NAG Library Routine Document

F07MRF (ZHETRF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

F07MRF (ZHETRF) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix.

2 Specification

SUBROUTINE F07MRF (

UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO)

INTEGER	N, LDA, IPIV(*), LWORK, INFO
COMPLEX (KIND=nag_wp)	A(LDA,*), WORK(max(1,LWORK))
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zhetrf.

3 Description

F07MRF (ZHETRF) factorizes a complex Hermitian matrix

A

, using the Bunch–Kaufman diagonal pivoting method.

A

is factorized either as

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: $A (LDA *)$ – COMPLEX (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least

\max (1, N)

On entry: the

n

n

Hermitian indefinite matrix

A

If $UPLO ='U'$ , the upper triangular part of $A$ must be stored and the elements of the array below the diagonal are not referenced.
If $UPLO ='L'$ , the lower triangular part of $A$ must be stored and the elements of the array above the diagonal are not referenced.

On exit: the upper or lower triangle of

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: $LDA$ – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07MRF (ZHETRF) is called.

Constraint:

LDA \geq \max (1, N)

5: $IPIV (*)$ – INTEGER arrayOutput

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

\max (1, N)

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.

6: $WORK (\max (1, LWORK))$ – COMPLEX (KIND=nag_wp) arrayWorkspace

On exit: if

INFO = 0

WORK (1)

contains the minimum value of LWORK required for optimum performance.

7: $LWORK$ – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F07MRF (ZHETRF) is called, unless

LWORK = - 1

, in which case a workspace query is assumed and the routine only calculates the optimal dimension of WORK (using the formula given below).

Suggested value: for optimum performance LWORK should be at least

N \times nb

, where

nb

is the block size.

Constraint:

LWORK \geq 1

LWORK = - 1

8: $INFO$ – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

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

$INFO > 0$: Element $〈value〉$ of the diagonal 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 Parallelism and Performance

F07MRF (ZHETRF) is not threaded by NAG in any implementation.

F07MRF (ZHETRF) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 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

L

are stored in the corresponding columns of the array A, 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

is stored explicitly (except for its 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 F07MRF (ZHETRF) may be followed by calls to the routines:

F07MSF (ZHETRS) to solve $A X = B$ ;
F07MUF (ZHECON) to estimate the condition number of $A$ ;
F07MWF (ZHETRI) to compute the inverse of $A$ .

The real analogue of this routine is F07MDF (DSYTRF).

10 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}) .

NAG Library Routine DocumentF07MRF (ZHETRF)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

F07MRF (ZHETRF)