NAG Library Routine Document

F08UNF (ZHBGV)

+− 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

F08UNF (ZHBGV) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form

A z = λ B z,

where

A

and

B

are Hermitian and banded, and

B

is also positive definite.

2 Specification

SUBROUTINE F08UNF (

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

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

The routine may be called by its LAPACK name zhbgv.

3 Description

The generalized Hermitian-definite band problem

A z = λ B z

is first reduced to a standard band Hermitian problem

C x = λ x,

where

C

is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian 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^{H} A Z = Λ and Z^{H} 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, $*$ ) – COMPLEX (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

Hermitian 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 F08UNF (ZHBGV) is called.

Constraint:

LDAB \geq KA + 1

8: BB(LDBB, $*$ ) – COMPLEX (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

Hermitian 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^{H} S

, as returned by F08UTF (ZPBSTF).

9: LDBB – INTEGERInput

On entry: the first dimension of the array BB as declared in the (sub)program from which F08UNF (ZHBGV) 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, $*$ ) – COMPLEX (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^{H} 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 F08UNF (ZHBGV) is called.

Constraints:

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

13: WORK(N) – COMPLEX (KIND=nag_wp) arrayWorkspace

14: RWORK( $3 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

15: 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 F08UTF (ZPBSTF) 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 real analogue of this routine is F08UAF (DSBGV).

9 Example

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

A z = λ B z

, where

A = (\begin{array}{r} - 1.13 & 1.94 - 2.10 i & - 1.40 + 0.25 i & 0 \\ 1.94 + 2.10 i & - 1.91 & - 0.82 - 0.89 i & - 0.67 + 0.34 i \\ - 1.40 - 0.25 i & - 0.82 + 0.89 i & - 1.87 & - 1.10 - 0.16 i \\ 0 & - 0.67 - 0.34 i & - 1.10 + 0.16 i & 0.50 \end{array})

and

B = (\begin{array}{r} 9.89 & 1.08 - 1.73 i & 0 & 0 \\ 1.08 + 1.73 i & 1.69 & - 0.04 + 0.29 i & 0 \\ 0 & - 0.04 - 0.29 i & 2.65 & - 0.33 + 2.24 i \\ 0 & 0 & - 0.33 - 2.24 i & 2.17 \end{array}) .

NAG Library Routine DocumentF08UNF (ZHBGV)

+− 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

F08UNF (ZHBGV)