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Quick reference guide for sparse matrices

In this page, we give a quick summary of the main operations available for sparse matrices in the class SparseMatrix. First, it is recommended to read the introductory tutorial at Sparse matrix manipulations. The important point to have in mind when working on sparse matrices is how they are stored : i.e either row major or column major. The default is column major. Most arithmetic operations on sparse matrices will assert that they have the same storage order.

Sparse Matrix Initialization

Category Operations Notes
Constructor 
SparseMatrix<double> sm1(1000,1000); 
SparseMatrix<std::complex<double>,RowMajor> sm2;
 Default is ColMajor
Resize/Reserve 
sm1.resize(m,n);      // Change sm1 to a m x n matrix.
sm1.reserve(nnz);     // Allocate room for nnz nonzeros elements.
Note that when calling reserve(), it is not required that nnz is the exact number of nonzero elements in the final matrix. However, an exact estimation will avoid multiple reallocations during the insertion phase.
Assignment 
 SparseMatrix<double,Colmajor> sm1;
// Initialize sm2 with sm1.
 SparseMatrix<double,Rowmajor> sm2(sm1), sm3;        
 // Assignment and evaluations modify the storage order.
 sm3 = sm1;
The copy constructor can be used to convert from a storage order to another
Element-wise Insertion 
// Insert a new element; 
 sm1.insert(i, j) = v_ij;  
 
// Update the value v_ij
 sm1.coeffRef(i,j) = v_ij;
 sm1.coeffRef(i,j) += v_ij;
 sm1.coeffRef(i,j) -= v_ij;
insert() assumes that the element does not already exist; otherwise, use coeffRef()
Batch insertion 
std::vector< Eigen::Triplet<double> > tripletList;
tripletList.reserve(estimation_of_entries);
// -- Fill tripletList with nonzero elements...
sm1.setFromTriplets(TripletList.begin(), TripletList.end());
A complete example is available at Triplet Insertion .
Constant or Random Insertion 
sm1.setZero();
Remove all non-zero coefficients

Matrix properties

Beyond the basic functions rows() and cols(), there are some useful functions that are available to easily get some information from the matrix.

sm1.rows();         // Number of rows
sm1.cols();         // Number of columns 
sm1.nonZeros();     // Number of non zero values   
sm1.outerSize();    // Number of columns (resp. rows) for a column major (resp. row major )
sm1.innerSize();    // Number of rows (resp. columns) for a row major (resp. column major)
sm1.norm();         // Euclidian norm of the matrix
sm1.squaredNorm();  // Squared norm of the matrix
sm1.blueNorm();
sm1.isVector();     // Check if sm1 is a sparse vector or a sparse matrix
sm1.isCompressed(); // Check if sm1 is in compressed form
...

Arithmetic operations

It is easy to perform arithmetic operations on sparse matrices provided that the dimensions are adequate and that the matrices have the same storage order. Note that the evaluation can always be done in a matrix with a different storage order. In the following, sm denotes a sparse matrix, dm a dense matrix and dv a dense vector.

Operations Code

Notes
add subtract 
sm3 = sm1 + sm2; 
sm3 = sm1 - sm2;
sm2 += sm1; 
sm2 -= sm1;

sm1 and sm2 should have the same storage order
scalar product 
sm3 = sm1 * s1;   sm3 *= s1; 
sm3 = s1 * sm1 + s2 * sm2; sm3 /= s1;

Many combinations are possible if the dimensions and the storage order agree.
Sparse Product 
sm3 = sm1 * sm2;
dm2 = sm1 * dm1;
dv2 = sm1 * dv1;

transposition, adjoint 
sm2 = sm1.transpose();
sm2 = sm1.adjoint();
Note that the transposition change the storage order. There is no support for transposeInPlace().
Permutation 
perm.indices();      // Reference to the vector of indices
sm1.twistedBy(perm); // Permute rows and columns
sm2 = sm1 * perm;    // Permute the columns
sm2 = perm * sm1;    // Permute the columns

Component-wise ops 
sm1.cwiseProduct(sm2);
sm1.cwiseQuotient(sm2);
sm1.cwiseMin(sm2);
sm1.cwiseMax(sm2);
sm1.cwiseAbs();
sm1.cwiseSqrt();
sm1 and sm2 should have the same storage order

Other supported operations

Code Notes
Sub-matrices 
sm1.block(startRow, startCol, rows, cols); 
sm1.block(startRow, startCol); 
sm1.topLeftCorner(rows, cols); 
sm1.topRightCorner(rows, cols);
sm1.bottomLeftCorner( rows, cols);
sm1.bottomRightCorner( rows, cols);
Contrary to dense matrices, here all these methods are read-only.
See Block operations and below for read-write sub-matrices.
Range 
sm1.innerVector(outer);           // RW
sm1.innerVectors(start, size);    // RW
sm1.leftCols(size);               // RW
sm2.rightCols(size);              // RO because sm2 is row-major
sm1.middleRows(start, numRows);   // RO because sm1 is column-major
sm1.middleCols(start, numCols);   // RW
sm1.col(j);                       // RW
A inner vector is either a row (for row-major) or a column (for column-major).
As stated earlier, for a read-write sub-matrix (RW), the evaluation can be done in a matrix with different storage order.
Triangular and selfadjoint views 
sm2 = sm1.triangularview<Lower>();
sm2 = sm1.selfadjointview<Lower>();
 Several combination between triangular views and blocks views are possible
Triangular solve 
dv2 = sm1.triangularView<Upper>().solve(dv1);
dv2 = sm1.topLeftCorner(size, size)
         .triangularView<Lower>().solve(dv1);
 For general sparse solve, Use any suitable module described at Solving Sparse Linear Systems
Low-level API 
sm1.valuePtr();      // Pointer to the values
sm1.innerIndexPtr();  // Pointer to the indices.
sm1.outerIndexPtr(); // Pointer to the beginning of each inner vector
If the matrix is not in compressed form, makeCompressed() should be called before.
Note that these functions are mostly provided for interoperability purposes with external libraries.
A better access to the values of the matrix is done by using the InnerIterator class as described in the Tutorial Sparse section
Mapping external buffers 
int outerIndexPtr[cols+1];
int innerIndices[nnz];
double values[nnz];
Map<SparseMatrix<double> > sm1(rows,cols,nnz,outerIndexPtr, // read-write
                               innerIndices,values);
Map<const SparseMatrix<double> > sm2(...);                  // read-only
As for dense matrices, class Map<SparseMatrixType> can be used to see external buffers as an Eigen's SparseMatrix object.

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