naginterfaces.library.lapacklin.dsgesv

naginterfaces.library.lapacklin.dsgesv(a, b)

dsgesv computes the solution to a real system of linear equations

$$AX=B\text{,}$$

where $A$ is an $n\times n$ matrix and $X$ and $B$ are $n\times r$ matrices.

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

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07acf.html

Parameters

afloat, array-like, shape $(n,n)$

The $n\times n$ coefficient matrix $A$.

bfloat, array-like, shape $(n,\text{nrhs})$

The $n\times r$ right-hand side matrix $B$.

Returns

afloat, ndarray, shape $(n,n)$

If iterative refinement has been successfully used (i.e., if no exception or warning is raised and $itera\ge 0$), then $A$ is unchanged. If double precision factorization has been used (when no exception or warning is raised and $itera<0$), $A$ contains the factors $L$ and $U$ from the factorization $A=PLU$; the unit diagonal elements of $L$ are not stored.

ipivint, ndarray, shape $\left(n\right)$

If no constraints are violated, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row $ipiv[i-1]$. $ipiv[i-1]=i$ indicates a row interchange was not required. $ipiv$ corresponds either to the single precision factorization (if no exception or warning is raised and $itera\ge 0$) or to the double precision factorization (if no exception or warning is raised and $itera<0$).

xfloat, ndarray, shape $(n,\text{nrhs})$

If no exception or warning is raised, the $n\times r$ solution matrix $X$.

iteraint

If $itera>0$, iterative refinement has been successfully used and $itera$ is the number of iterations carried out.

If $itera<0$, iterative refinement has failed for one of the reasons given below and double precision factorization has been carried out instead.

$itera=-1$

Taking into account machine parameters, and the values of $\text{n}$ and $\text{nrhs}$, it is not worth working in single precision.

$itera=-2$

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

$itera=-3$

An intermediate single precision factorization failed.

$itera=-31$

The maximum permitted number of iterations was exceeded.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $\text{n}$.

Constraint: $n\ge 0$.

(errno $-2$)

On entry, error in parameter $\text{nrhs}$.

Constraint: $\text{nrhs}\ge 0$.

Warns

NagAlgorithmicWarning

(errno $i>0$)

Element $⟨value⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ 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

$$itera>\text{itermax}\text{,}$$

where $itera$ is the number of iterations carried out thus far and $\text{itermax}$ is the maximum number of iterations allowed, which is fixed at $30$ iterations. The process is also stopped if for all right-hand sides we have

$$\parallel \text{resid}\parallel <\sqrt{n}\parallel x\parallel \parallel A\parallel ϵ\text{,}$$

where $\parallel \text{resid}\parallel$ is the $\infty$-norm of the residual, $\parallel x\parallel$ is the $\infty$-norm of the solution, $\parallel A\parallel$ is the $\infty$-operator-norm of the matrix $A$ and $ϵ$ 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