Implementation and Analysis of the Solution of Equations
Recall that solving an equation of the form f (x) = means finding its roots. The root operation locates the real root to which Newton’s method converges given an initial iteration point.
The root operation revolves around a single
loop (see Example 13-3), which
calculates successive approximations using the Newton iteration
formula. In the implementation presented here,
f is the function for which we are
approximating the root, and g is the
derivative of f. After each iteration, we
determine whether the current approximation of the root is
satisfactory. An approximation is deemed satisfactory when the
difference between it and that of the previous iteration is less than
delta. If after
n iterations a satisfactory root still has
not been found, root terminates.
The runtime complexity of root is O (n), where n is the maximum number of iterations the caller wishes to perform. The worst case occurs when we do not find the root we are looking for.
/*****************************************************************************
* *
* -------------------------------- root.c -------------------------------- *
* *
*****************************************************************************/
#include <math.h>
#include "nummeths.h"
/*****************************************************************************
* *
* --------------------------------- root --------------------------------- *
* *
*****************************************************************************/
int root(double (*f)(double x), double (*g)(double x), double *x, int *n,
double delta) {
int satisfied,
i;
/*****************************************************************************
* *
* Use Newton's method to find a root of f. *
* *
*****************************************************************************/
i = 0;
satisfied = 0;
while (!satisfied && i + 1 < *n) {
/**************************************************************************
* *
* Determine the next iteration of x. *
* *
**************************************************************************/
x[i + 1] = x[i] - (f(x[i]) / g(x[i]));
/**************************************************************************
* *
* Determine whether the desired approximation has been obtained. *
* *
**************************************************************************/
if (fabs(x[i + 1] - x[i]) < delta)
satisfied = 1;
/**************************************************************************
* *
* Prepare for the next iteration. *
* *
**************************************************************************/
i++;
}
/*****************************************************************************
* *
* Even without iterating, indicate that one value has been stored in x. *
* *
*****************************************************************************/
if (i == 0)
*n = 1;
else
*n = i + 1;
/*****************************************************************************
* *
* Return whether a root was found or the maximum iterations were reached. *
* *
*****************************************************************************/
if (satisfied)
return 0;
else
return -1;
}