The Pardiso.jl package provides an interface for using Panua Pardiso, it's predecessors from pardiso-project.org, and Intel MKL PARDISO from the Julia language.

You cannot use Pardiso.jl without either having a valid license for Panua Pardiso or having the MKL library installed. This package is available free of charge and in no way replaces or alters any functionality of the linked libraries.

The package itself is installed with Pkg.add("Pardiso") but you also need to follow the installation instructions below to install a working PARDISO library.

By default, when adding "Pardiso.jl" to the active environmnent, Julia will automatically install a suitable MKL for your platform by loading MKL_jll.jl. Note that if you use a mac you will need to pin MKL_jll to version 2023.

If you instead use a self installed MKL, follow these instructions:

• Set the MKLROOT environment variable. See the MKL set environment variables manual for a thorough guide how to set this variable correctly, typically done by executing something like source /opt/intel/oneapi/setvars.sh intel64 or running "C:\Program Files (x86)\IntelSWTools\compilers_and_libraries\windows\mkl\bin\mklvars.bat" intel64
• Run Pkg.build("Pardiso", verbose=true)
• Eventually, run Pardiso.show_build_log() to see the build log for additional information.
• Note that the MKLROOT environment variable must be set, and LD_LIBRARY_PATH must contain $MKLROOT/lib whenever using the library this way.

• Unzip the download file panua-pardiso-yyyymmdd-os.zip to some folder and set the environment variable JULIA_PARDISO to the lib subdirectory of this folder. For example, create an entry ENV["JULIA_PARDISO"] = "/Users/Someone/panua-pardiso-yyyymmdd-os/lib" in .julia/config/startup.jl. If you have a valid license for the predecessor from pardiso-project.org, put the PARDISO library to a subdirectory denoted by ENV["JULIA_PARDISO"] and evenutally rename it to libpardiso.so.
• Perform the platform specific steps described below
• Run Pkg.build("Pardiso", verbose=true)
• Eventually, run Pardiso.show_build_log() to see the build log for additional information.

Note: In the past, weird errors and problems with MKL Pardiso had been observed when PanuaPardiso is enabled (likely because some library that is needed by PanauaPardiso was problematic with MKL). In that case, if you want to use MKL Pardiso it is better to just disable PanuaPardiso by not setting the environment variable JULIA_PARDISO (and rerunning Pkg.build("Pardiso")).

• Make sure that the version of gfortran corresponding to the pardiso library is installed.
• Make sure OpenMP is installed.
• Install a (fast) installation of a BLAS and LAPACK (this should preferably be single threaded since PARDISO handles threading itself), using for example OpenBLAS

gfortran and OpenMP usually come with recent version of gcc/gfortran. On Linux, Panua Pardiso looks for libraries libgfortran.so and libgomp.so . They may be named differently on your system. In this situation you may try to create links to them with names known to Pardiso.jl (bash; pathnames serve as examples here):

$ mkdir $HOME/extralibs
$ ln -s /usr/lib64/libgomp.so.1 $HOME/extralibs/libgomp.so
$ ln -s /usr/lib64/libgfortran.so.5 $HOME/extralibs/libgfortran.so
$ export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:$HOME/extralibs/

This section will explain how to solve equations using Pardiso.jl with the default settings of the library. For more advanced users there is a section further down.

A PardisoSolver is created with PardisoSolver() for solving with PanuaPardiso or MKLPardisoSolver() for solving with MKL PARDISO. This object will hold the settings of the solver and will be passed into the solve functions. In the following sections an instance of a PardisoSolver or an MKLPardisoSolver() will be referred to as ps.

Solving equations is done with the solve and solve! functions. They have the following signatures:

• solve(ps, A, B) solves AX=B and returns X
• solve!(ps, X, A, B) solves AX=B and stores it in X

The symbols :T or :C can be added as an extra argument to solve the transposed or the conjugate transposed system of equations, respectively.

Here is an example of solving a system of real equations with two right-hand sides:

ps = PardisoSolver()

A = sparse(rand(10, 10))
B = rand(10, 2)
X = zeros(10, 2)
solve!(ps, X, A, B)

which happened to give the result

julia> X
10x2 Array{Float64,2}:
 -0.487361  -0.715372
 -0.644219  -3.38342
  0.465575   4.4838
  1.14448   -0.103854
  2.00892   -7.04965
  0.870507   1.7014
  0.590723  -5.74338
 -0.843841  -0.903796
 -0.279381   7.24754
 -1.17295    8.47922

Given a partitioned matrix M = [A B; C D], the Schur complement of A in M is S = D-CA⁻¹B. This can be found with the function schur_complement with the following signatures:

• schur_complement(ps, M, n) returns Schur complement of submatrix A in M, where n is the size of submatrix D (and therefore also of Schur complement)
• schur_complement(ps, M, x) returns Schur complement of submatrix A in M, where submatrix D is defined by nonzero rows of SparseVector or SparseMatrix x.

The symbols :T or :C can be added as an extra argument to solve the transposed or the conjugate transposed system of equations, respectively.

Here is an example of finding the Schur complement:

ps = PardisoSolver()
m = 100; n = 5; p = .5; T = Float64
rng = MersenneTwister(1234);
A = I + sprand(rng,T,m,m,p)
A⁻¹ = inv(Matrix(A))
B = sprand(rng,T,m,n,p)
C = sprand(rng,T,n,m,p)
D = sprand(rng,T,n,n,p)
M = [A B; C D]
S = schur_complement(ps,M,n)

which gives

julia> S
5×5 Array{Float64,2}:
  -0.121404    1.49473  -1.25965    7.40326    0.571538
 -19.4928     -7.71151  12.9496    -7.13646  -20.4194
   9.88029     3.35502  -7.2346     1.70651   13.9759
  -9.06094    -5.86454   7.44917   -2.54985   -9.17327
 -33.7006    -17.8323   20.2588   -19.5863   -37.6132

We can check the validity by comparing to explicity form:

julia> norm(D - C*A⁻¹*B - S)
5.033075778861378e-13

At present there seems to be an instability in the Schur complement computation for complex matrices.

The number of threads to use is set in different ways for MKL PARDISO and PanuaPardiso.

set_nprocs!(ps, i) # Sets the number of threads to use
get_nprocs(ps) # Gets the number of threads being used

The number of threads are set at the creation of the PardisoSolver by looking for the environment variable OMP_NUM_THREADS. This can be done in Julia with for example ENV["OMP_NUM_THREADS"] = 2. Note: OMP_NUM_THREADS must be set before Pardiso is loaded and can not be changed during runtime.

The number of threads used by a PardisoSolver can be retrieved with get_nprocs(ps)

This section discusses some more advanced usage of Pardiso.jl.

For terminology in this section please refer to the PanuaPardiso manual and the oneMKL PARDISO manual.

After using functionality in this section, calls should no longer be made to the solve functions but instead directly to the function

pardiso(ps, X, A, B)

This will ensure that the properties you set will not be overwritten.

If you want, you can use get_matrix(ps, A, T) to return a matrix that is suitable to use with pardiso depending on the matrix type that ps has set. The parameter T is a symbol representing if you will solve the normal, transposed or conjugated system. These are represented by :N, :T, :C) respectively.

For ease of use, Pardiso.jl provides enums for most options. These are not exported so has to either be explicitly imported or qualified with the module name first. It is possible to both use the enum as an input key to the options or the corresponding integer as given in the manuals.

The matrix type can be explicitly set with set_matrixtype!(ps, key) where the key has the following meaning:

The matrix type for a solver can be retrieved with get_matrixtype(ps).

PanuatPardiso supports direct and iterative solvers. The solver is set with set_solver!(ps, key) where the key has the following meaning:

Depending on the phase calls to solve (and pardiso which is mentioned later) does different things. The phase is set with set_phase!(ps, key) where key has the meaning:

Advanced users likely want to explicitly set and retrieve the IPARM and DPARM (PanuaPardiso only) parameters. This can be done with the getters and setters:

get_iparm(ps, i) # Gets IPARM[i]
get_iparms(ps) # Gets IPARM
set_iparm!(ps, i, v) # Sets IPARM[i] = v

# PanuaPardiso only
get_dparm(ps, i) # Gets DPARM[i]
get_dparms(ps) # Gets DPARM
set_dparm!(ps, i, v) # Sets DPARM[i] = v

To set the default values of the IPARM and DPARM call pardisoinit(ps). The default values depend on what solver and matrix type is set.

It is possible for Pardiso to print out timings and statistics when solving. This is done by set_msglvl!(ps, key) where key has the meaning:

PanuaPardiso comes with a few matrix and vector checkers to check the consistency and integrity of the input data. These can be called with the functions:

printstats(ps, A, B)
checkmatrix(ps, A)
checkvec(ps, B)

In MKL PARDISO this is instead done by setting IPARM[27] to 1 before calling pardiso.

These are set and retrieved with the functions

set_mnum!(ps, i)
get_mnum(ps)

set_maxfct!(ps, i)
get_maxfct(ps)

get_perm(ps)
set_perm!(ps, perm) # Perm is a Vector{Int}

The pardiso(ps,...) syntax can be used to compute the Schur compelement (as described below). The answer can be retrieved with pardisogetschur(ps).

To use the low-level API to compute the Schur complement:

• use custom IPARMS (set_iparm!(ps,1,1)), set the Schur complement block size to n (set_iparm!(ps,38,n)), and set the phase to analyze & factorize (set_phase!(ps,12)).
• compute the Schur complement by calling pardiso(ps,X,M,X), where B is a dummy vector with length(X)=size(M,1) that shares element type with M.
• retrieve with pardisogetschur(ps)

• Julia uses CSC sparse matrices while PARDISO expects a CSR matrix. These can be seen as transposes of each other so to solve AX = B the transpose flag (IPARAM[12]) should be set to 1.
• For symmetric matrices, PARDISO needs to have the diagonal stored in the sparse structure even if the diagonal element happens to be 0. The manual recommends adding an eps to the diagonal when you suspect you might have 0 values diagonal elements that are not stored in the sparse structure.
• Unless IPARM[1] = 1, all values in IPARM will be ignored and default values are used.
• When solving a symmetric matrix, Pardiso expects only the upper triangular part. Since Julia has CSC matrices this means you should pass in tril(A) to the pardiso function. Use checkmatrix to see that you managed to get the matrix in a valid format.

If you have suggestions or idea of improving this package, please file an issue or even better, create a PR!