NAG Library Routine Document

Integer, Intent (In)	::	itype, n
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	bp(*)
Real (Kind=nag_wp), Intent (Inout)	::	ap(*)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nagmk26.h>

void	f08tef_ (const Integer itype, const char uplo, const Integer n, double ap[], const double bp[], Integer info, const Charlen length_uplo)

The routine may be called by its LAPACK name dspgst.

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

using packed storage, f08tef (dspgst) must be preceded by a call to f07gdf (dpptrf) which computes the Cholesky factorization of

B

;

B

must be positive definite.

The different problem types are specified by the argument 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

Arguments

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: $ap (*)$ – Real (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

symmetric 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: $bp (*)$ – Real (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 f07gdf (dpptrf).

6: $info$ – 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.

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 f08tef (dspgst) 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

Parallelism and Performance

f08tef (dspgst) 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 total number of floating-point operations is approximately

n^{3}

The complex analogue of this routine is f08tsf (zhpgst).

10

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}),

using packed storage. Here

B

is symmetric positive definite and must first be factorized by f07gdf (dpptrf). The program calls f08tef (dspgst) to reduce the problem to the standard form

C y = λ y

; then f08gef (dsptrd) to reduce

C

to tridiagonal form, and f08jff (dsterf) to compute the eigenvalues.

NAG Library Routine Document

f08tef (dspgst)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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

2

Specification

3

Description

4

References

5

Arguments

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