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# Muller

Calculates the zeros of a function using Muller's method.
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Contents

C++

## Muller

 doublemuller( double (*f)(double)[function pointer] double x0 = 0 double d = 3 double eps = 1E-10 int maxit = 1000 )
This method extends the idea of the secant method which works with a linear polynomial, to a quadratic polynomial. Given three previous estimates ,  and , for an unknown root, a new value is computed by

where

and






The values  may yield too large changes for  which possibly leads to another root and causes slow convergence. This can be circumvented by allowing a fixed maximum increase of  from one iteration to another. Care must also be taken when computing  which is necessary to compute ,  and . If an estimate of  indicates a value greater than the maximum possible number, we choose

in place of the original relation, and repeat this until no overflow occurs. The algorithm stops whenever the actual value  is smaller than the smallest value  until now and

holds where  is some small number depending on the computer accuracy. To avoid a lot of iterations where the above condition fails, we allow only a fixed maximum number of iterations.

Convergence for this method is superlinear. However, it is one of those methods that will converge to both real and complex roots from a real initial approximation.

This algorithm finds the roots of the user-defined function f starting with an initial guess x0 and iterating the sequence above until either the accuracy eps is achieved or the maximum number of iterations maxit is exceeded. Another required parameter is the bound on the error of the initial guess d.

## References:

• Jean-Pierre Moreau's Home Page, http://perso.wanadoo.fr/jean-pierre.moreau/
• F.R. Ruckdeschel, "BASIC Scientific Subroutines", Vol. II, BYTE/McGRAWW-HILL, 1981
• Wankere R. Mekwi, "Iterative Methods for Roots of Polynomials", Exeter College, University of Oxford

### Example 1

#include <codecogs/maths/rootfinding/muller.h>

#include <iostream>
#include <iomanip>

// user-defined function
double f(double x) {
return sin(x);
}

int main()
{
double x = Maths::RootFinding::muller(f, 4);

std::cout << "The calculated zero is X = " << std::setprecision(15) << x << std::endl;
std::cout << "The associated ordinate value is Y = " << f(x) << std::endl;
return 0;
}
Output:
The calculated zero is X = 3.14159265358979
The associated ordinate value is Y = 1.22460635382238e-016

### Parameters

 f the user-defined function x0 Default value = 0 d Default value = 3 eps Default value = 1E-10 maxit Default value = 1000

### Authors

Lucian Bentea (August 2005)
##### Source Code

Source code is available when you buy a Commercial licence.

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