NAG Library Routine Document

F08UAF (DSBGV)

+− 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

F08UAF (DSBGV) computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form

A z = λ B z,

where

A

and

B

are symmetric and banded, and

B

is also positive definite.

2 Specification

SUBROUTINE F08UAF (

JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)

INTEGER	N, KA, KB, LDAB, LDBB, LDZ, INFO
REAL (KIND=nag_wp)	AB(LDAB,), BB(LDBB,), W(N), Z(LDZ,), WORK(3N)
CHARACTER(1)	JOBZ, UPLO

The routine may be called by its LAPACK name dsbgv.

3 Description

The generalized symmetric-definite band problem

A z = λ B z

is first reduced to a standard band symmetric problem

C x = λ x,

where

C

is a symmetric band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The symmetric eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.

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.

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

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44

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

Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

5 Parameters

1: 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'

2: 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'

3: N – INTEGERInput

On entry:

n

, the order of the matrices

A

and

B

Constraint:

N \geq 0

4: KA – INTEGERInput

On entry: if

UPLO ='U'

, the number of superdiagonals,

k_{a}

, of the matrix

A

UPLO ='L'

, the number of subdiagonals,

k_{a}

, of the matrix

A

Constraint:

KA \geq 0

5: KB – INTEGERInput

On entry: if

UPLO ='U'

, the number of superdiagonals,

k_{b}

, of the matrix

B

UPLO ='L'

, the number of subdiagonals,

k_{b}

, of the matrix

B

Constraint:

KA \geq KB \geq 0

6: AB(LDAB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the upper or lower triangle of the

n

n

symmetric band matrix

A

The matrix is stored in rows

1

k_{a} + 1

, more precisely,

if $UPLO ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (k_{a} + 1 + i - j, j) for \max (1, j - k_{a}) \leq i \leq j$ ;
if $UPLO ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $AB (1 + i - j, j) for j \leq i \leq \min (n, j + k_{a}) .$

On exit: the contents of AB are overwritten.

7: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F08UAF (DSBGV) is called.

Constraint:

LDAB \geq KA + 1

8: BB(LDBB, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, N)

On entry: the upper or lower triangle of the

n

n

symmetric band matrix

B

The matrix is stored in rows

1

k_{b} + 1

, more precisely,

if $UPLO ='U'$ , the elements of the upper triangle of $B$ within the band must be stored with element $B_{i j}$ in $BB (k_{b} + 1 + i - j, j) for \max (1, j - k_{b}) \leq i \leq j$ ;
if $UPLO ='L'$ , the elements of the lower triangle of $B$ within the band must be stored with element $B_{i j}$ in $BB (1 + i - j, j) for j \leq i \leq \min (n, j + k_{b}) .$

On exit: the factor

S

from the split Cholesky factorization

B = S^{T} S

, as returned by F08UFF (DPBSTF).

9: LDBB – INTEGERInput

On entry: the first dimension of the array BB as declared in the (sub)program from which F08UAF (DSBGV) is called.

Constraint:

LDBB \geq KB + 1

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

On exit: the eigenvalues in ascending order.

11: 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, with the

i

th column of

Z

holding the eigenvector associated with

W (i)

. The eigenvectors are normalized so that

Z^{T} B Z = I

JOBZ ='N'

, Z is not referenced.

12: LDZ – INTEGERInput

On entry: the first dimension of the array Z as declared in the (sub)program from which F08UAF (DSBGV) is called.

Constraints:

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

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

14: 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.

$INFO > 0$: If $INFO = i$ and $i \leq N$ , the algorithm failed to converge; $i$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

If $INFO = i$ and $i > N$ , if $INFO = N + i$ , for $1 \leq i \leq N$ , then F08UFF (DPBSTF) returned $INFO = i$ : $B$ 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.

8 Further Comments

The total number of floating point operations is proportional to

n^{3}

JOBZ ='V'

and, assuming that

n ≫ k_{a}

, is approximately proportional to

n^{2} k_{a}

otherwise.

The complex analogue of this routine is F08UNF (ZHBGV).

9 Example

This example finds all the eigenvalues of the generalized band symmetric eigenproblem

A z = λ B z

, where

A = (\begin{array}{r} 0.24 & 0.39 & 0.42 & 0 \\ 0.39 & - 0.11 & 0.79 & 0.63 \\ 0.42 & 0.79 & - 0.25 & 0.48 \\ 0 & 0.63 & 0.48 & - 0.03 \end{array}) and B = (\begin{array}{r} 2.07 & 0.95 & 0 & 0 \\ 0.95 & 1.69 & - 0.29 & 0 \\ 0 & - 0.29 & 0.65 & - 0.33 \\ 0 & 0 & - 0.33 & 1.17 \end{array}) .

NAG Library Routine DocumentF08UAF (DSBGV)

+− 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

F08UAF (DSBGV)