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NAG Toolbox: nag_lapack_zheevd (f08fq)

Purpose

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 QLQL or QRQR 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 AA. In other words, it can compute the spectral factorization of AA as
A = ZΛZH,
A=ZΛZH,
where ΛΛ is a real diagonal matrix whose diagonal elements are the eigenvalues λiλi, and ZZ is the (complex) unitary matrix whose columns are the eigenvectors zizi. Thus
Azi = λizi,  i = 1,2,,n.
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'job='N'
Only eigenvalues are computed.
job = 'V'job='V'
Eigenvalues and eigenvectors are computed.
Constraint: job = 'N'job='N' or 'V''V'.
2:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of AA is stored.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn Hermitian matrix AA.
  • If uplo = 'U'uplo='U', the upper triangular part of aa must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', the lower triangular part of aa 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 The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work lwork rwork lrwork iwork liwork

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
If job = 'V'job='V', a stores the unitary matrix ZZ which contains the eigenvectors of AA.
2:     w( : :) – double array
Note: the dimension of the array w must be at least max (1,n)max(1,n).
The eigenvalues of the matrix AA in ascending order.
3:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii 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 > 0INFO>0
if info = iinfo=i and job = 'N'job='N', the algorithm failed to converge; ii elements of an intermediate tridiagonal form did not converge to zero; if info = iinfo=i and job = 'V'job='V', then the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and column i / (n + 1)i/(n+1) through i  mod  (n + 1)i mod (n+1).

Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrix (A + E)(A+E), where
E2 = O(ε) A2 ,
E2 = O(ε) A2 ,
and εε 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

function nag_lapack_zheevd_example
job = 'V';
uplo = 'L';
a = [complex(1),  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];
[aOut, w, info] = nag_lapack_zheevd(job, uplo, a)
 

aOut =

  -0.4836 + 0.0000i  -0.6470 + 0.0000i  -0.4456 + 0.0000i  -0.3859 + 0.0000i
  -0.2912 + 0.3618i   0.4984 + 0.1130i  -0.0230 - 0.5702i  -0.4441 + 0.0156i
   0.3163 + 0.3696i  -0.2949 - 0.3165i   0.5331 + 0.1317i  -0.5173 - 0.0844i
   0.4447 - 0.3406i   0.2241 + 0.2878i  -0.3510 + 0.2261i  -0.5277 - 0.3168i


w =

   -4.2443
   -0.6886
    1.1412
   13.7916


info =

                    0


function f08fq_example
job = 'V';
uplo = 'L';
a = [complex(1),  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];
[aOut, w, info] = f08fq(job, uplo, a)
 

aOut =

  -0.4836 + 0.0000i  -0.6470 + 0.0000i  -0.4456 + 0.0000i  -0.3859 + 0.0000i
  -0.2912 + 0.3618i   0.4984 + 0.1130i  -0.0230 - 0.5702i  -0.4441 + 0.0156i
   0.3163 + 0.3696i  -0.2949 - 0.3165i   0.5331 + 0.1317i  -0.5173 - 0.0844i
   0.4447 - 0.3406i   0.2241 + 0.2878i  -0.3510 + 0.2261i  -0.5277 - 0.3168i


w =

   -4.2443
   -0.6886
    1.1412
   13.7916


info =

                    0



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Chapter Contents
Chapter Introduction
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