Integer, Intent (In)	::	n, ka, kb, ldab, ldbb, ldz
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	ab(ldab,), bb(ldbb,), z(ldz,*)
Real (Kind=nag_wp), Intent (Out)	::	w(n), work(3*n)
Character (1), Intent (In)	::	jobz, uplo

C Header Interface

#include <nag.h>

void	f08uaf_ (const char jobz, const char uplo, const Integer n, const Integer ka, const Integer kb, double ab[], const Integer ldab, double bb[], const Integer ldbb, double w[], double z[], const Integer ldz, double work[], Integer *info, const Charlen length_jobz, const Charlen length_uplo)

The routine may be called by the names f08uaf, nagf_lapackeig_dsbgv or 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 https://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 Arguments

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: $n$ – Integer Input

On entry:

n

, the order of the matrices

A

and

B

Constraint:

n \geq 0

4: $ka$ – Integer Input

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: $kb$ – Integer Input

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) array Input/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: $ldab$ – Integer Input

On entry: the first dimension of the array ab as declared in the (sub)program from which f08uaf is called.

Constraint:

ldab \geq ka + 1

8: $bb (ldbb *)$ – Real (Kind=nag_wp) array Input/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.

9: $ldbb$ – Integer Input

On entry: the first dimension of the array bb as declared in the (sub)program from which f08uaf is called.

Constraint:

ldbb \geq kb + 1

10: $w (n)$ – Real (Kind=nag_wp) array Output

On exit: the eigenvalues in ascending order.

11: $z (ldz *)$ – Real (Kind=nag_wp) array Output

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: $ldz$ – Integer Input

On entry: the first dimension of the array z as declared in the (sub)program from which f08uaf is called.

Constraints:

if $jobz ='V'$ , $ldz \geq \max (1, n)$ ;
otherwise $ldz \geq 1$ .

13: $work (3 \times n)$ – Real (Kind=nag_wp) array Workspace

14: $info$ – Integer Output

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.

$info = 1, \dots, n$: The algorithm failed to converge; $〈value〉$ off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

$info > n$: If $info = n + 〈value〉$ , for $1 \leq 〈value〉 \leq n$ , f08uff returned $info = 〈value〉$ : $B$ 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 Parallelism and Performance

f08uaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08uaf 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 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.

10 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}) .

Interfaces: FL CL AD

NAG FL Interface Introduction

F08 (Lapackeig) Chapter Contents

F08 (Lapackeig) Chapter Introduction

f08ua: FL CL AD

NAG FL Interfacef08uaf (dsbgv)

▸▿ 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

NAG FL Interface
f08uaf (dsbgv)