Sparse linear algebra » Solving Sparse Linear Systems module

List of sparse solvers

Eigen currently provides a wide set of built-in solvers, as well as wrappers to external solver libraries. They are summarized in the following tables:

Sparse solver concept

All these solvers follow the same general concept. Here is a typical and general example:

#include <Eigen/RequiredModuleName>
// ...
SparseMatrix<double> A;
// fill A
VectorXd b, x;
// fill b
// solve Ax = b
SolverClassName<SparseMatrix<double> > solver;
solver.compute(A);
if(solver.info()!=Success) {
  // decomposition failed
  return;
}
x = solver.solve(b);
if(solver.info()!=Success) {
  // solving failed
  return;
}
// solve for another right hand side:
x1 = solver.solve(b1);

For SPD solvers, a second optional template argument allows to specify which triangular part have to be used, e.g.:

#include <Eigen/IterativeLinearSolvers>

ConjugateGradient<SparseMatrix<double>, Eigen::Upper> solver;
x = solver.compute(A).solve(b);

In the above example, only the upper triangular part of the input matrix A is considered for solving. The opposite triangle might either be empty or contain arbitrary values.

In the case where multiple problems with the same sparsity pattern have to be solved, then the "compute" step can be decomposed as follow:

SolverClassName<SparseMatrix<double> > solver;
solver.analyzePattern(A);   // for this step the numerical values of A are not used
solver.factorize(A);
x1 = solver.solve(b1);
x2 = solver.solve(b2);
...
A = ...;                    // modify the values of the nonzeros of A, the nonzeros pattern must stay unchanged
solver.factorize(A);
x1 = solver.solve(b1);
x2 = solver.solve(b2);
...

The compute() method is equivalent to calling both analyzePattern() and factorize().

Each solver provides some specific features, such as determinant, access to the factors, controls of the iterations, and so on. More details are available in the documentations of the respective classes.

Finally, most of the iterative solvers, can also be used in a matrix-free context, see the following example.

The Compute Step

In the compute() function, the matrix is generally factorized: LLT for self-adjoint matrices, LDLT for general hermitian matrices, LU for non hermitian matrices and QR for rectangular matrices. These are the results of using direct solvers. For this class of solvers precisely, the compute step is further subdivided into analyzePattern() and factorize().

The goal of analyzePattern() is to reorder the nonzero elements of the matrix, such that the factorization step creates less fill-in. This step exploits only the structure of the matrix. Hence, the results of this step can be used for other linear systems where the matrix has the same structure. Note however that sometimes, some external solvers (like SuperLU) require that the values of the matrix are set in this step, for instance to equilibrate the rows and columns of the matrix. In this situation, the results of this step should not be used with other matrices.

Eigen provides a limited set of methods to reorder the matrix in this step, either built-in (COLAMD, AMD) or external (METIS). These methods are set in template parameter list of the solver : DirectSolverClassName<SparseMatrix<double>, OrderingMethod<IndexType> > solver;

See the OrderingMethods module for the list of available methods and the associated options.

In factorize(), the factors of the coefficient matrix are computed. This step should be called each time the values of the matrix change. However, the structural pattern of the matrix should not change between multiple calls.

For iterative solvers, the compute step is used to eventually setup a preconditioner. For instance, with the ILUT preconditioner, the incomplete factors L and U are computed in this step. Remember that, basically, the goal of the preconditioner is to speedup the convergence of an iterative method by solving a modified linear system where the coefficient matrix has more clustered eigenvalues. For real problems, an iterative solver should always be used with a preconditioner. In Eigen, a preconditioner is selected by simply adding it as a template parameter to the iterative solver object. IterativeSolverClassName<SparseMatrix<double>, PreconditionerName<SparseMatrix<double> > solver; The member function preconditioner() returns a read-write reference to the preconditioner to directly interact with it. See the Iterative solvers module and the documentation of each class for the list of available methods.

The Solve step

The solve() function computes the solution of the linear systems with one or many right hand sides. X = solver.solve(B); Here, B can be a vector or a matrix where the columns form the different right hand sides. The solve() function can be called several times as well, for instance when all the right hand sides are not available at once.

x1 = solver.solve(b1);
// Get the second right hand side b2
x2 = solver.solve(b2); 
//  ...

For direct methods, the solution are computed at the machine precision. Sometimes, the solution need not be too accurate. In this case, the iterative methods are more suitable and the desired accuracy can be set before the solve step using setTolerance(). For all the available functions, please, refer to the documentation of the Iterative solvers module.

BenchmarkRoutine

Most of the time, all you need is to know how much time it will take to solve your system, and hopefully, what is the most suitable solver. In Eigen, we provide a benchmark routine that can be used for this purpose. It is very easy to use. In the build directory, navigate to bench/spbench and compile the routine by typing make spbenchsolver. Run it with –help option to get the list of all available options. Basically, the matrices to test should be in MatrixMarket Coordinate format, and the routine returns the statistics from all available solvers in Eigen.

To export your matrices and right-hand-side vectors in the matrix-market format, you can the the unsupported SparseExtra module:

#include <unsupported/Eigen/SparseExtra>
...
Eigen::saveMarket(A, "filename.mtx");
Eigen::saveMarket(A, "filename_SPD.mtx", Eigen::Symmetric); // if A is symmetric-positive-definite
Eigen::saveMarketVector(B, "filename_b.mtx");

The following table gives an example of XML statistics from several Eigen built-in and external solvers.