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NAG Toolbox: nag_lapack_zheevx (f08fp)

Purpose

nag_lapack_zheevx (f08fp) computes selected eigenvalues and, optionally, eigenvectors of a complex nn by nn Hermitian matrix AA. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[a, m, w, z, jfail, info] = f08fp(jobz, range, uplo, a, vl, vu, il, iu, abstol, 'n', n)
[a, m, w, z, jfail, info] = nag_lapack_zheevx(jobz, range, uplo, a, vl, vu, il, iu, abstol, 'n', n)

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:     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.
5:     vl – double scalar
6:     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.
7:     il – int64int32nag_int scalar
8:     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 .
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] [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

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.)
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda ldz work lwork rwork iwork

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).
The lower triangle (if uplo = 'L'uplo='L') or the upper triangle (if uplo = 'U'uplo='U') of a, including the diagonal, is overwritten.
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( : :) – double array
Note: the dimension of the array w must be at least max (1,n)max(1,n).
The first m elements contain 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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: a, 6: lda, 7: vl, 8: vu, 9: il, 10: iu, 11: abstol, 12: m, 13: w, 14: z, 15: ldz, 16: work, 17: lwork, 18: rwork, 19: iwork, 20: jfail, 21: 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.
W 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_dsyevx (f08fb).

Example

function nag_lapack_zheevx_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'Upper';
a = [1,  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];
vl = -2;
vu = 2;
il = int64(134537040);
iu = int64(0);
abstol = 0;
[aOut, m, w, z, jfail, info] = nag_lapack_zheevx(jobz, range, uplo, a, vl, vu, il, iu, abstol)
 

aOut =

  -0.2187 + 0.0000i   1.0422 + 0.0000i   0.4448 + 0.4277i   0.3367 + 0.0008i
   0.0000 + 0.0000i  -0.3942 + 0.0000i  -3.4564 + 0.0000i   0.3567 - 0.0783i
   0.0000 + 0.0000i   0.0000 + 0.0000i   6.6129 + 0.0000i  -7.8740 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 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 f08fp_example
jobz = 'Vectors';
range = 'Values in range';
uplo = 'Upper';
a = [1,  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];
vl = -2;
vu = 2;
il = int64(134537040);
iu = int64(0);
abstol = 0;
[aOut, m, w, z, jfail, info] = f08fp(jobz, range, uplo, a, vl, vu, il, iu, abstol)
 

aOut =

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