nag_lapack_zheevd (f08fq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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

[a, w, info] = f08fq(job, uplo, a, 'n', n)

[a, w, info] = nag_lapack_zheevd(job, uplo, a, 'n', n)

Description

nag_lapack_zheevd (f08fq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian 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: $a (lda :)$ – complex array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

Hermitian matrix

A

If $uplo ='U'$ , the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

If $job ='V'$ , a stores the unitary matrix $Z$ which contains the eigenvectors of $A$ .
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: $info$ – 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: a, 5: lda, 6: w, 7: work, 8: lwork, 9: rwork, 10: lrwork, 11: iwork, 12: liwork, 13: 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_dsyevd (f08fc).

Example

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

A

, where

A = (\begin{array}{r} 1.0 + 0.0 i & 2.0 - 1.0 i & 3.0 - 1.0 i & 4.0 - 1.0 i \\ 2.0 + 1.0 i & 2.0 + 0.0 i & 3.0 - 2.0 i & 4.0 - 2.0 i \\ 3.0 + 1.0 i & 3.0 + 2.0 i & 3.0 + 0.0 i & 4.0 - 3.0 i \\ 4.0 + 1.0 i & 4.0 + 2.0 i & 4.0 + 3.0 i & 4.0 + 0.0 i \end{array}) .

The example program for nag_lapack_zheevd (f08fq) illustrates the computation of error bounds for the eigenvalues and eigenvectors.

Open in the MATLAB editor: f08fq_example

function f08fq_example


fprintf('f08fq example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'L';
n = 4;
a = [ 1 + 0i,  0 + 0i,  0 + 0i,  0 + 0i;
      2 + 1i,  2 + 0i,  0 + 0i,  0 + 0i;
      3 + 1i,  3 + 2i,  3 + 0i,  0 + 0i;
      4 + 1i,  4 + 2i,  4 + 3i,  4 + 0i];

% Calculate all the eigenvalues and eigenvectors of A
job = 'Vectors';
[z, w, info] = f08fq( ...
                      job, uplo, a);

% Normalize vectors, largest element is real and positive.
for i = 1:n
  [~,k] = max(abs(real(z(:,i)))+abs(imag(z(:,i))));
  z(:,i) = z(:,i)*conj(z(k,i))/abs(z(k,i));
end

% Display results
fprintf('Eigenvalues:\n');
disp(w);

ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
                 'General', ' ', z, 'Bracketed', 'F7.4', ...
                 'Eigenvectors', 'Integer', 'Integer', ...
                 ncols, indent);

f08fq example results

Eigenvalues:
   -4.2443
   -0.6886
    1.1412
   13.7916

 Eigenvectors
                    1                 2                 3                 4
 1  (-0.3839,-0.2941) ( 0.6470, 0.0000) (-0.4326, 0.1068) ( 0.3309,-0.1986)
 2  (-0.4512, 0.1102) (-0.4984,-0.1130) (-0.1590,-0.5480) ( 0.3728,-0.2419)
 3  ( 0.0263, 0.4857) ( 0.2949, 0.3165) ( 0.5491, 0.0000) ( 0.4870,-0.1938)
 4  ( 0.5602, 0.0000) (-0.2241,-0.2878) (-0.2865, 0.3037) ( 0.6155, 0.0000)

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

Chapter Contents

Chapter Introduction

NAG Toolbox