Cholesky factorization of complex Hermitian positive-definite band matrix

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NAG[f07hrc] NAG[nag_zpbtrf] - Cholesky factorization of complex Hermitian positive-definite band matrix

Calling Sequence

f07hrc(uplo, kd, ab, 'n'=n, 'fail'=fail)

nag_zpbtrf(. . .)

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". .

kd - integer;

On entry: , the number of superdiagonals or subdiagonals of the matrix .

Constraint: . .

ab - 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 band matrix . This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements depends on the storage order and uplo arguments as follows:

if and , is stored in , for and ;

if and , is stored in , for and .

On exit: the upper or lower triangle of is overwritten by the Cholesky factor or as specified by uplo, using the same storage format as described above.

'n'=n - integer; (optional)

Default value: the second dimension of the array ab.

On entry: , the order of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zpbtrf (f07hrc) computes the Cholesky factorization of a complex Hermitian positive-definite band matrix.

	Description
	nag_zpbtrf (f07hrc) forms the Cholesky factorization of a complex Hermitian positive-definite band matrix either as if or if , where (or ) is an upper (or lower) triangular band matrix with the same number of superdiagonals (or subdiagonals) as .

	Error Indicators and Warnings
	"NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . 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 matrix is not positive-definite.

	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 , assuming .

A call to nag_zpbtrf (f07hrc) may be followed by calls to the functions:

f07hsc (nag_zpbtrs) to solve ;

f07huc (nag_zpbcon) to estimate the condition number of .

The real analogue of this function is f07hdc (nag_dpbtrf).

Examples

>	uplo := "Nag_Lower": n := 4: kd := 1: ab := Matrix([[0 +0I , 9.390000000000001 +0I , 1.08 +1.73I , 1.69 +0I ], [-0.04 -0.29I , 2.65 +0I , -0.33 -2.24I , 2.17 +0I ]], datatype=complex[8], order='C_order'): NAG:-f07hrc(uplo, kd, ab, 'n' = n):

ab;

	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.