Bunch–Kaufman factorization of complex Hermitian indefinite matrix

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NAG[f07mrc] NAG[nag_zhetrf] - Bunch–Kaufman factorization of complex Hermitian indefinite matrix

Calling Sequence

f07mrc(uplo, a, ipiv, 'n'=n, 'fail'=fail)

nag_zhetrf(. . .)

Parameters

uplo - String;

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

a - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by Hermitian matrix .

If , the upper triangle of must be stored and the elements of the array below the diagonal are not referenced.

If , the lower triangle of must be stored and the elements of the array above the diagonal are not referenced.

On exit: the upper or lower triangle of 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.

'n'=n - integer; (optional)

Default value: the dimension of the array a.

On entry: , the order of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zhetrf (f07mrc) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix.

Description

nag_zhetrf (f07mrc) factorizes a complex Hermitian matrix , using the Bunch–Kaufman diagonal pivoting method. 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_ALLOC_FAIL" Dynamic memory allocation failed. "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 or are stored in the corresponding columns of the array a, 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 is stored explicitly (except for its unit diagonal elements which are equal to 1).

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

A call to nag_zhetrf (f07mrc) may be followed by calls to the functions:

f07msc (nag_zhetrs) to solve ;

f07muc (nag_zhecon) to estimate the condition number of ;

f07mwc (nag_zhetri) to compute the inverse of .

The real analogue of this function is f07mdc (nag_dsytrf).

Examples

uplo := "Nag_Upper":
n := 4:
a := Matrix([[-1.36 +0*I , 1.58 +0.9*I , 2.21 -0.21*I , 3.91 +1.5*I ], [0 +0*I , -8.869999999999999 +0*I , -1.84 -0.03*I , -1.78 +1.18*I ], [0 +0*I , 0 +0*I , -4.63 +0*I , 0.11 +0.11*I ], [0 +0*I , 0 +0*I , 0 +0*I , -1.84 +0*I ]], datatype=complex[8], order='C_order'):
ipiv := Vector(4, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f07mrc(uplo, a, ipiv, 'n' = n):

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.