nag_lapack_dsbevd (f08hc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix. If the eigenvectors are requested, then it uses a divide-and-conquer algorithm to compute eigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal–Walker–Kahan variant of the

Q L

Q R

algorithm.

Syntax

[ab, w, z, info] = f08hc(job, uplo, kd, ab, 'n', n)

[ab, w, z, info] = nag_lapack_dsbevd(job, uplo, kd, ab, 'n', n)

Description

nag_lapack_dsbevd (f08hc) computes all the eigenvalues and, optionally, all the eigenvectors of a real symmetric band matrix

A

. In other words, it can compute the spectral factorization of

A

A = Z Λ Z^{T},

where

Λ

is a diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the orthogonal matrix whose columns are the eigenvectors

z_{i}

. Thus

A z_{i} = λ_{i} z_{i}, i = 1, 2, \dots, n .

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

Parameters

Compulsory Input Parameters

1: $job$ – string (length ≥ 1)

Indicates whether eigenvectors are computed.

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

Constraint:

job ='N'

'V'

2: $uplo$ – string (length ≥ 1)

Indicates whether the upper or lower triangular part of

A

is stored.

$uplo ='U'$: The upper triangular part of $A$ is stored.
$uplo ='L'$: The lower triangular part of $A$ is stored.

Constraint:

uplo ='U'

'L'

3: $kd$ – int64int32nag_int scalar

uplo ='U'

, the number of superdiagonals,

k_{d}

, of the matrix

A

uplo ='L'

, the number of subdiagonals,

k_{d}

, of the matrix

A

Constraint:

kd \geq 0

4: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kd + 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_{d} + 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 $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

Optional Input Parameters

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

Constraint: $n \geq 0$ .

Output Parameters

1: $ab (ldab :)$ – double array

The first dimension of the array ab will be

kd + 1

The second dimension of the array ab will be

\max (1, n)

ab stores values generated during the reduction to tridiagonal form.

The first superdiagonal or subdiagonal and the diagonal of the tridiagonal matrix

T

are returned in ab using the same storage format as described above.

2: $w (:)$ – double array

The dimension of the array w will be

\max (1, n)

The eigenvalues of the matrix

A

in ascending order.

3: $z (ldz :)$ – double array

The first dimension,

ldz

, of the array z will be

if $job ='V'$ , $ldz = \max (1, n)$ ;
if $job ='N'$ , $ldz = 1$ .

The second dimension of the array z will be

\max (1, n)

job ='V'

and at least

1

job ='N'

job ='V'

, z stores the orthogonal matrix

Z

which contains the eigenvectors of

A

. The

i

th column of

Z

contains the eigenvector which corresponds to the eigenvalue

w (i)

job ='N'

, z is not referenced.

4: $info$ – 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: job, 2: uplo, 3: n, 4: kd, 5: ab, 6: ldab, 7: w, 8: z, 9: ldz, 10: work, 11: lwork, 12: iwork, 13: liwork, 14: 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 $job ='N'$ , the algorithm failed to converge; $i$ elements of an intermediate tridiagonal form did not converge to zero; if $info = i$ and $job ='V'$ , then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column $i / (n + 1)$ through $i mod (n + 1)$ .

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix

(A + E)

, where

{‖E‖}_{2} = O (ε) {‖A‖}_{2},

and

ε

is the machine precision. See Section 4.7 of Anderson et al. (1999) for further details.

Further Comments

The complex analogue of this function is nag_lapack_zhbevd (f08hq).

Example

This example computes all the eigenvalues and eigenvectors of the symmetric band matrix

A

, where

A = (\begin{matrix} 1 & 2 & 3 & 0 & 0 \\ 2 & 2 & 3 & 4 & 0 \\ 3 & 3 & 3 & 4 & 5 \\ 0 & 4 & 4 & 4 & 5 \\ 0 & 0 & 5 & 5 & 5 \end{matrix}) .

Open in the MATLAB editor: f08hc_example

function f08hc_example


fprintf('f08hc example results\n\n');

% Symmetric band matrix A, stored in symmetric banded format
uplo = 'L';
kd = int64(2);
n  = 5;
ab = [1, 2, 3, 4, 5;
      2, 3, 4, 5, 0;
      3, 4, 5, 0, 0];

% Calculate all the eigenvalues and eigenvectors of A
job = 'V';
[abf, w, z, info] = f08hc( ...
                           job, uplo, kd, ab);

% Normalize eigenvectors: largest element positive
for j = 1:n
  [~,k] = max(abs(z(:,j)));
  if z(k,j) < 0;
    z(:,j) = -z(:,j);
  end
end                            

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

[ifail] = x04ca( ...
		 'General', ' ', z, 'Eigenvectors');

f08hc example results

Eigenvalues
   -3.2474   -2.6633    1.7511    4.1599   14.9997

 Eigenvectors
          1       2       3       4       5
 1   0.0394  0.6238  0.5635 -0.5165  0.1582
 2   0.5721 -0.2575 -0.3896 -0.5955  0.3161
 3  -0.4372 -0.5900  0.4008 -0.1470  0.5277
 4  -0.4424  0.4308 -0.5581  0.0470  0.5523
 5   0.5332  0.1039  0.2421  0.5956  0.5400

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

Chapter Contents

Chapter Introduction

NAG Toolbox