File:NewtonHook.f90

$ \renewcommand{\vec}[1]{ {\bf #1} } \newcommand{\bnabla}{ \vec{\nabla} } \newcommand{\Rey}{Re} \def\vechat#1{ \hat{ \vec{#1} } } \def\mat#1{#1} $

The method

Implements the Newton-Raphson-Krylov method for solving \[\vec{F}(\vec{x})=\vec{0},\] where $\vec{x}$ is an $n$-vector, using a Hookstep-Trust-region approach.

For each iteration, $\vec{x}_{i+1} = \vec{x}_i + \vec{\delta x}$, the linear problem \[\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\vec{\delta x}=-\vec{x}_i \] is solved for the step $\vec{\delta x}$ subject to the condition that it be smaller than some magnitude $\delta$, where $\delta$ is the size of the trust region. This problem is in the form $A\vec{x}=\vec{b}$, and is solved using the Krylov-subspace method, GMRES(m). See File:GMRESm.f90.

The code

To download, click the link above.

new_x(:) and new_fx(:) need to be allocated the dimension $n$, and new_x(:) assigned the initial guess, prior to calling the subroutine newtonhook(...). The vector new_x(:) is updated with improved vectors (smaller new_f(:)) as the calculation proceeds.