naginterfaces.library.lapacklin.dsgesv

naginterfaces.library.lapacklin.dsgesv(a, b)[source]

dsgesv computes the solution to a real system of linear equations

where is an matrix and and are matrices.

For full information please refer to the NAG Library document for f07ac

https://www.nag.com/numeric/nl/nagdoc_27.3/flhtml/f07/f07acf.html

Parameters
afloat, array-like, shape

The coefficient matrix .

bfloat, array-like, shape

The right-hand side matrix .

Returns
afloat, ndarray, shape

If iterative refinement has been successfully used (i.e., if no exception or warning is raised and ), then is unchanged. If double precision factorization has been used (when no exception or warning is raised and ), contains the factors and from the factorization ; the unit diagonal elements of are not stored.

ipivint, ndarray, shape

If no constraints are violated, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required. corresponds either to the single precision factorization (if no exception or warning is raised and ) or to the double precision factorization (if no exception or warning is raised and ).

xfloat, ndarray, shape

If no exception or warning is raised, the solution matrix .

iteraint

If , iterative refinement has been successfully used and is the number of iterations carried out.

If , iterative refinement has failed for one of the reasons given below and double precision factorization has been carried out instead.

Taking into account machine parameters, and the values of and , it is not worth working in single precision.

Overflow of an entry occurred when moving from double to single precision.

An intermediate single precision factorization failed.

The maximum permitted number of iterations was exceeded.

Raises
NagValueError
(errno )

On entry, error in parameter .

Constraint: .

(errno )

On entry, error in parameter .

Constraint: .

Warns
NagAlgorithmicWarning
(errno )

Element of the diagonal is exactly zero. The factorization has been completed, but the factor is exactly singular, so the solution could not be computed.

Notes

dsgesv first attempts to factorize the matrix in single precision and use this factorization within an iterative refinement procedure to produce a solution with full double precision accuracy. If the approach fails the method switches to a double precision factorization and solve.

The iterative refinement process is stopped if

where is the number of iterations carried out thus far and is the maximum number of iterations allowed, which is fixed at iterations. The process is also stopped if for all right-hand sides we have

where is the -norm of the residual, is the -norm of the solution, is the -operator-norm of the matrix and is the machine precision returned by machine.precision.

The iterative refinement strategy used by dsgesv can be more efficient than the corresponding direct full precision algorithm. Since this strategy must perform iterative refinement on each right-hand side, any efficiency gains will reduce as the number of right-hand sides increases. Conversely, as the matrix size increases the cost of these iterative refinements become less significant relative to the cost of factorization. Thus, any efficiency gains will be greatest for a very small number of right-hand sides and for large matrix sizes. The cut-off values for the number of right-hand sides and matrix size, for which the iterative refinement strategy performs better, depends on the relative performance of the reduced and full precision factorization and back-substitution. For now, dsgesv always attempts the iterative refinement strategy first; you are advised to compare the performance of dsgesv with that of its full precision counterpart dgesv() to determine whether this strategy is worthwhile for your particular problem dimensions.

References

Anderson, E, Bai, Z, Bischof, C, Blackford, S, Demmel, J, Dongarra, J J, Du Croz, J J, Greenbaum, A, Hammarling, S, McKenney, A and Sorensen, D, 1999, LAPACK Users’ Guide, (3rd Edition), SIAM, Philadelphia, https://www.netlib.org/lapack/lug

Buttari, A, Dongarra, J, Langou, J, Langou, J, Luszczek, P and Kurzak, J, 2007, Mixed precision iterative refinement techniques for the solution of dense linear systems, International Journal of High Performance Computing Applications

Golub, G H and Van Loan, C F, 1996, Matrix Computations, (3rd Edition), Johns Hopkins University Press, Baltimore