solve_matrixT_solve_matrixSolveMatrixSolveMatrixsolve_matrix (Operator)

Name

solve_matrixT_solve_matrixSolveMatrixSolveMatrixsolve_matrix — Compute the solution of a system of equations.

Signature

Description

The operator solve_matrixsolve_matrixSolveMatrixSolveMatrixSolveMatrixsolve_matrix computes the solution of a system of linear equations or of a linear least squares problem. The input matrices MatrixLHS and MatrixRHS are defined by the matrix handles MatrixLHSIDMatrixLHSIDMatrixLHSIDMatrixLHSIDmatrixLHSIDmatrix_lhsid and MatrixRHSIDMatrixRHSIDMatrixRHSIDMatrixRHSIDmatrixRHSIDmatrix_rhsid. The number of rows of matrices MatrixLHS and MatrixRHS must be identical. The operator returns the matrix handle MatrixResultIDMatrixResultIDMatrixResultIDMatrixResultIDmatrixResultIDmatrix_result_id of the matrix MatrixResult. Access to the elements of the matrix is possible e.g., with the operator get_full_matrixget_full_matrixGetFullMatrixGetFullMatrixGetFullMatrixget_full_matrix.

For linear equation systems, the equations \texttt{MatrixLHS} * \texttt{MatrixResult} = \texttt{MatrixRHS} are solved. Therefore, the matrix MatrixLHS must be a square matrix and the parameter EpsilonEpsilonEpsilonEpsilonepsilonepsilon must be 0. The type of the matrix MatrixLHS can be selected via the parameter MatrixLHSTypeMatrixLHSTypeMatrixLHSTypeMatrixLHSTypematrixLHSTypematrix_lhstype. The following values are supported: 'general'"general""general""general""general""general" for general, 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric" for symmetric, 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite" for symmetric positive definite, 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal" for tridiagonal, 'upper_triangular'"upper_triangular""upper_triangular""upper_triangular""upper_triangular""upper_triangular" for upper triangular, 'permuted_upper_triangular'"permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular""permuted_upper_triangular" for permuted upper triangular, 'lower_triangular'"lower_triangular""lower_triangular""lower_triangular""lower_triangular""lower_triangular" for lower triangular, and 'permuted_lower_triangular'"permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular""permuted_lower_triangular" for permuted lower triangular matrices.

Example:

For linear least squares problems or if EpsilonEpsilonEpsilonEpsilonepsilonepsilon is not 0, the matrix MatrixLHS need not be a square matrix. The linear least squares problem is solved using the singular value decomposition (SVD) of the matrix MatrixLHS by minimizing ||\texttt{MatrixRHS} - \texttt{MatrixLHS} * \texttt{MatrixResult}||. All singular values less than the value Epsilon * the largest singular value are set to 0. For these values no internal division is done to prevent a division by zero. Also, the matrix MatrixLHS may be rank-deficient. The type of matrix must be selected via MatrixLHSTypeMatrixLHSTypeMatrixLHSTypeMatrixLHSTypematrixLHSTypematrix_lhstype = 'general'"general""general""general""general""general".

Example:

Note: The relative accuracy of the floating point representation of the used data type (double) is EpsilonEpsilonEpsilonEpsilonepsilonepsilon = 2.2204e-16.

Attention

For MatrixLHSTypeMatrixLHSTypeMatrixLHSTypeMatrixLHSTypematrixLHSTypematrix_lhstype = 'symmetric'"symmetric""symmetric""symmetric""symmetric""symmetric", 'positive_definite'"positive_definite""positive_definite""positive_definite""positive_definite""positive_definite", or 'upper_triangular'"upper_triangular""upper_triangular""upper_triangular""upper_triangular""upper_triangular" the upper triangular part of the input MatrixLHS must contain the relevant information of the matrix. The strictly lower triangular part of the matrix is not referenced. For MatrixLHSTypeMatrixLHSTypeMatrixLHSTypeMatrixLHSTypematrixLHSTypematrix_lhstype = 'lower_triangular'"lower_triangular""lower_triangular""lower_triangular""lower_triangular""lower_triangular" the lower triangular part of the input MatrixLHS must contain the relevant information of the matrix. The strictly upper triangular part of the matrix is not referenced. For MatrixLHSTypeMatrixLHSTypeMatrixLHSTypeMatrixLHSTypematrixLHSTypematrix_lhstype = 'tridiagonal'"tridiagonal""tridiagonal""tridiagonal""tridiagonal""tridiagonal", only the main diagonal, the superdiagonal, and the subdiagonal of the input MatrixLHS are used. The other parts of the matrix are not referenced. If the referenced part of the input MatrixLHS is not of the specified type, an exception is raised.

Execution Information

• Multithreading type: reentrant (runs in parallel with non-exclusive operators).
• Multithreading scope: global (may be called from any thread).
• Processed without parallelization.

Parameters

Result

If the parameters are valid, the operator solve_matrixsolve_matrixSolveMatrixSolveMatrixSolveMatrixsolve_matrix returns the value TRUE. If necessary, an exception is raised.

Possible Predecessors

create_matrixcreate_matrixCreateMatrixCreateMatrixCreateMatrixcreate_matrix

Possible Successors

get_full_matrixget_full_matrixGetFullMatrixGetFullMatrixGetFullMatrixget_full_matrix, get_value_matrixget_value_matrixGetValueMatrixGetValueMatrixGetValueMatrixget_value_matrix

References

David Poole: “Linear Algebra: A Modern Introduction”; Thomson; Belmont; 2006.
Gene H. Golub, Charles F. van Loan: “Matrix Computations”; The Johns Hopkins University Press; Baltimore and London; 1996.

Module

Foundation