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Muller's Method

Generalizes the Secant Method of root finding by using quadratic 3-point interpolation

q\equiv {x_n-x_{n-1}\over x_{n-1}-x_{n-2}}.
\end{displaymath} (1)

Then define
$\displaystyle A$ $\textstyle \equiv$ $\displaystyle qP(x_n)-q(1+q)P(x_{n-1})+q^2P(x_{n-2})$ (2)
$\displaystyle B$ $\textstyle \equiv$ $\displaystyle (2q+1)P(x_n)-(1+q)^2P(x_{n-1})+q^2P(x_{n-2})$  
$\displaystyle C$ $\textstyle \equiv$ $\displaystyle (1+q)P(x_n),$ (4)

and the next iteration is
x_{n+1} = x_n-(x_n-x_{n-1}){2C\over\max(B\pm\sqrt{B^2-4AC}\,)}.
\end{displaymath} (5)

This method can also be used to find Complex zeros of Analytic Functions.


Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 364, 1992.

© 1996-9 Eric W. Weisstein