Muller’s method is an iterative generalization of the secant method for locating the complex roots of a function. It doesn’t require derivative of the function. The C program for Muller’s method requires three initial guesses and, mathematically, the approximation is done by a second degree parabola passing through these points. (Derivation)
The programming effort for Muller’s Method in C language is a bit tedious as it requires three initial approximations. However, the method converges quadratically for almost all the initial guesses. Overall, this method is more effective for locating complex roots – it can effectively generate complex roots of the function, even though the initial approximations or guesses are real.
Muller’s method converges slower than the Newton-Raphson method, but faster than the secant method. The convergence is linear and the overall approach used in the method is quadratic interpolation of the three guesses.
Below is a short and simple source code in C program for Muller’s method to find the root of cos(x) – x*e^x .
Variables:
- itr – a counter which keeps track of the no. of iterations performed
- maxmitr – maximum number of iterations to be performed
f(x) = cos(x) – x*e^x
Source Code for Muller’s Method in C:
#include<stdio.h>
#include<math.h>
#define I 2
float f(float x)
{
return cos(x) - x*exp(x);
}
main ()
{
int i, itr, maxmitr;
float x[4], li, di, mu, s, l, allerr;
printf("\nEnter the three initial guesses:\n");
for (i=I-2; i<3; i++)
scanf("%f", &x[i]);
printf("Enter allowed error and maximum iterations:\n");
scanf("%f %d", &allerr, &maxmitr);
for (itr=1; itr<=maxmitr; itr++)
{
li = (x[I] - x[I-1]) / (x[I-1] - x[I-2]);
di = (x[I] - x[I-2]) / (x[I-1] - x[I-2]);
mu = f(x[I-2])*li*li - f(x[I-1])*di*di + f(x[I])*(di+li);
s = sqrt ((mu*mu - 4*f(x[I])*di*li*(f(x[I-2])*li - f(x[I-1])*di + f(x[I]))));
if (mu < 0)
l = (2*f(x[I])*di)/(-mu+s);
else
l = (2*f(x[I])*di)/(-mu-s);
x[I+1] = x[I] + l*(x[I] - x[I-1]);
printf("At iteration no. %3d, x = %7.5f\n", itr, x[I+1]);
if (fabs (x[I+1] - x[I]) < allerr)
{
printf("After %3d iterations, the required root is %6.4f\n", itr, x[I+1]);
return 0;
}
for (i=I-2; i<3; i++)
x[i] = x[i+1];
}
printf("The required solution does not converge or iterations are insufficient\n");
return 1;
}
Here, x is an array which holds the three initial guesses for the root and the new improved value obtained as I has been defined as 2 in the program. It is done in this way in this program because array subscripts always start from zero and are never negative. Further, the use of I facilitates more readable and understandable expressions.
Input/Output:
Also see,
Muller’s Method Algorithm/Flowchart
Numerical Methods Tutorial Compilation
Muller’s Method was developed by David Muller in 1956. Considered not that useful to find the real roots of simple functions, it is a very effective method for finding complex roots. You can compare the above result obtained from the Muller’s method C program with that obtained from the Regula-Falsi Method.
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