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Revision as of 03:48, 26 June 2019
$ \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 code uses the LAPACK package.
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
Click on a date/time to view the file as it appeared at that time.
Date/Time | Dimensions | User | Comment | |
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current | 03:48, 26 June 2019 | (5 KB) | Apwillis (talk | contribs) | Minor edits in comments only. |
03:30, 13 December 2016 | (5 KB) | Apwillis (talk | contribs) | Solve Ax=b for x, subject to constraint |x|<delta. |
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File usage
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