Cholesky factorization of complex Hermitian positive-definite matrix, packed storage

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NAG[f07grc] NAG[nag_zpptrf] - Cholesky factorization of complex Hermitian positive-definite matrix, packed storage

Calling Sequence

f07grc(uplo, n, ap, 'fail'=fail)

nag_zpptrf(. . .)

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 positive-definite 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: the upper or lower triangle of is overwritten by the Cholesky factor or as specified by uplo, using the same packed storage format as described above.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zpptrf (f07grc) computes the Cholesky factorization of a complex Hermitian positive-definite matrix, using packed storage.

	Description
	nag_zpptrf (f07grc) forms the Cholesky factorization of a complex Hermitian positive-definite matrix either as if or if , where is an upper triangular matrix and is lower triangular, using packed storage.

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_POS_DEF"

The leading minor of order is not positive-definite and the factorization could not be completed. Hence itself is not positive-definite. This may indicate an error in forming the matrix . To factorize a matrix which is not positive-definite, call f07prc (nag_zhptrf) instead.

	Accuracy
	If , the computed factor is the exact factor of a perturbed matrix , where is a modest linear function of , and is the machine precision. If , a similar statement holds for the computed factor . It follows that .

Further Comments

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

A call to nag_zpptrf (f07grc) may be followed by calls to the functions:

f07gsc (nag_zpptrs) to solve ;

f07guc (nag_zppcon) to estimate the condition number of ;

f07gwc (nag_zpptri) to compute the inverse of .

The real analogue of this function is f07gdc (nag_dpptrf).

Examples

>	uplo := "Nag_Lower": n := 4: ap := Vector([3.23 +0I, 1.51 +1.92I, 3.58 +0I, 1.9 -0.84I, -0.23 -1.11I, 4.09 +0I, 0.42 -2.5I, -1.18 -1.37I, 2.33 +0.14I, 4.29 +0I], datatype=complex[8], order='C_order'): NAG:-f07grc(uplo, n, ap):

ap;

	See Also
	Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville 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.