Appendix c

Modified Newton–Raphson method

Consider function f of x with root x0 at f (x) = 0, and with an approximation, xapp, to x0. Expanding about the approximation


The standard Newton-Raphson method considers the first two terms of the expansion and inverts to make the unknown x0xapp explicit


More rapid convergence might be expected when the first three terms are taken into account. These represent a quadratic equation in the unknown x0xapp, with solutions


This must tend to the previous expression as ∂2f/∂x2 tends to zero, so that, by inspection on expanding the root by the binomial theorem, the positive option applies.

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