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} $ Click link just above to download. The code requires File:GMRESm.f90 and LAPACK.

See also Template/Example for FORTRAN / MATLAB at Newton-Krylov_method

*** THE DESCRIPTION OF THE METHOD HAS MOVED TO Newton-Krylov_method ***

The Jacobian-free Newton-Krylov (JFNK) Method

The code above implements the Jacobian-free Newton-Krylov (JFNK) method for solving \begin{equation}\label{eq:ROOTF}\vec{F}(\vec{x})=\vec{0},\end{equation} where $\vec{x}$ and $\vec{F}$ are $n$-vectors, supplemented with a Hookstep--Trust-region approach.

The code

To download, click the links at top of page. A template for calling the routine: File:Newton template.f90. The code also requires File:GMRESm.f90 and LAPACK.

• Prior to calling the subroutine newtonhook(...), the arrays new_x(:)$\equiv\vec{x}_i$ and new_fx(:)$\equiv\vec{F}(\vec{x}_i)$ need to be allocated the dimension $n$.
• Also prior to calling newtonhook(...), new_x(:) must be assigned the initial guess $\vec{x}_0$. (As the calculation proceeds, the vector new_x(:) is updated with improved vectors, having smaller corresponding new_fx(:).)

• In addition to scalar and array variables, newtonhook(...) needs to be passed
  • a function that calculates the dot product of two vectors,
  • a function that calculates $\vec{F}(\vec{x})$ for a given $\vec{x}$,
  • a function that calculates $\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\,\vec{\delta x}$ for a given $\vec{\delta x}$. See possible approximation above, where $\vec{F}(\vec{x}_i)$ is already stored in new_fx(:).
  • a subroutine that replaces a vector $\vec{x}$ with the solution of $M\vec{x}'=\vec{x}$ for $\vec{x}'$, where $M$ is a preconditioner matrix. This may simply be an empty subroutine if no preconditioner is required, i.e. $M=I$.
  • a subroutine that is called at the end of each iteration. This may be used to save the current state after each iteration, if desired.

• The functions above may require auxiliary data, in addition to the given $\vec{x}$ or $\vec{\delta x}$. A few ways to access the data:
  • place the required data in a module and access via 'use mymodule'.
  • place the data in a common block.
  • write the vector to disk, call a script that runs an external programme, then load the result.

• Extra constraints may be necessary, e.g. that determine an update to the period of an orbit. The function that evaluates $\left.\frac{\vec{dF}}{\vec{dx}}\right|_{\vec{x}_i}\,\vec{\delta x}$ for a given $\vec{\delta x}$ should append to the result the evaluations of the constraints. Correspondingly, for each constraint an extra 0 should be appended to $\vec{F}(\vec{x}_i)$ (the evaluated constraint should equal zero when converged).

Parallel use

It is NOT necessary to edit this code for parallel (MPI) use:

• let each thread pass its subsection of the vector $\vec{x}$,
• make the dot product function mpi_allreduce the result of the dot product.
• to avoid multiple outputs to the terminal, set info=1 on rank 0 and info=0 for the other ranks.