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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhpevd (f08gq)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

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 QL or QR 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 as
A=ZΛZH,  
where Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex) unitary matrix whose columns are the eigenvectors zi. Thus
Azi=λizi,  i=1,2,,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' or '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' or 'L'.
3:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
4:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The upper or lower triangle of the n by n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×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 max1,n
The eigenvalues of the matrix A in ascending order.
3:     zldz: – complex array
The first dimension, ldz, of the array z will be
  • if job='V', ldz= max1,n ;
  • if job='N', ldz=1.
The second dimension of the array z will be max1,n if job='V' and at least 1 if job='N'.
If job='V', z stores the unitary matrix Z which contains the eigenvectors of A.
If 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
E2 = Oε A2 ,  
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 = 1.0+0.0i 2.0-1.0i 3.0-1.0i 4.0-1.0i 2.0+1.0i 2.0+0.0i 3.0-2.0i 4.0-2.0i 3.0+1.0i 3.0+2.0i 3.0+0.0i 4.0-3.0i 4.0+1.0i 4.0+2.0i 4.0+3.0i 4.0+0.0i .  
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)

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Chapter Contents
Chapter Introduction
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