# File:GMRESm.f90

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

## The GMRES(m) Method

This implements the classic GMRES(m) method for solving the system \[A\vec{x}=\vec{b}\] for $\vec{x}$. This implementation minimises the error $\|A\vec{x}-\vec{b}\|$ subject to the additional constraint $\|\vec{x}\|<\delta$. (This constraint may be ignored by supplying $\delta<0$ in the implementation.)

The main advantage of the GMRES method is that it only requires calculations of multiplies by $A$ for a given $\vec{x}$ -- it does not need to know $A$ itself. This means that $A$ need not even be stored, and could correspond to a very complex linear 'action' on $\vec{x}$, e.g. a time integral with initial condition $\vec{x}$. For a given starting vector $\vec{x}_0$, the method seeks solutions for $\vec{x}$ in $\mathrm{span}\{\vec{x}_0,\,A\vec{x}_0,\,A^2\vec{x}_0,...\}$, but uses Gram-Schmidt orthogonalisation to improve the suitability of this basis set. The set of orthogonalised vectors is called the Krylov-subspace, and m is the maximum number of vectors stored.

Whereas m is traditionally a small number, e.g. 3 or 4, the added constraint renders restarts difficult. If the constraint is important, then m should be chosen sufficiently large to solve within m iterations.

GMRES is closely related to the Arnoldi method. See the remarks at File:Arnoldi.f on improved suitability via timestepping or exponentiation of the matrix.

## The code

To download, click the link above. The Lapack package is also required.

This constraint $\|\vec{x}\|<\delta$ may be ignored by supplying negative '`del`'.

In addition to scalar and array variables, the routine needs to be passed

- an external function that calculates dot products,
- an external subroutine that calculates the action of multiplication by $A$,
- an external 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. $M=I$.

The functions above may require auxiliary data in addition to $\vec{x}$ or $\vec{\delta x}$. Place this data in a module and access via '`use mymodule`' in the function/subroutine.

## Parallel use

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

- let each thread pass its subsection for 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.

## File history

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Date/Time | Dimensions | User | Comment | |
---|---|---|---|---|

current | 03:30, 13 December 2016 | (5 KB) | Apwillis (Talk | contribs) | Solve Ax=b for x, subject to constraint |x|<delta. |

- You cannot overwrite this file.

## File usage

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