naginterfaces.library.lapacklin.zcposv

naginterfaces.library.lapacklin.zcposv(uplo, n, nrhs, a, b)

zcposv uses the Cholesky factorization

$$A=U^{H}U\text{or}A=L{L}^H$$

to compute the solution to a complex system of linear equations

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

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

For full information please refer to the NAG Library document for f07fq

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f07/f07fqf.html

Parameters

uplostr, length 1

Specifies whether the upper or lower triangular part of $A$ is stored.

$uplo=\text{'U'}$

The upper triangular part of $A$ is stored.

$uplo=\text{'L'}$

The lower triangular part of $A$ is stored.

nint

$n$, the number of linear equations, i.e., the order of the matrix $A$.

nrhsint

$r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.

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

The $n\times n$ Hermitian positive definite matrix $A$.

bcomplex, array-like, shape $(n,nrhs)$

The right-hand side matrix $B$.

Returns

acomplex, ndarray, shape $(n,n)$

If iterative refinement has been successfully used (no exception or warning is raised and $itera\ge 0$, see $itera$), then $a$ is unchanged. If full precision factorization has been used (no exception or warning is raised and $itera<0$, see $itera$), then the array $A$ contains the factor $U$ or $L$ from the Cholesky factorization $A=U^{H}U$ or $A=L{L}^H$.

xcomplex, ndarray, shape $(n,nrhs)$

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

iteraint

Information on the progress of the interative refinement process.

$itera<0$

Iterative refinement has failed for one of the reasons given below, full precision factorization has been performed instead.

$-1$	The function fell back to full precision for implementation - or machine-specific reasons.
$-2$	Narrowing the precision induced an overflow, the function fell back to full precision.
$-3$	An intermediate reduced precision factorization failed.
$-31$	The maximum permitted number of iterations was exceeded.

$itera>0$

Iterative refinement has been sucessfully used. $itera$ returns the number of iterations.

Raises

NagValueError

(errno $-1$)

On entry, error in parameter $uplo$.

Constraint: $uplo=\text{'U'}$ or $\text{'L'}$.

(errno $-2$)

On entry, error in parameter $n$.

Constraint: $n\ge 0$.

(errno $-3$)

On entry, error in parameter $nrhs$.

Constraint: $nrhs\ge 0$.

(errno $(i>0)\text{and}(i\le n)$)

The leading minor of order $⟨value⟩$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

zcposv first attempts to factorize the matrix in reduced precision and use this factorization within an iterative refinement procedure to produce a solution with full precision normwise backward error quality (see below). If the approach fails the method switches to a full precision factorization and solve.

The iterative refinement can be more efficient than the corresponding direct full precision algorithm. Since the strategy implemented by zcposv 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. zcposv always attempts the iterative refinement strategy first; you are advised to compare the performance of zcposv with that of its full precision counterpart zposv() to determine whether this strategy is worthwhile for your particular problem dimensions.

The iterative refinement process is stopped if $itera>30$ where $itera$ is the number of iterations carried out thus far. 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 $\infty$-norm of the matrix $A$ and $ϵ$ is the machine precision returned by machine.precision.

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

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

Higham, N J, 2002, Accuracy and Stability of Numerical Algorithms, (2nd Edition), SIAM, Philadelphia