nag_lapack_dsbgvd (f08uc) 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. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

Syntax

[ab, bb, w, z, info] = f08uc(jobz, uplo, ka, kb, ab, bb, 'n', n)

[ab, bb, w, z, info] = nag_lapack_dsbgvd(jobz, uplo, ka, kb, ab, bb, 'n', n)

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.

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

Parameters

Compulsory Input Parameters

1: $jobz$ – string (length ≥ 1)

Indicates whether eigenvectors are computed.

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

Constraint:

jobz ='N'

'V'

2: $uplo$ – string (length ≥ 1)

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: $ka$ – int64int32nag_int scalar

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

4: $kb$ – int64int32nag_int scalar

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

5: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

ka + 1

The second dimension of the array ab must be at least

\max (1, n)

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 $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{a}) .$

6: $bb (ldbb :)$ – double array

The first dimension of the array bb must be at least

kb + 1

The second dimension of the array bb must be at least

\max (1, n)

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 $bb (1 + i - j, j) for j \leq i \leq \min (n, j + k_{b}) .$

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the arrays ab, bb. (An error is raised if these dimensions are not equal.)
$n$ , the order of the matrices $A$ and $B$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – double array

The first dimension of the array ab will be

ka + 1

The second dimension of the array ab will be

\max (1, n)

The contents of ab are overwritten.

2: $bb (ldbb :)$ – double array

The first dimension of the array bb will be

kb + 1

The second dimension of the array bb will be

\max (1, n)

The factor

S

from the split Cholesky factorization

B = S^{T} S

, as returned by nag_lapack_dpbstf (f08uf).

3: $w (n)$ – double array

The eigenvalues in ascending order.

4: $z (ldz :)$ – double array

The first dimension,

ldz

, of the array z will be

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

The second dimension of the array z will be

\max (1, n)

jobz ='V'

and

1

otherwise.

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.

5: $info$ – int64int32nag_int scalar

info = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: uplo, 3: n, 4: ka, 5: kb, 6: ab, 7: ldab, 8: bb, 9: ldbb, 10: w, 11: z, 12: ldz, 13: work, 14: lwork, 15: iwork, 16: liwork, 17: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

$info > 0$: 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 nag_lapack_dpbstf (f08uf) returned $info = i$ : $B$ is not positive definite. The factorization of $B$ could not be completed and no eigenvalues or eigenvectors were computed.

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.

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 function is nag_lapack_zhbgvd (f08uq).

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}) .

Open in the MATLAB editor: f08uc_example

function f08uc_example


fprintf('f08uc example results\n\n');

% Symmetric banded matrices A and B stored in symmetric banded format
uplo = 'U';
ka = int64(2);
ab = [0,     0,     0.42,  0.63;
      0,     0.39,  0.79,  0.48;
      0.24, -0.11, -0.25, -0.03];
kb = int64(1);
bb = [0,     0.95, -0.29, -0.33;
      2.07,  1.69,  0.65,  1.17];

% Eigenvalues only of Ax = lmbda Bx
jobz = 'No vectors';
[~, ~, w, ~, info] = f08uc( ...
			    jobz, uplo, ka, kb, ab, bb);

disp('Eigenvalues');
disp(w');

f08uc example results

Eigenvalues
   -0.8305   -0.6401    0.0992    1.8525

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox