Stirling and Pulse-tube Cryo-coolers

Appendix c

Modified Newton–Raphson method

Consider function f of x with root x₀ at f (x) = 0, and with an approximation, x_app, 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 x₀ – x_app explicit

More rapid convergence might be expected when the first three terms are taken into account. These represent a quadratic equation in the unknown x₀ – x_app, 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.