NAG Library Routine Document

F08TAF (DSPGV)

which is solved for the eigenvalues and, optionally, the eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.

For the problem

A z = λ B z

, the eigenvectors are normalized so that the matrix of eigenvectors,

Z

, satisfies

Z^{T} A Z = Λ and Z^{T} B Z = I,

where

Λ

is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem

A B z = λ z

we correspondingly have

Z^{- 1} A Z^{- T} = Λ and Z^{T} B Z = I,

and for

B A z = λ z

we have

Z^{T} A Z = Λ and Z^{T} B^{- 1} Z = I .

4 References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

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: specifies the problem type to be solved.

$ITYPE = 1$: $A z = λ B z$ .
$ITYPE = 2$: $A B z = λ z$ .
$ITYPE = 3$: $B A z = λ z$ .

Constraint:

ITYPE = 1

2

3

2: JOBZ – CHARACTER(1)Input

On entry: indicates whether eigenvectors are computed.

$JOBZ ='N'$: Only eigenvalues are computed.
$JOBZ ='V'$: Eigenvalues and eigenvectors are computed.

Constraint:

JOBZ ='N'

'V'

3: UPLO – CHARACTER(1)Input

On entry: if

UPLO ='U'

, the upper triangles of

A

and

B

are stored.

UPLO ='L'

, the lower triangles of

A

and

B

are stored.

Constraint:

UPLO ='U'

'L'

4: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

5: 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 contents of AP are destroyed.

6: BP( $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

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

On entry: the upper or lower triangle of the

n

n

symmetric matrix

B

, packed by columns.

More precisely,

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

On exit: the triangular factor

U

L

from the Cholesky factorization

B = U^{T} U

B = L L^{T}

, in the same storage format as

B

7: W(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the eigenvalues in ascending order.

8: Z(LDZ, $*$ ) – REAL (KIND=nag_wp) arrayOutput

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

\max (1, N)

JOBZ ='V'

, and at least

1

otherwise.

On exit: if

JOBZ ='V'

, Z contains the matrix

Z

of eigenvectors. The eigenvectors are normalized as follows:

if $ITYPE = 1$ or $2$ , $Z^{T} B Z = I$ ;
if $ITYPE = 3$ , $Z^{T} B^{- 1} Z = I$ .

JOBZ ='N'

, Z is not referenced.

9: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08TAF (DSPGV) is called.

Constraints:

if $JOBZ ='V'$ , $LDZ \geq \max (1, N)$ ;
otherwise $LDZ \geq 1$ .

10: WORK( $3 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

11: 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.

$INFO > 0$

F07GDF (DPPTRF) or F08GAF (DSPEV) returned an error code:

$\leq N$	if $INFO = i$ , F08GAF (DSPEV) failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero;
$> N$	if $INFO = N + i$ , for $1 \leq i \leq N$ , then the leading minor of order $i$ of $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

7 Accuracy

B

is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of

B

differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of

B

would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

The example program below illustrates the computation of approximate error bounds.

8 Further Comments

The total number of floating point operations is proportional to

n^{3}

The complex analogue of this routine is F08TNF (ZHPGV).

9 Example

This example finds all the eigenvalues and eigenvectors of the generalized symmetric eigenproblem

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}),

together with an estimate of the condition number of

B

, and approximate error bounds for the computed eigenvalues and eigenvectors.

The example program for F08TCF (DSPGVD) illustrates solving a generalized symmetric eigenproblem of the form

A B z = λ z

NAG Library Routine DocumentF08TAF (DSPGV)

+− 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

F08TAF (DSPGV)