Bunch–Kaufman factorization of complex Hermitian indefinite matrix, packed storage

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NAG[f07prc] NAG[nag_zhptrf] - Bunch–Kaufman factorization of complex Hermitian indefinite matrix, packed storage

Calling Sequence

f07prc(uplo, n, ap, ipiv, 'fail'=fail)

nag_zhptrf(. . .)

Parameters

uplo - String;

On entry: indicates whether the upper or lower triangular part of is stored and how is factorized.

The upper triangular part of is stored and is factorized as where is upper triangular.

The lower triangular part of is stored and is factorized as where is lower triangular.

Constraint: "Nag_Upper" or "Nag_Lower". .

n - integer;

On entry: , the order of the matrix .

Constraint: . .

ap - Vector(1..dim, datatype=complex[8]);

Note: the dimension, dim, of the array ap must be at least .

On entry: the Hermitian indefinite matrix , packed by rows or columns. The storage of elements depends on the storage order and uplo arguments as follows:

if and , is stored in , for ;

if and , is stored in , for .

On exit: is overwritten by details of the block diagonal matrix and the multipliers used to obtain the factor or as specified by uplo.

ipiv - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);

Note: the dimension, dim, of the array ipiv must be at least .

On exit: details of the interchanges and the block structure of . More precisely,

if , is a by pivot block and the th row and column of were interchanged with the th row and column;

if and , is a by pivot block and the th row and column of were interchanged with the th row and column;

if and , $Matrix(2, 2, {(1, 1) = d[i^2], (1, 2) = d[i+1, i], (2, 1) = d[i+1, i], (2, 2) = d[i+1, i+1]})$ is a by pivot block and the th row and column of were interchanged with the th row and column.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zhptrf (f07prc) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

Description

nag_zhptrf (f07prc) factorizes a complex Hermitian matrix , using the Bunch–Kaufman diagonal pivoting method and packed storage. is factorized as either if or if , where is a permutation matrix, (or ) is a unit upper (or lower) triangular matrix and is an Hermitian block diagonal matrix with 1 by 1 and 2 by 2 diagonal blocks; (or ) has 2 by 2 unit diagonal blocks corresponding to the 2 by 2 blocks of . Row and column interchanges are performed to ensure numerical stability while keeping the matrix Hermitian.

This method is suitable for Hermitian matrices which are not known to be positive-definite. If is in fact positive-definite, no interchanges are performed and no 2 by 2 blocks occur in .

	Error Indicators and Warnings
	"NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . "NE_INTERNAL_ERROR" An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance. "NE_SINGULAR" The block diagonal matrix is exactly singular.

	Accuracy
	If , the computed factors and are the exact factors of a perturbed matrix , where is a modest linear function of , and is the machine precision. If , a similar statement holds for the computed factors and .

Further Comments

The elements of overwrite the corresponding elements of ; if has 2 by 2 blocks, only the upper or lower triangle is stored, as specified by uplo.

The unit diagonal elements of or and the 2 by 2 unit diagonal blocks are not stored. The remaining elements of and are stored in the corresponding columns of the array ap, but additional row interchanges must be applied to recover or explicitly (this is seldom necessary). If , for (as is the case when is positive-definite), then or are stored explicitly in packed form (except for their unit diagonal elements which are equal to 1).

The total number of real floating-point operations is approximately .

A call to nag_zhptrf (f07prc) may be followed by calls to the functions:

f07psc (nag_zhptrs) to solve ;

f07puc (nag_zhpcon) to estimate the condition number of ;

f07pwc (nag_zhptri) to compute the inverse of .

The real analogue of this function is f07pdc (nag_dsptrf).

Examples

uplo := "Nag_Upper":
n := 4:
ap := Vector([-1.36 +0*I, 1.58 +0.9*I, 2.21 -0.21*I, 3.91 +1.5*I, -8.869999999999999 +0*I, -1.84 -0.03*I, -1.78 +1.18*I, -4.63 +0*I, 0.11 +0.11*I, -1.84 +0*I], datatype=complex[8], order='C_order'):
ipiv := Vector(4, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f07prc(uplo, n, ap, ipiv):

ap;

ipiv;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f07 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.