Sparse Matrix class daecpp::sparse_matrix defines sparse matrix holder in three-array format. It is used to define the mass matrix and, optionally, the Jacobian matrix of the DAE system. The class provides a set of helper tools for the user to allocate memory, define and convert sparse matrices, etc., in a very straightforward way, e.g.:

sparse_matrix M;                 // Creates empty sparse matrix M
M.reserve(2);                    // Reserves memory for 2 elements (optional)
M.add_element(0, 1, 12.0);       // Fills row 0, column 1 with value 12.0
M.add_element(1, 1, 42.0);       // Fills row 1, column 1 with value 42.0
std::cout << M.dense(3) << '\n'; // Prints matrix M on screen assuming it's a 3x3 matrix

The three-array format defines a sparse matrix using two (unsigned) integer vectors of indices i and j and one floating-point vector A of the corresponding non-zero elements of the matrix. Only non-zero elements of the sparse matrix should be defined and stored. Theoretically, all three arrays can be empty. This will define a zero matrix where all elements are zeros.

{: .note } All three arrays i, j, and A should be the same size. This can be (and will be) checked by calling the void check() method of the class daecpp::sparse_matrix.

Here is an example of a sparse matrix:

$$ \mathbf{M} = \begin{vmatrix} 1 & 0 & 3 \\ 0 & 2 & 0 \\ 0 & 0 & 0 \end{vmatrix}. $$

This matrix contains only 3 non-zero elements: 1, 2, and 3. Here is the matrix $$\mathbf{M}$$ definition in three-array format:

The indices i and j define the position of the corresponding non-zero element A in the matrix. Index i is the row index and index j is the column index.

It doesn't matter in which order you define each element of the matrix.

{: .note } The row and column numeration starts from 0.

For float_type and int_type definition, refer to the Prerequisites section.

{: .note } All three vectors are initially empty and not preallocated.

• ind_i - row index of the element (int_type)
• ind_j - column index of the element (int_type)
• A_ij- non-zero element (float_type)

sparse_matrix M; // Creates empty sparse matrix M
M(0, 1, 42.0);   // Fills row 0, column 1 with value 42.0
M(1, 2, 10.0);   // Defines next non-zero element: row 1, column 2, value 10.0
M(0, 1, 2.0);    // Duplicated element is OK, it will be summed up, i.e., the value will be 44.0

• ind_i - row index of the element (int_type)
• ind_j - column index of the element (int_type)
• A_ij- non-zero element (float_type)

sparse_matrix M;           // Creates empty sparse matrix M
M.add_element(0, 1, 42.0); // Fills row 0, column 1 with value 42.0
M.add_element(1, 2, 10.0); // Defines next non-zero element: row 1, column 2, value 10.0
M.add_element(0, 1, 2.0);  // Duplicated element will be summed up, i.e., the value will be 44.0

• N_elements - estimated number of non-zero elements (int_type)

sparse_matrix M; // Creates empty sparse matrix M
M.reserve(16);   // Pre-allocates memory for 16 non-zero elements

• N - matrix size (square matrix N x N) (int_type)

• Dense matrix representation in Eigen format.

sparse_matrix M;                 // Creates empty sparse matrix M
M(0, 1, 42.0);                   // Fills row 0, column 1 with value 42.0
std::cout << M.dense(3) << '\n'; // Prints matrix on screen assuming it's a 3x3 matrix

• N - matrix size (square matrix N x N) (int_type)

• Eigen::SparseMatrix<float_type> (core::eimat) sparse matrix.