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Eigen  3.2.5
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Quick reference guide for sparse matrices
[Sparse linear algebra]


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 informations 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

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);

Range

  sm1.innerVector(outer); 
  sm1.innerVectors(start, size);
  sm1.leftCols(size);
  sm2.rightCols(size);
  sm1.middleRows(start, numRows);
  sm1.middleCols(start, numCols);
  sm1.col(j);

A inner vector is either a row (for row-major) or a column (for column-major). As stated earlier, 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.innerIndextr(); // 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