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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zheevr (f08fr)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zheevr (f08fr) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.


[a, m, w, z, isuppz, info] = f08fr(jobz, range, uplo, a, vl, vu, il, iu, abstol, 'n', n)
[a, m, w, z, isuppz, info] = nag_lapack_zheevr(jobz, range, uplo, a, vl, vu, il, iu, abstol, 'n', n)


The Hermitian matrix is first reduced to a real tridiagonal matrix T, using unitary similarity transformations. Then whenever possible, nag_lapack_zheevr (f08fr) computes the eigenspectrum using Relatively Robust Representations. nag_lapack_zheevr (f08fr) computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various ‘good’ LDLT representations (also known as Relatively Robust Representations). Gram–Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the ith unreduced block of T:
(a) compute T - σi I = Li Di LiT , such that Li Di LiT  is a relatively robust representation,
(b) compute the eigenvalues, λj, of Li Di LiT  to high relative accuracy by the dqds algorithm,
(c) if there is a cluster of close eigenvalues, ‘choose’ σi close to the cluster, and go to (a),
(d) given the approximate eigenvalue λj of Li Di LiT , compute the corresponding eigenvector by forming a rank-revealing twisted factorization.
The desired accuracy of the output can be specified by the argument abstol. For more details, see Dhillon (1997) and Parlett and Dhillon (2000).


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
Barlow J and Demmel J W (1990) Computing accurate eigensystems of scaled diagonally dominant matrices SIAM J. Numer. Anal. 27 762–791
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Dhillon I (1997) A new On2 algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem Computer Science Division Technical Report No. UCB//CSD-97-971 UC Berkeley
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Parlett B N and Dhillon I S (2000) Relatively robust representations of symmetric tridiagonals Linear Algebra Appl. 309 121–151


Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
Only eigenvalues are computed.
Eigenvalues and eigenvectors are computed.
Constraint: jobz='N' or 'V'.
2:     range – string (length ≥ 1)
If range='A', all eigenvalues will be found.
If range='V', all eigenvalues in the half-open interval vl,vu will be found.
If range='I', the ilth to iuth eigenvalues will be found.
For range='V' or 'I' and iu-il<n-1, nag_lapack_dstebz (f08jj) and nag_lapack_zstein (f08jx) are called.
Constraint: range='A', 'V' or 'I'.
3:     uplo – string (length ≥ 1)
If uplo='U', the upper triangular part of A is stored.
If uplo='L', the lower triangular part of A is stored.
Constraint: uplo='U' or 'L'.
4:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by 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.
5:     vl – double scalar
6:     vu – double scalar
If range='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If range='A' or 'I', vl and vu are not referenced.
Constraint: if range='V', vl<vu.
7:     il int64int32nag_int scalar
8:     iu int64int32nag_int scalar
If range='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range='A' or 'V', il and iu are not referenced.
  • if range='I' and n=0, il=1 and iu=0;
  • if range='I' and n>0, 1 il iu n .
9:     abstol – double scalar
The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval a,b  of width less than or equal to
abstol+ε maxa,b ,  
where ε  is the machine precision. If abstol is less than or equal to zero, then ε T1  will be used in its place, where T is the real tridiagonal matrix obtained by reducing A to tridiagonal form. See Demmel and Kahan (1990).
If high relative accuracy is important, set abstol to x02am  , although doing so does not currently guarantee that eigenvalues are computed to high relative accuracy. See Barlow and Demmel (1990) for a discussion of which matrices can define their eigenvalues to high relative accuracy.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
The lower triangle (if uplo='L') or the upper triangle (if uplo='U') of a, including the diagonal, is overwritten.
2:     m int64int32nag_int scalar
The total number of eigenvalues found. 0mn.
If range='A', m=n.
If range='I', m=iu-il+1.
3:     w: – double array
The dimension of the array w will be max1,n
The first m elements contain the selected eigenvalues in ascending order.
4:     zldz: – complex array
The first dimension, ldz, of the array z will be
  • if jobz='V', ldz= max1,n ;
  • otherwise ldz=1.
The second dimension of the array z will be max1,m if jobz='V' and 1 otherwise.
If jobz='V', the first m columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the ith column of Z holding the eigenvector associated with wi.
If jobz='N', z is not referenced.
5:     isuppz: int64int32nag_int array
The dimension of the array isuppz will be max1,2×m
The support of the eigenvectors in z, i.e., the indices indicating the nonzero elements in z. The ith eigenvector is nonzero only in elements isuppz2×i-1 through isuppz2×i. Implemented only for range='A' or 'I' and iu-il=n-1.
6:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: jobz, 2: range, 3: uplo, 4: n, 5: a, 6: lda, 7: vl, 8: vu, 9: il, 10: iu, 11: abstol, 12: m, 13: w, 14: z, 15: ldz, 16: isuppz, 17: work, 18: lwork, 19: rwork, 20: lrwork, 21: iwork, 22: liwork, 23: 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.
nag_lapack_zheevr (f08fr) failed to converge.


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 total number of floating-point operations is proportional to n3.
The real analogue of this function is nag_lapack_dsyevr (f08fd).


This example finds the eigenvalues with indices in the range 2,3 , and the corresponding eigenvectors, of the Hermitian matrix
A = 1 2-i 3-i 4-i 2+i 2 3-2i 4-2i 3+i 3+2i 3 4-3i 4+i 4+2i 4+3i 4 .  
Information on required and provided workspace is also output.
function f08fr_example

fprintf('f08fr example results\n\n');

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

% Compute 2nd and 3rd eigenvalues and coresponding vectors
jobz  = 'Vectors';
range = 'I';
vl = 0;
vu = 0;
il = int64(2);
iu = int64(3);
abstol = 0;
[~, m, w, z, ~, info] = ...
  f08fr( ...
         jobz, range, uplo, a, vl, vu, il, iu, abstol);

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

% Display
fprintf(' Eigenvalues numbered 2 to 3 are:\n');

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

f08fr example results

 Eigenvalues numbered 2 to 3 are:

 Corresponding eigenvectors
                    1                 2
 1  ( 0.6470, 0.0000) ( 0.0179,-0.4453)
 2  (-0.4984,-0.1130) ( 0.5706, 0.0000)
 3  ( 0.2949, 0.3165) (-0.1530, 0.5273)
 4  (-0.2241,-0.2878) (-0.2118,-0.3598)

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Chapter Contents
Chapter Introduction
NAG Toolbox

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