NAG Library Routine Document

F07PRF (ZHPTRF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07PRF (ZHPTRF) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.

2 Specification

SUBROUTINE F07PRF (

UPLO, N, AP, IPIV, INFO)

INTEGER	N, IPIV(N), INFO
COMPLEX (KIND=nag_wp)	AP(*)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zhptrf.

3 Description

F07PRF (ZHPTRF) factorizes a complex Hermitian matrix

A

, using the Bunch–Kaufman diagonal pivoting method and packed storage.

A

is factorized as either

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

UPLO ='U'

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

UPLO ='L'

, where

P

is a permutation matrix,

U

(or

L

) is a unit upper (or lower) triangular matrix and

D

is an Hermitian block diagonal matrix with

1

1

and

2

2

diagonal blocks;

U

(or

L

) has

2

2

unit diagonal blocks corresponding to the

2

2

blocks of

D

. 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

A

is in fact positive definite, no interchanges are performed and no

2

2

blocks occur in

D

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Parameters

1: UPLO – CHARACTER(1)Input

On entry: specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$UPLO ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $P U D U^{H} P^{T}$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $P L D L^{H} P^{T}$ , where $L$ is lower triangular.

Constraint:

UPLO ='U'

'L'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: AP( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array AP must be at least

\max (1, N \times (N + 1) / 2)

On entry: the

n

n

Hermitian matrix

A

, packed by columns.

More precisely,

if $UPLO ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $UPLO ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $AP (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

On exit:

A

is overwritten by details of the block diagonal matrix

D

and the multipliers used to obtain the factor

U

L

as specified by UPLO.

4: IPIV(N) – INTEGER arrayOutput

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

D

. More precisely,

if $IPIV (i) = k > 0$ , $d_{i i}$ is a $1$ by $1$ pivot block and the $i$ th row and column of $A$ were interchanged with the $k$ th row and column;
if $UPLO ='U'$ and $IPIV (i - 1) = IPIV (i) = - l < 0$ , $(\begin{matrix} d_{i - 1, i - 1} & {\bar{d}}_{i, i - 1} \\ {\bar{d}}_{i, i - 1} & d_{i i} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i - 1)$ th row and column of $A$ were interchanged with the $l$ th row and column;
if $UPLO ='L'$ and $IPIV (i) = IPIV (i + 1) = - m < 0$ , $(\begin{matrix} d_{i i} & d_{i + 1, i} \\ d_{i + 1, i} & d_{i + 1, i + 1} \end{matrix})$ is a $2$ by $2$ pivot block and the $(i + 1)$ th row and column of $A$ were interchanged with the $m$ th row and column.

5: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0$: If $INFO = i$ , $d (i, i)$ is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

7 Accuracy

UPLO ='U'

, the computed factors

U

and

D

are the exact factors of a perturbed matrix

A + E

, where

|E| \leq c (n) ε P |U| |D| |U^{H}| P^{T},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

UPLO ='L'

, a similar statement holds for the computed factors

L

and

D

8 Further Comments

The elements of

D

overwrite the corresponding elements of

A

; if

D

has

2

2

blocks, only the upper or lower triangle is stored, as specified by UPLO.

The unit diagonal elements of

U

L

and the

2

2

unit diagonal blocks are not stored. The remaining elements of

U

and

L

are stored in the corresponding columns of the array AP, but additional row interchanges must be applied to recover

U

L

explicitly (this is seldom necessary). If

IPIV (i) = i

, for

i = 1, 2, \dots, n

(as is the case when

A

is positive definite), then

U

L

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

\frac{4}{3} n^{3}

A call to F07PRF (ZHPTRF) may be followed by calls to the routines:

F07PSF (ZHPTRS) to solve $A X = B$ ;
F07PUF (ZHPCON) to estimate the condition number of $A$ ;
F07PWF (ZHPTRI) to compute the inverse of $A$ .

The real analogue of this routine is F07PDF (DSPTRF).

9 Example

This example computes the Bunch–Kaufman factorization of the matrix

A

, where

A = (\begin{array}{r} - 1.36 + 0.00 i & 1.58 + 0.90 i & 2.21 - 0.21 i & 3.91 + 1.50 i \\ 1.58 - 0.90 i & - 8.87 + 0.00 i & - 1.84 - 0.03 i & - 1.78 + 1.18 i \\ 2.21 + 0.21 i & - 1.84 + 0.03 i & - 4.63 + 0.00 i & 0.11 + 0.11 i \\ 3.91 - 1.50 i & - 1.78 - 1.18 i & 0.11 - 0.11 i & - 1.84 + 0.00 i \end{array}),

using packed storage.

NAG Library Routine DocumentF07PRF (ZHPTRF)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07PRF (ZHPTRF)