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Sparse matrix manipulations

Manipulating and solving sparse problems involves various modules which are summarized below:

ModuleHeader fileContents
SparseCore
#include <Eigen/SparseCore>
SparseMatrix and SparseVector classes, matrix assembly, basic sparse linear algebra (including sparse triangular solvers)
SparseCholesky
#include <Eigen/SparseCholesky>
Direct sparse LLT and LDLT Cholesky factorization to solve sparse self-adjoint positive definite problems
SparseLU
#include<Eigen/SparseLU>
Sparse LU factorization to solve general square sparse systems
SparseQR
#include<Eigen/SparseQR>
Sparse QR factorization for solving sparse linear least-squares problems
IterativeLinearSolvers
#include <Eigen/IterativeLinearSolvers>
Iterative solvers to solve large general linear square problems (including self-adjoint positive definite problems)
Sparse
#include <Eigen/Sparse>
Includes all the above modules

Sparse matrix format

In many applications (e.g., finite element methods) it is common to deal with very large matrices where only a few coefficients are different from zero. In such cases, memory consumption can be reduced and performance increased by using a specialized representation storing only the nonzero coefficients. Such a matrix is called a sparse matrix.

The SparseMatrix class

The class SparseMatrix is the main sparse matrix representation of Eigen's sparse module; it offers high performance and low memory usage. It implements a more versatile variant of the widely-used Compressed Column (or Row) Storage scheme. It consists of four compact arrays:

This storage scheme is better explained on an example. The following matrix

03000
2200017
75010
00000
001408

and one of its possible sparse, column major representation:

Values: 227_3514__1_178
InnerIndices: 12_024__2_14
OuterStarts:03581012
InnerNNZs: 22112

Currently the elements of a given inner vector are guaranteed to be always sorted by increasing inner indices. The "_" indicates available free space to quickly insert new elements. Assuming no reallocation is needed, the insertion of a random element is therefore in O(nnz_j) where nnz_j is the number of nonzeros of the respective inner vector. On the other hand, inserting elements with increasing inner indices in a given inner vector is much more efficient since this only requires to increase the respective InnerNNZs entry that is a O(1) operation.

The case where no empty space is available is a special case, and is refered as the compressed mode. It corresponds to the widely used Compressed Column (or Row) Storage schemes (CCS or CRS). Any SparseMatrix can be turned to this form by calling the SparseMatrix::makeCompressed() function. In this case, one can remark that the InnerNNZs array is redundant with OuterStarts because we the equality: InnerNNZs[j] = OuterStarts[j+1]-OuterStarts[j]. Therefore, in practice a call to SparseMatrix::makeCompressed() frees this buffer.

It is worth noting that most of our wrappers to external libraries requires compressed matrices as inputs.

The results of Eigen's operations always produces compressed sparse matrices. On the other hand, the insertion of a new element into a SparseMatrix converts this later to the uncompressed mode.

Here is the previous matrix represented in compressed mode:

Values: 22735141178
InnerIndices: 12024214
OuterStarts:024568

A SparseVector is a special case of a SparseMatrix where only the Values and InnerIndices arrays are stored. There is no notion of compressed/uncompressed mode for a SparseVector.

First example

Before describing each individual class, let's start with the following typical example: solving the Laplace equation $ \Delta u = 0 $ on a regular 2D grid using a finite difference scheme and Dirichlet boundary conditions. Such problem can be mathematically expressed as a linear problem of the form $ Ax=b $ where $ x $ is the vector of m unknowns (in our case, the values of the pixels), $ b $ is the right hand side vector resulting from the boundary conditions, and $ A $ is an $ m \times m $ matrix containing only a few non-zero elements resulting from the discretization of the Laplacian operator.

#include <Eigen/Sparse>
#include <vector>
typedef Eigen::SparseMatrix<double> SpMat; // declares a column-major sparse matrix type of double
void buildProblem(std::vector<T>& coefficients, Eigen::VectorXd& b, int n);
void saveAsBitmap(const Eigen::VectorXd& x, int n, const char* filename);
int main(int argc, char** argv)
{
assert(argc==2);
int n = 300; // size of the image
int m = n*n; // number of unknows (=number of pixels)
// Assembly:
std::vector<T> coefficients; // list of non-zeros coefficients
Eigen::VectorXd b(m); // the right hand side-vector resulting from the constraints
buildProblem(coefficients, b, n);
SpMat A(m,m);
A.setFromTriplets(coefficients.begin(), coefficients.end());
// Solving:
Eigen::SimplicialCholesky<SpMat> chol(A); // performs a Cholesky factorization of A
Eigen::VectorXd x = chol.solve(b); // use the factorization to solve for the given right hand side
// Export the result to a file:
saveAsBitmap(x, n, argv[1]);
return 0;
}
Tutorial_sparse_example.jpeg

In this example, we start by defining a column-major sparse matrix type of double SparseMatrix<double>, and a triplet list of the same scalar type Triplet<double>. A triplet is a simple object representing a non-zero entry as the triplet: row index, column index, value.

In the main function, we declare a list coefficients of triplets (as a std vector) and the right hand side vector $ b $ which are filled by the buildProblem function. The raw and flat list of non-zero entries is then converted to a true SparseMatrix object A. Note that the elements of the list do not have to be sorted, and possible duplicate entries will be summed up.

The last step consists of effectively solving the assembled problem. Since the resulting matrix A is symmetric by construction, we can perform a direct Cholesky factorization via the SimplicialLDLT class which behaves like its LDLT counterpart for dense objects.

The resulting vector x contains the pixel values as a 1D array which is saved to a jpeg file shown on the right of the code above.

Describing the buildProblem and save functions is out of the scope of this tutorial. They are given here for the curious and reproducibility purpose.

The SparseMatrix class

Matrix and vector properties
The SparseMatrix and SparseVector classes take three template arguments: the scalar type (e.g., double) the storage order (ColMajor or RowMajor, the default is ColMajor) the inner index type (default is int).

As for dense Matrix objects, constructors takes the size of the object. Here are some examples:

SparseMatrix<std::complex<float> > mat(1000,2000); // declares a 1000x2000 column-major compressed sparse matrix of complex<float>
SparseMatrix<double,RowMajor> mat(1000,2000); // declares a 1000x2000 row-major compressed sparse matrix of double
SparseVector<std::complex<float> > vec(1000); // declares a column sparse vector of complex<float> of size 1000
SparseVector<double,RowMajor> vec(1000); // declares a row sparse vector of double of size 1000

In the rest of the tutorial, mat and vec represent any sparse-matrix and sparse-vector objects, respectively.

The dimensions of a matrix can be queried using the following functions:

Standard
dimensions
mat.rows()
mat.cols()
vec.size()
Sizes along the
inner/outer dimensions
mat.innerSize()
mat.outerSize()
Number of non
zero coefficients
mat.nonZeros()
vec.nonZeros()

Iterating over the nonzero coefficients
Random access to the elements of a sparse object can be done through the coeffRef(i,j) function. However, this function involves a quite expensive binary search. In most cases, one only wants to iterate over the non-zeros elements. This is achieved by a standard loop over the outer dimension, and then by iterating over the non-zeros of the current inner vector via an InnerIterator. Thus, the non-zero entries have to be visited in the same order than the storage order. Here is an example:

SparseMatrix<double> mat(rows,cols);
for (int k=0; k<mat.outerSize(); ++k)
for (SparseMatrix<double>::InnerIterator it(mat,k); it; ++it)
{
it.value();
it.row(); // row index
it.col(); // col index (here it is equal to k)
it.index(); // inner index, here it is equal to it.row()
}
SparseVector<double> vec(size);
for (SparseVector<double>::InnerIterator it(vec); it; ++it)
{
it.value(); // == vec[ it.index() ]
it.index();
}

For a writable expression, the referenced value can be modified using the valueRef() function. If the type of the sparse matrix or vector depends on a template parameter, then the typename keyword is required to indicate that InnerIterator denotes a type; see The template and typename keywords in C++ for details.

Filling a sparse matrix

Because of the special storage scheme of a SparseMatrix, special care has to be taken when adding new nonzero entries. For instance, the cost of a single purely random insertion into a SparseMatrix is O(nnz), where nnz is the current number of non-zero coefficients.

The simplest way to create a sparse matrix while guaranteeing good performance is thus to first build a list of so-called triplets, and then convert it to a SparseMatrix.

Here is a typical usage example:

std::vector<T> tripletList;
tripletList.reserve(estimation_of_entries);
for(...)
{
// ...
tripletList.push_back(T(i,j,v_ij));
}
SparseMatrixType mat(rows,cols);
mat.setFromTriplets(tripletList.begin(), tripletList.end());
// mat is ready to go!

The std::vector of triplets might contain the elements in arbitrary order, and might even contain duplicated elements that will be summed up by setFromTriplets(). See the SparseMatrix::setFromTriplets() function and class Triplet for more details.

In some cases, however, slightly higher performance, and lower memory consumption can be reached by directly inserting the non-zeros into the destination matrix. A typical scenario of this approach is illustrated bellow:

1: SparseMatrix<double> mat(rows,cols); // default is column major
2: mat.reserve(VectorXi::Constant(cols,6));
3: for each i,j such that v_ij != 0
4: mat.insert(i,j) = v_ij; // alternative: mat.coeffRef(i,j) += v_ij;
5: mat.makeCompressed(); // optional

Supported operators and functions

Because of their special storage format, sparse matrices cannot offer the same level of flexibility than dense matrices. In Eigen's sparse module we chose to expose only the subset of the dense matrix API which can be efficiently implemented. In the following sm denotes a sparse matrix, sv a sparse vector, dm a dense matrix, and dv a dense vector.

Basic operations

Sparse expressions support most of the unary and binary coefficient wise operations:

sm1.real() sm1.imag() -sm1 0.5*sm1
sm1+sm2 sm1-sm2 sm1.cwiseProduct(sm2)

However, a strong restriction is that the storage orders must match. For instance, in the following example:

sm4 = sm1 + sm2 + sm3;

sm1, sm2, and sm3 must all be row-major or all column-major. On the other hand, there is no restriction on the target matrix sm4. For instance, this means that for computing $ A^T + A $, the matrix $ A^T $ must be evaluated into a temporary matrix of compatible storage order:

SparseMatrix<double> A, B;
B = SparseMatrix<double>(A.transpose()) + A;

Binary coefficient wise operators can also mix sparse and dense expressions:

sm2 = sm1.cwiseProduct(dm1);
dm2 = sm1 + dm1;
dm2 = dm1 - sm1;

Performance-wise, the adding/subtracting sparse and dense matrices is better performed in two steps. For instance, instead of doing dm2 = sm1 + dm1, better write:

dm2 = dm1;
dm2 += sm1;

This version has the advantage to fully exploit the higher performance of dense storage (no indirection, SIMD, etc.), and to pay the cost of slow sparse evaluation on the few non-zeros of the sparse matrix only.

Sparse expressions also support transposition:

sm1 = sm2.transpose();
sm1 = sm2.adjoint();

However, there is no transposeInPlace() method.

Matrix products

Eigen supports various kind of sparse matrix products which are summarize below:

Block operations

Regarding read-access, sparse matrices expose the same API than for dense matrices to access to sub-matrices such as blocks, columns, and rows. See Block operations for a detailed introduction. However, for performance reasons, writing to a sub-sparse-matrix is much more limited, and currently only contiguous sets of columns (resp. rows) of a column-major (resp. row-major) SparseMatrix are writable. Moreover, this information has to be known at compile-time, leaving out methods such as block(...) and corner*(...). The available API for write-access to a SparseMatrix are summarized below:

SparseMatrix<double,ColMajor> sm1;
sm1.col(j) = ...;
sm1.leftCols(ncols) = ...;
sm1.middleCols(j,ncols) = ...;
sm1.rightCols(ncols) = ...;
SparseMatrix<double,RowMajor> sm2;
sm2.row(i) = ...;
sm2.topRows(nrows) = ...;
sm2.middleRows(i,nrows) = ...;
sm2.bottomRows(nrows) = ...;

In addition, sparse matrices expose the SparseMatrixBase::innerVector() and SparseMatrixBase::innerVectors() methods, which are aliases to the col/middleCols methods for a column-major storage, and to the row/middleRows methods for a row-major storage.

Triangular and selfadjoint views

Just as with dense matrices, the triangularView() function can be used to address a triangular part of the matrix, and perform triangular solves with a dense right hand side:

dm2 = sm1.triangularView<Lower>(dm1);
dv2 = sm1.transpose().triangularView<Upper>(dv1);

The selfadjointView() function permits various operations:

Please, refer to the Quick Reference guide for the list of supported operations. The list of linear solvers available is here.