NAG Library Routine Document

F08SEF (DSYGST)

+− 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

F08SEF (DSYGST) reduces a real symmetric-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 real symmetric matrix and

B

has been factorized by F07FDF (DPOTRF).

2 Specification

SUBROUTINE F08SEF (

ITYPE, UPLO, N, A, LDA, B, LDB, INFO)

INTEGER	ITYPE, N, LDA, LDB, INFO
REAL (KIND=nag_wp)	A(LDA,), B(LDB,)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dsygst.

3 Description

To reduce the real symmetric-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

, F08SEF (DSYGST) must be preceded by a call to F07FDF (DPOTRF) 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'	$U^{T} U$ $L L^{T}$	$U^{- T} A U^{- 1}$ $L^{- 1} A L^{- T}$	$U^{- 1} y$ $L^{- T} y$
$2$	$A B z = λ z$	'U' 'L'	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{- 1} y$ $L^{- T} y$
$3$	$B A z = λ z$	'U' 'L'	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{T} y$ $L y$

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 $UPLO ='U'$ , $C = U^{- T} A U^{- 1}$ ;
if $UPLO ='L'$ , $C = L^{- 1} A L^{- T}$ .

$ITYPE = 2$ or $3$

if $UPLO ='U'$ , $C = U A U^{T}$ ;
if $UPLO ='L'$ , $C = L^{T} A L$ .

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 $B = U^{T} U$ .
$UPLO ='L'$: The lower triangular part of $A$ is stored and $B = L L^{T}$ .

Constraint:

UPLO ='U'

'L'

3: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

4: A(LDA, $*$ ) – REAL (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

symmetric 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 the corresponding upper or lower triangle of

C

as specified by ITYPE and UPLO.

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08SEF (DSYGST) is called.

Constraint:

LDA \geq \max (1, N)

6: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput

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

\max (1, N)

On entry: the Cholesky factor of

B

as specified by UPLO and returned by F07FDF (DPOTRF).

7: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F08SEF (DSYGST) is called.

Constraint:

LDB \geq \max (1, N)

8: 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 F08SEF (DSYGST) 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 F08SAF (DSYGV) for further details.

8 Further Comments

The total number of floating point operations is approximately

n^{3}

The complex analogue of this routine is F08SSF (ZHEGST).

9 Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} 0.24 & 0.39 & 0.42 & - 0.16 \\ 0.39 & - 0.11 & 0.79 & 0.63 \\ 0.42 & 0.79 & - 0.25 & 0.48 \\ - 0.16 & 0.63 & 0.48 & - 0.03 \end{array}) and B = (\begin{array}{r} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.09 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.09 & 0.34 & 1.18 \end{array}) .

Here

B

is symmetric positive definite and must first be factorized by F07FDF (DPOTRF). The program calls F08SEF (DSYGST) to reduce the problem to the standard form

C y = λ y

; then F08FEF (DSYTRD) to reduce

C

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

NAG Library Routine DocumentF08SEF (DSYGST)

+− 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

F08SEF (DSYGST)