nag_lapack_zhbevd (f08hq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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] = f08hq(job, uplo, kd, ab, 'n', n)

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

Description

nag_lapack_zhbevd (f08hq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian band matrix

A

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

A

A = Z Λ Z^{H},

where

Λ

is a real diagonal matrix whose diagonal elements are the eigenvalues

λ_{i}

, and

Z

is the (complex) unitary 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: $ab (ldab :)$ – complex 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

Hermitian 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: $ab (ldab :)$ – complex 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: $z (ldz :)$ – complex 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 unitary 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: rwork, 13: lrwork, 14: iwork, 15: liwork, 16: 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 real analogue of this function is nag_lapack_dsbevd (f08hc).

Example

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

A

, where

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

Open in the MATLAB editor: f08hq_example

function f08hq_example


fprintf('f08hq example results\n\n');

% Hermitian band matrix A, stored on symmetric banded format
uplo = 'L';
kd = int64(2);
ab = [1 + 0i,  2 + 0i,  3 + 0i,  4 + 0i,  5 + 0i;
      2 + 1i,  3 + 2i,  4 + 3i,  5 + 4i,  0 + 0i;
      3 + 1i,  4 + 2i,  5 + 3i,  0 + 0i,  0 + 0i];

% All eigenvalues and eigenvectors of A
job = 'V';
[abf, w, z, info] = f08hq( ...
                           job, uplo, kd, ab);

% Normalize: largest elements are real
for i = 1:5
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

disp('Eigenvalues');
disp(w');
[ifail] = x04da( ...
                 'General', ' ', z, 'Eigenvectors');

f08hq example results

Eigenvalues
   -6.4185   -1.4094    1.4421    4.4856   16.9002

 Eigenvectors
          1       2       3       4       5
 1  -0.2534 -0.4188 -0.2560  0.4767  0.1051
    -0.0538  0.4797  0.3721 -0.2748 -0.0983

 2  -0.0662 -0.0122  0.5344  0.5524  0.2516
     0.4301 -0.3529  0.0000  0.0000 -0.1789

 3   0.5274  0.4621 -0.4245  0.2076  0.4994
     0.0000  0.0000  0.0915 -0.0660 -0.1513

 4   0.1061 -0.1642  0.4964 -0.1379  0.5611
    -0.4981  0.3146 -0.1546  0.1026  0.0000

 5  -0.4519 -0.0360 -0.1979 -0.2651  0.4837
     0.0424 -0.3593 -0.1114 -0.4948  0.2509

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

Chapter Contents

Chapter Introduction

NAG Toolbox