NAG C Library, Mark 7 : f07mwc

#include <nag.h> #include <nagf07.h>
void nag_zhetri	(Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, const Integer ipiv[], NagError *fail)

3 Description

To compute the inverse of a complex Hermitian indefinite matrix

A

, this function must be preceded by a call to nag_zhetrf (f07mrc), which computes the Bunch–Kaufman factorization of

A

uplo

=

Nag_Upper,

A = P U D U^{H} P^{T}

and

A^{- 1}

is computed by solving

U^{H} P^{T} X P U = D^{- 1}

for

X

uplo

=

Nag_Lower,

A = P L D L^{H} P^{T}

and

A^{- 1}

is computed by solving

L^{H} P^{T} X P L = D^{- 1}

for

X

4 References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

5 Parameters

1: order – Nag_OrderTypeInput

On entry: the order parameter specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order

=

Nag_RowMajor. See Section 2.2.1.4 of the Essential Introduction for a more detailed explanation of the use of this parameter.

Constraint: order

=

Nag_RowMajor or Nag_ColMajor.

2: uplo – Nag_UploType Input

On entry: indicates how

A

has been factorized as follows:

if $uplo$ $=$ Nag_Upper, $A = P U D U^{H} P^{T}$ , where $U$ is upper triangular;
if $uplo$ $=$ Nag_Lower, $A = P L D L^{H} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo

=

Nag_Upper or Nag_Lower.

3: n – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: a[ $\dim$ ] – Complex Input/Output

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

\max (1, pda \times \max (1, n))

On entry: details of the factorization of

A

, as returned by nag_zhetrf (f07mrc).

On exit: the factorization is overwritten by the

n

n

Hermitian matrix

A^{- 1}

. If

uplo

=

Nag_Upper, the upper triangle of

A^{- 1}

is stored in the upper triangular part of the array; if

uplo

=

Nag_Lower, the lower triangle of

A^{- 1}

is stored in the lower triangular part of the array.

5: pda – Integer Input

On entry: the stride separating row or column elements (depending on the value of order ) of the matrix in the array a .

Constraint:

pda \geq \max (1, n)

6: ipiv[ $\dim$ ] – const Integer Input

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

\max (1, n)

On entry: details of the interchanges and the block structure of

D

, as returned by nag_zhetrf (f07mrc).

7: fail – NagError *Input/Output

The NAG error parameter (see the Essential Introduction).

6 Error Indicators and Warnings

NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .; On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ , $n = 〈value〉$ .
Constraint: $pda \geq \max (1, n)$ .
NE_SINGULAR: The block diagonal matrix $D$ is exactly singular.
NE_ALLOC_FAIL: Memory allocation failed.
NE_BAD_PARAM: On entry, parameter $〈value〉$ had an illegal value.
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.

7 Accuracy

The computed inverse

X

satisfies a bound of the form

if $uplo$ $=$ Nag_Upper, $|D U^{H} P^{T} X P U - I| \leq c (n) ε (|D| |U^{H}| P^{T} |X| P |U| + |D| |D^{- 1}|)$ ;
if $uplo$ $=$ Nag_Lower, $|D L^{H} P^{T} X P L - I| \leq c (n) ε (|D| |L^{H}| P^{T} |X| P |L| + |D| |D^{- 1}|)$ ,