nag_lapack_zhpevd (f08gq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix held in packed storage. 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

[ap, w, z, info] = f08gq(job, uplo, n, ap)

[ap, w, z, info] = nag_lapack_zhpevd(job, uplo, n, ap)

Description

nag_lapack_zhpevd (f08gq) computes all the eigenvalues and, optionally, all the eigenvectors of a complex Hermitian matrix

A

(held in packed storage). 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: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

4: $ap (:)$ – complex array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The upper or lower triangle of the

n

n

Hermitian matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

Optional Input Parameters

None.

Output Parameters

1: $ap (:)$ – complex array

The dimension of the array ap will be

\max (1, n \times (n + 1) / 2)

ap stores the values generated during the reduction to tridiagonal form. The elements of the diagonal and the off-diagonal of the tridiagonal matrix overwrite the corresponding elements 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: $z (ldz :)$ – 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

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: ap, 5: w, 6: z, 7: ldz, 8: work, 9: lwork, 10: rwork, 11: lrwork, 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 real analogue of this function is nag_lapack_dspevd (f08gc).

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}) .

Open in the MATLAB editor: f08gq_example

function f08gq_example


fprintf('f08gq example results\n\n');

% Lower triangular part of Hermitian matrix A
uplo = 'L';
n = int64(4);
ap = [1;      2 + 1i;      3 + 1i;      4 + 1i;
              2 + 0i;      3 + 2i;      4 + 2i;
                           3 + 0i;      4 + 3i;
                                        4 + 0i];

% Calculate all the eigenvalues and eigenvectors of A
job = 'Vectors';
[apf, w, z, info] = f08gq( ...
                             job, uplo, n, ap);

% 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);

f08gq 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