solve_matrix — Compute the solution of a system of equations.
solve_matrix( : : MatrixLHSID, MatrixLHSType, Epsilon, MatrixRHSID : MatrixResultID)
The operator solve_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 MatrixLHSID and MatrixRHSID.
The number of rows of matrices MatrixLHS and
MatrixRHS must be identical. The operator returns the
matrix handle MatrixResultID of the matrix
MatrixResult. Access to the elements of the matrix is
possible e.g., with the operator get_full_matrix.
For linear equation systems, the equations
are solved. Therefore, the matrix MatrixLHS must be a
square matrix and the parameter Epsilon must be 0. The
type of the matrix MatrixLHS can be selected via the
parameter MatrixLHSType. The following values are
supported: 'general' for general, 'symmetric' for
symmetric, 'positive_definite' for symmetric positive
definite, 'tridiagonal' for tridiagonal,
'upper_triangular' for upper triangular,
'permuted_upper_triangular' for permuted upper triangular,
'lower_triangular' for lower triangular, and
'permuted_lower_triangular' for permuted lower triangular
matrices.
Example:
MatrixLHSType = 'positive_definite',
Epsilon = 0
For linear least squares problems or if Epsilon 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
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 MatrixLHSType =
'general'.
Example:
MatrixLHSType = 'general',
Epsilon = 2.2204e-16
Note: The relative accuracy of the floating point representation of
the used data type (double) is Epsilon = 2.2204e-16.
For MatrixLHSType = 'symmetric',
'positive_definite', or '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
MatrixLHSType = '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 MatrixLHSType =
'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.
MatrixLHSID (input_control) matrix → (handle)
Matrix handle of the input matrix of the left hand side.
MatrixLHSType (input_control) string → (string)
The type of the input matrix of the left hand side.
Default: 'general'
List of values: 'general', 'lower_triangular', 'permuted_lower_triangular', 'permuted_upper_triangular', 'positive_definite', 'symmetric', 'tridiagonal', 'upper_triangular'
Epsilon (input_control) real → (real)
Type of solving and limitation to set singular values to be 0.
Default: 0.0
Suggested values: 0.0, 2.2204e-16
MatrixRHSID (input_control) matrix → (handle)
Matrix handle of the input matrix of right hand side.
MatrixResultID (output_control) matrix → (handle)
New matrix handle with the solution.
If the parameters are valid, the operator solve_matrix
returns the value 2 (
H_MSG_TRUE)
. If necessary, an exception is raised.
get_full_matrix,
get_value_matrix
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.
Foundation