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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zhpevx (f08gp)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zhpevx (f08gp) computes selected eigenvalues and, optionally, eigenvectors of a complex n by n Hermitian matrix A in packed storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.


[ap, m, w, z, jfail, info] = f08gp(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
[ap, m, w, z, jfail, info] = nag_lapack_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)


The Hermitian matrix A is first reduced to real tridiagonal form, using unitary similarity transformations. The required eigenvalues and eigenvectors are then computed from the tridiagonal matrix; the method used depends upon whether all, or selected, eigenvalues and eigenvectors are required.


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
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore


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.
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:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
5:     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.
6:     vl – double scalar
7:     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.
8:     il int64int32nag_int scalar
9:     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 .
10:   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 tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am   , not zero. If this function returns with info>0, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am   . See Demmel and Kahan (1990).

Optional Input Parameters


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:     m int64int32nag_int scalar
The total number of eigenvalues found. 0mn.
If range='A', m=n.
If range='I', m=iu-il+1.
3:     wn – double array
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', then
  • if info=0, 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 an eigenvector fails to converge (info>0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz='N', z is not referenced.
5:     jfail: int64int32nag_int array
The dimension of the array jfail will be max1,n
If jobz='V', then
  • if info=0, the first m elements of jfail are zero;
  • if info>0, jfail contains the indices of the eigenvectors that failed to converge.
If jobz='N', jfail is not referenced.
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, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The algorithm failed to converge; _ eigenvectors did not converge. Their indices are stored in array jfail.


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_dspevx (f08gb).


This example finds the eigenvalues in the half-open interval -2,2 , 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 .  
function f08gp_example

fprintf('f08gp example results\n\n');

% Hermitian A stored in packed format.
uplo = 'U';
n = int64(4);
ap = [1;
      2 - 1i;      2 + 0i;
      3 - 1i;      3 - 2i;      3 + 0i;
      4 - 1i;      4 - 2i;      4 - 3i;      4 + 0i];

% Eigenvalues in range [-2,2] of Hermitian A stored in packed format.
jobz  = 'Vectors';
range = 'Values in range';
vl = -2;
vu =  2;
il = int64(0);
iu = int64(0);
abstol = 0;
[~, m, w, z, jfail, info] = ...
f08gp( ...
       jobz, range, uplo, n, ap, vl, vu, il, iu, abstol);

% Normalize: largest elements are real
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));

fprintf('Number of eigenvalues in [-2,2] is %2d\n',m);
fprintf('\n Eigenvalues are:\n');

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

f08gp example results

Number of eigenvalues in [-2,2] is  2

 Eigenvalues 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)

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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