 3n");
scanf("%lf, %lf", &A, &B);
printf("You entered [%lf, %lf]n", A, B);
printf("---n");
printf("Please enter number of steps: (Not more than 500 !)n");
scanf("%d",&M);
if(M > MAX)
{
printf(" Not prepared for more than %d steps. Terminating. Sorryn",MAX);
exit(0);
}

printf("---n");
printf("Please enter initial value Y[0] :n");
printf("For used Example type : 1n");
scanf("%lf", &Y[0]);
printf("You entered Y[0] = %lfn", Y[0]);

/* Compute the step size */

H = (B - A) / M;

/* Initialize the variable */

T[0] = A;

for(J = 0; J <= M-1; J++)
{
t = T[J];
y = Y[J];
K1 = H * ffunction(t,y);
K2 = H * ffunction(t + H/2.0, y + 0.5 * K1);
K3 = H * ffunction(t + H/2.0, y + 0.5 * K2);
K4 = H * ffunction(t + H, y + K3);
Y[J+1] = y + ( K1 + 2.0 * K2 + 2.0 * K3 + K4 ) / 6.0;
T[J+1] = A + H * (J+1);
}

/* Output */

for ( J = 0; J <= M; J++)
{
printf("J = %d, T[J] = %lf, Y[J] = %lfn", J, T[J], Y[J]);
}

} /* End of main program */