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NAG Toolbox: nag_lapack_zhpevx (f08gp)

Purpose

nag_lapack_zhpevx (f08gp) computes selected eigenvalues and, optionally, eigenvectors of a complex nn by nn Hermitian matrix AA 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.

Syntax

[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)

Description

The Hermitian matrix AA 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.

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
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

Parameters

Compulsory Input Parameters

1:     jobz – string (length ≥ 1)
Indicates whether eigenvectors are computed.
jobz = 'N'jobz='N'
Only eigenvalues are computed.
jobz = 'V'jobz='V'
Eigenvalues and eigenvectors are computed.
Constraint: jobz = 'N'jobz='N' or 'V''V'.
2:     range – string (length ≥ 1)
If range = 'A'range='A', all eigenvalues will be found.
If range = 'V'range='V', all eigenvalues in the half-open interval (vl,vu](vl,vu] will be found.
If range = 'I'range='I', the ilth to iuth eigenvalues will be found.
Constraint: range = 'A'range='A', 'V''V' or 'I''I'.
3:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangular part of AA is stored.
If uplo = 'L'uplo='L', the lower triangular part of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
4:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
5:     ap( : :) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The upper or lower triangle of the nn by nn Hermitian matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.
6:     vl – double scalar
7:     vu – double scalar
If range = 'V'range='V', the lower and upper bounds of the interval to be searched for eigenvalues.
If range = 'A'range='A' or 'I''I', vl and vu are not referenced.
Constraint: if range = 'V'range='V', vl < vuvl<vu.
8:     il – int64int32nag_int scalar
9:     iu – int64int32nag_int scalar
If range = 'I'range='I', the indices (in ascending order) of the smallest and largest eigenvalues to be returned.
If range = 'A'range='A' or 'V''V', il and iu are not referenced.
Constraints:
  • if range = 'I'range='I' and n = 0n=0, il = 1il=1 and iu = 0iu=0;
  • if range = 'I'range='I' and n > 0n>0, 1 il iu n 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] [a,b]  of width less than or equal to
abstol + ε max (|a|,|b|) ,
abstol+ε max(|a|,|b|) ,
where ε ε  is the machine precision. If abstol is less than or equal to zero, then ε T1 ε T1  will be used in its place, where TT is the tridiagonal matrix obtained by reducing AA to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set to twice the underflow threshold 2 × x02am (   ) 2 × x02am ( ) , not zero. If this function returns with INFO > 0INFO>0, indicating that some eigenvectors did not converge, try setting abstol to 2 × x02am (   ) 2 × x02am ( ) . See Demmel and Kahan (1990).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

ldz work rwork iwork

Output Parameters

1:     ap( : :) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,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 AA.
2:     m – int64int32nag_int scalar
The total number of eigenvalues found. 0mn0mn.
If range = 'A'range='A', m = nm=n.
If range = 'I'range='I', m = iuil + 1m=iu-il+1.
3:     w(n) – double array
The selected eigenvalues in ascending order.
4:     z(ldz, : :) – complex array
The first dimension, ldz, of the array z will be
  • if jobz = 'V'jobz='V', ldz max (1,n) ldz max(1,n) ;
  • otherwise ldz1ldz1.
The second dimension of the array will be max (1,m)max(1,m) if jobz = 'V'jobz='V', and at least 11 otherwise
If jobz = 'V'jobz='V', then
  • if INFO = 0INFO=0, the first m columns of ZZ contain the orthonormal eigenvectors of the matrix AA corresponding to the selected eigenvalues, with the iith column of ZZ holding the eigenvector associated with w(i)wi;
  • if an eigenvector fails to converge (INFO > 0INFO>0), then that column of ZZ contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If jobz = 'N'jobz='N', z is not referenced.
5:     jfail( : :) – int64int32nag_int array
Note: the dimension of the array jfail must be at least max (1,n)max(1,n).
If jobz = 'V'jobz='V', then
  • if INFO = 0INFO=0, the first m elements of jfail are zero;
  • if INFO > 0INFO>0, jfail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N'jobz='N', jfail is not referenced.
6:     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: jobz, 2: range, 3: uplo, 4: n, 5: ap, 6: vl, 7: vu, 8: il, 9: iu, 10: abstol, 11: m, 12: w, 13: z, 14: ldz, 15: work, 16: rwork, 17: iwork, 18: jfail, 19: 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, then ii eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.

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 total number of floating point operations is proportional to n3n3.
The real analogue of this function is nag_lapack_dspevx (f08gb).

Example

function nag_lapack_zhpevx_example
jobz = 'Vectors';
range = 'Values in range';
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];
vl = -2;
vu = 2;
il = int64(-1208245600);
iu = int64(121387641);
abstol = 0;
[apOut, m, w, z, jfail, info] = nag_lapack_zhpevx(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
 

apOut =

  -0.2187 + 0.0000i
   1.0422 + 0.0000i
  -0.3942 + 0.0000i
   0.4448 + 0.4277i
  -3.4564 + 0.0000i
   6.6129 + 0.0000i
   0.3367 + 0.0008i
   0.3567 - 0.0783i
  -7.8740 + 0.0000i
   4.0000 + 0.0000i


m =

                    2


w =

   -0.6886
    1.1412
         0
         0


z =

  -0.3975 + 0.5105i  -0.3746 - 0.2414i
   0.3953 - 0.3238i   0.2895 - 0.4917i
  -0.4309 + 0.0383i   0.3768 + 0.3994i
   0.3648 + 0.0000i  -0.4175 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0


function f08gp_example
jobz = 'Vectors';
range = 'Values in range';
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];
vl = -2;
vu = 2;
il = int64(-1208245600);
iu = int64(121387641);
abstol = 0;
[apOut, m, w, z, jfail, info] = f08gp(jobz, range, uplo, n, ap, vl, vu, il, iu, abstol)
 

apOut =

  -0.2187 + 0.0000i
   1.0422 + 0.0000i
  -0.3942 + 0.0000i
   0.4448 + 0.4277i
  -3.4564 + 0.0000i
   6.6129 + 0.0000i
   0.3367 + 0.0008i
   0.3567 - 0.0783i
  -7.8740 + 0.0000i
   4.0000 + 0.0000i


m =

                    2


w =

   -0.6886
    1.1412
         0
         0


z =

  -0.3975 + 0.5105i  -0.3746 - 0.2414i
   0.3953 - 0.3238i   0.2895 - 0.4917i
  -0.4309 + 0.0383i   0.3768 + 0.3994i
   0.3648 + 0.0000i  -0.4175 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0



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Chapter Contents
Chapter Introduction
NAG Toolbox

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