NAG Library Routine Document

F08TSF (ZHPGST)

+− 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

F08TSF (ZHPGST) reduces a complex Hermitian-definite generalized eigenproblem

A z = λ B z

A B z = λ z

B A z = λ z

to the standard form

C y = λ y

, where

A

is a complex Hermitian matrix and

B

has been factorized by F07GRF (ZPPTRF), using packed storage.

2 Specification

SUBROUTINE F08TSF (

ITYPE, UPLO, N, AP, BP, INFO)

INTEGER	ITYPE, N, INFO
COMPLEX (KIND=nag_wp)	AP(), BP()
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zhpgst.

3 Description

To reduce the complex Hermitian-definite generalized eigenproblem

A z = λ B z

A B z = λ z

B A z = λ z

to the standard form

C y = λ y

using packed storage, F08TSF (ZHPGST) must be preceded by a call to F07GRF (ZPPTRF) which computes the Cholesky factorization of

B

;

B

must be positive definite.

The different problem types are specified by the parameter ITYPE, as indicated in the table below. The table shows how

C

is computed by the routine, and also how the eigenvectors

z

of the original problem can be recovered from the eigenvectors of the standard form.

ITYPE	Problem	UPLO	$B$	$C$	$z$
$1$	$A z = λ B z$	'U' 'L'	$U^{H} U$ $L L^{H}$	$U^{- H} A U^{- 1}$ $L^{- 1} A L^{- H}$	$U^{- 1} y$ $L^{- H} y$
$2$	$A B z = λ z$	'U' 'L'	$U^{H} U$ $L L^{H}$	$U A U^{H}$ $L^{H} A L$	$U^{- 1} y$ $L^{- H} y$
$3$	$B A z = λ z$	'U' 'L'	$U^{H} U$ $L L^{H}$	$U A U^{H}$ $L^{H} A L$	$U^{H} y$ $L y$

4 References

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

5 Parameters

1: ITYPE – INTEGERInput

On entry: indicates how the standard form is computed.

$ITYPE = 1$

if $UPLO ='U'$ , $C = U^{- H} A U^{- 1}$ ;
if $UPLO ='L'$ , $C = L^{- 1} A L^{- H}$ .

$ITYPE = 2$ or $3$

if $UPLO ='U'$ , $C = U A U^{H}$ ;
if $UPLO ='L'$ , $C = L^{H} A L$ .

Constraint:

ITYPE = 1

2

3

2: UPLO – CHARACTER(1)Input

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

A

is stored and how

B

has been factorized.

$UPLO ='U'$: The upper triangular part of $A$ is stored and $B = U^{H} U$ .
$UPLO ='L'$: The lower triangular part of $A$ is stored and $B = L L^{H}$ .

Constraint:

UPLO ='U'

'L'

3: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

4: 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 upper or lower triangle of 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: the upper or lower triangle of AP is overwritten by the corresponding upper or lower triangle of

C

as specified by ITYPE and UPLO, using the same packed storage format as described above.

5: BP( $*$ ) – COMPLEX (KIND=nag_wp) arrayInput

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

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

On entry: the Cholesky factor of

B

as specified by UPLO and returned by F07GRF (ZPPTRF).

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

7 Accuracy

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

if (

ITYPE = 1

) or

B

(if

ITYPE = 2

3

). When F08TSF (ZHPGST) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if

B

is ill-conditioned with respect to inversion. See the document for F08SNF (ZHEGV) for further details.

8 Further Comments

The total number of real floating point operations is approximately

4 n^{3}

The real analogue of this routine is F08TEF (DSPGST).

9 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} - 7.36 + 0.00 i & 0.77 - 0.43 i & - 0.64 - 0.92 i & 3.01 - 6.97 i \\ 0.77 + 0.43 i & 3.49 + 0.00 i & 2.19 + 4.45 i & 1.90 + 3.73 i \\ - 0.64 + 0.92 i & 2.19 - 4.45 i & 0.12 + 0.00 i & 2.88 - 3.17 i \\ 3.01 + 6.97 i & 1.90 - 3.73 i & 2.88 + 3.17 i & - 2.54 + 0.00 i \end{array})

and

B = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}),

using packed storage. Here

B

is Hermitian positive definite and must first be factorized by F07GRF (ZPPTRF). The program calls F08TSF (ZHPGST) to reduce the problem to the standard form

C y = λ y

; then F08GSF (ZHPTRD) to reduce

C

to tridiagonal form, and F08JFF (DSTERF) to compute the eigenvalues.

NAG Library Routine DocumentF08TSF (ZHPGST)

+− 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

F08TSF (ZHPGST)