MUMPS.jl documentation
MUMPS is a library for the solution of large linear systems using a factorization. Structure can be exploited, such as symmetry, or symmetry and definiteness. The factorization and solve phases can be performed in parallel via MPI by way of MPI.jl.
How to Install
julia> ]
pkg> add MUMPS
pkg> test MUMPS
How to Use
The main data type holding information on a factorization is Mumps
. Remember to initialize MPI before attempting to create a Mumps
object. A simple session is as follows:
using MUMPS, MPI, SparseArrays, LinearAlgebra
MPI.Init()
A = sprand(10, 10, 0.2) + I
rhs = rand(10)
x = solve(A, rhs)
norm(x - A \ rhs) / norm(x)
MPI.Finalize()
It is possible to separate the initialization, the analysis/factorization, and the solve phases. It is also possible to access the information reported by MUMPS after the factorization and solve phases, and to modify this information (e.g., to perform iterative refinement).
When creating an instance of a Mumps
object explicitly, it is important to specify in advance what arithmetic should be used. Single and double precision real (Float32
and Float64
) and complex (ComplexF32
and ComplexF64
) arithmetics are supported.
For instance,
MPI.Init()
mumps = Mumps{Float64}(mumps_unsymmetric, default_icntl, default_cntl64)
A = sparse(rand(4,4))
rhs = rand(4)
associate_matrix!(mumps, A)
factorize!(mumps)
associate_rhs!(mumps, rhs)
solve!(mumps)
x = get_solution(mumps)
finalize(mumps)
MPI.Finalize()
Once the arithmetic of the Mumps
instance has been specified, it cannot be changed. The module is flexible in that various data types may be used to define the matrix to be factorized and the right-hand side, and appropriate conversions will take place. Dense matrices may be used, and they will be converted to sparse format.
For intance,
mumps = Mumps{ComplexF64}(mumps_unsymmetric, default_icntl, default_cntl64)
A = rand(Int16, 4, 4)
rhs = rand(Float32, 4)
associate_matrix!(mumps, A) # A is converted to a sparse ComplexF64 matrix
associate_rhs!(mumps, rhs) # rhs is converted to a Complex64 vector
See test for more examples.
Constants and Methods Exposed
Constants
The following convenience constants may be used when initializing a Mumps
object:
Constant | Meaning |
---|---|
mumps_unsymmetric | matrix is general unsymmetric (or symmetry is unknown) |
mumps_definite | matrix is symmetric and (positive or negative) definite |
mumps_symmetric | matrix is symmetric but indefinite (or definiteness is unknown) |
default_icntl | array of default integer parameters |
default_cntl32 | array of default real parameters in single precision |
default_cntl64 | array of default real parameters in double precision |
See Section 5 of the MUMPS User's Manual for a description of the integer and real control arrays.
Methods
A Mumps
object is created using the default constructor, which must be supplied with:
- the data type for the arithmetic to be used, as a type parameter, i.e.,
Mumps{Float64}(...)
orMumps{ComplexF64}(...)
sym
: one of the constantsmumps_unsymmetric
,mumps_definite
ormumps_symmetric
. Note that there is no support for Hermitian complex matrices in MUMPS. Therefore, we recommend to always usemumps_unsymmetric
for complex data.icntl
: an integer parameters array (see the MUMPS Users's Manual)cntl
: a real parameters array (see the MUMPS Users's Manual)
The convenience function get_icntl()
returns an array of integer parameters corresponding to certain commonly-used options. Its arguments are all optional:
det
: a boolean indicating whether the determinant should be computed (default:false
)verbose
: a boolean (default:false
)ooc
: a boolean indicating whether factors should be stored out of core (default:false
)itref
: the number of iterative refinement steps (default: 0).
A Mumps
object is destroyed by calling the finalize()
method. Because finalize
still issues MPI commands, it is important to call finalize()
before calling MPI.Finalize()
.
Method | Description |
---|---|
finalize | Finalize a Mumps object. Must be done before calling MPI.Finalize() |
associate_matrix! | Register a matrix with the Mumps object. This function makes it possible to define the data on the host only. |
factorize! | Factorize the matrix registered with the Mumps object. |
associate_rhs! | Register right-hand sides with the Mumps object. This function makes it possible to define the data on the host only. |
solve! | Solve the linear system for the given right-hand side. |
get_solution | Retrieve the solution from the Mumps object. This function makes it possible for the solution to be assembled on the host only. |
Parallel Execution
MPI is controled by way of MPI.jl. Look for the lines that say NUMBER OF WORKING PROCESSES
in the output of
mpirun -np 4 julia examples/mumps_mpi.jl
Custom Installation
Note: MUMPS is already precompiled with Yggdrasil for all platforms except Windows.
To use your custom MUMPS, set the environment variable JULIA_MUMPS_LIBRARY_PATH
to point to the shared library before using MUMPS
. Note that the same version of MUMPS as used by the MUMPS_jll
artifact is needed.
For example, macOS users may install precompiled MUMPS binaries from the Homebrew tap dpo/mumps-jl
as follows:
brew tap dpo/mumps-jl
brew install mpich-mumps
export JULIA_MUMPS_LIBRARY_PATH=$(brew --prefix)/opt/mpich-mumps/lib
Apple Silicon users should remember to use arch x86_64 brew
to refer to Intel binaries run through Rosetta, as we do not (yet) ship Silicon binaries of MUMPS via Homebrew.
The JULIA_MUMPS_LIBRARY_PATH
environment variable may be set permanently in the shell's startup file, or in $HOME/.julia/config/startup.jl
.