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NAG Toolbox: nag_lapack_zhbgvd (f08uq)

Purpose

nag_lapack_zhbgvd (f08uq) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
Az = λBz ,
Az=λBz ,
where AA and BB are Hermitian and banded, and BB is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

Syntax

[ab, bb, w, z, info] = f08uq(jobz, uplo, ka, kb, ab, bb, 'n', n)
[ab, bb, w, z, info] = nag_lapack_zhbgvd(jobz, uplo, ka, kb, ab, bb, 'n', n)

Description

The generalized Hermitian-definite band problem
Az = λ Bz
Az = λ Bz
is first reduced to a standard band Hermitian problem
Cx = λx ,
Cx = λx ,
where CC is a Hermitian band matrix, using Wilkinson's modification to Crawford's algorithm (see Crawford (1973) and Wilkinson (1977)). The Hermitian eigenvalue problem is then solved for the eigenvalues and the eigenvectors, if required, which are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that the matrix of eigenvectors, ZZ, satisfies
ZH A Z = Λ   and   ZH B Z = I ,
ZH A Z = Λ   and   ZH B Z = I ,
where Λ Λ  is the diagonal matrix whose diagonal elements are the eigenvalues.

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
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1977) Some recent advances in numerical linear algebra The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press

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:     uplo – string (length ≥ 1)
If uplo = 'U'uplo='U', the upper triangles of AA and BB are stored.
If uplo = 'L'uplo='L', the lower triangles of AA and BB are stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     ka – int64int32nag_int scalar
If uplo = 'U'uplo='U', the number of superdiagonals, kaka, of the matrix AA.
If uplo = 'L'uplo='L', the number of subdiagonals, kaka, of the matrix AA.
Constraint: ka0ka0.
4:     kb – int64int32nag_int scalar
If uplo = 'U'uplo='U', the number of superdiagonals, kbkb, of the matrix BB.
If uplo = 'L'uplo='L', the number of subdiagonals, kbkb, of the matrix BB.
Constraint: kakb0kakb0.
5:     ab(ldab, : :) – complex array
The first dimension of the array ab must be at least ka + 1ka+1
The second dimension of the array must be at least max (1,n)max(1,n)
The upper or lower triangle of the nn by nn Hermitian band matrix AA.
The matrix is stored in rows 11 to ka + 1ka+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of AA within the band must be stored with element AijAij in ab(ka + 1 + ij,j)​ for ​max (1,jka)ijabka+1+i-jj​ for ​max(1,j-ka)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of AA within the band must be stored with element AijAij in ab(1 + ij,j)​ for ​jimin (n,j + ka).ab1+i-jj​ for ​jimin(n,j+ka).
6:     bb(ldbb, : :) – complex array
The first dimension of the array bb must be at least kb + 1kb+1
The second dimension of the array must be at least max (1,n)max(1,n)
The upper or lower triangle of the nn by nn Hermitian band matrix BB.
The matrix is stored in rows 11 to kb + 1kb+1, more precisely,
  • if uplo = 'U'uplo='U', the elements of the upper triangle of BB within the band must be stored with element BijBij in bb(kb + 1 + ij,j)​ for ​max (1,jkb)ijbbkb+1+i-jj​ for ​max(1,j-kb)ij;
  • if uplo = 'L'uplo='L', the elements of the lower triangle of BB within the band must be stored with element BijBij in bb(1 + ij,j)​ for ​jimin (n,j + kb).bb1+i-jj​ for ​jimin(n,j+kb).

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The second dimension of the arrays ab, bb.
nn, the order of the matrices AA and BB.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

ldab ldbb ldz work lwork rwork lrwork iwork liwork

Output Parameters

1:     ab(ldab, : :) – complex array
The first dimension of the array ab will be ka + 1ka+1
The second dimension of the array will be max (1,n)max(1,n)
ldabka + 1ldabka+1.
The contents of ab are overwritten.
2:     bb(ldbb, : :) – complex array
The first dimension of the array bb will be kb + 1kb+1
The second dimension of the array will be max (1,n)max(1,n)
ldbbkb + 1ldbbkb+1.
The factor SS from the split Cholesky factorization B = SHSB=SHS, as returned by nag_lapack_zpbstf (f08ut).
3:     w(n) – double array
The 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,n)max(1,n) if jobz = 'V'jobz='V', and at least 11 otherwise
If jobz = 'V'jobz='V', z contains the matrix ZZ of eigenvectors, with the iith column of ZZ holding the eigenvector associated with w(i)wi. The eigenvectors are normalized so that ZHBZ = IZHBZ=I.
If jobz = 'N'jobz='N', z is not referenced.
5:     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: uplo, 3: n, 4: ka, 5: kb, 6: ab, 7: ldab, 8: bb, 9: ldbb, 10: w, 11: z, 12: ldz, 13: work, 14: lwork, 15: rwork, 16: lrwork, 17: iwork, 18: liwork, 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 and inin, the algorithm failed to converge; ii off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
If info = iinfo=i and i > ni>n, if info = n + iinfo=n+i, for 1in1in, then nag_lapack_zpbstf (f08ut) returned info = iinfo=i: BB is not positive definite. The factorization of BB could not be completed and no eigenvalues or eigenvectors were computed.

Accuracy

If BB is ill-conditioned with respect to inversion, then the error bounds for the computed eigenvalues and vectors may be large, although when the diagonal elements of BB differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of BB would suggest. See Section 4.10 of Anderson et al. (1999) for details of the error bounds.

Further Comments

The total number of floating point operations is proportional to n3 n3  if jobz = 'V'jobz='V' and, assuming that nka nka , is approximately proportional to n2 ka n2 ka  otherwise.
The real analogue of this function is nag_lapack_dsbgvd (f08uc).

Example

function nag_lapack_zhbgvd_example
jobz = 'No vectors';
uplo = 'U';
ka = int64(2);
kb = int64(1);
ab = [0,  0 + 0i,  -1.4 + 0.25i,  -0.67 + 0.34i;
      0 + 0i,  1.94 - 2.1i,  -0.82 - 0.89i,  -1.1 - 0.16i;
      -1.13 + 0i,  -1.91 + 0i,  -1.87 + 0i,  0.5 + 0i];
bb = [0,  1.08 - 1.73i,  -0.04 + 0.29i,  -0.33 + 2.24i;
      9.89 + 0i,  1.69 + 0i,  2.65 + 0i,  2.17 + 0i];
[abOut, bbOut, w, z, info] = nag_lapack_zhbgvd(jobz, uplo, ka, kb, ab, bb)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.8302 + 0.1482i  -0.8561 + 1.5855i
   0.0000 + 0.0000i   1.5456 + 0.0000i   2.7633 + 0.0000i   2.4855 + 0.0000i
  -0.1143 + 0.0000i  -1.7545 + 0.0000i  -3.7946 + 0.0000i  -1.0557 + 0.0000i


bbOut =

   0.0000 + 0.0000i   0.3434 - 0.5501i  -0.0746 + 0.5408i  -0.2240 + 1.5206i
   3.1448 + 0.0000i   0.9856 + 0.0000i   0.5362 + 0.0000i   1.4731 + 0.0000i


w =

   -6.6089
   -2.0416
    0.1603
    1.7712


z =

   0.0000 + 0.0000i


info =

                    0


function f08uq_example
jobz = 'No vectors';
uplo = 'U';
ka = int64(2);
kb = int64(1);
ab = [0,  0 + 0i,  -1.4 + 0.25i,  -0.67 + 0.34i;
      0 + 0i,  1.94 - 2.1i,  -0.82 - 0.89i,  -1.1 - 0.16i;
      -1.13 + 0i,  -1.91 + 0i,  -1.87 + 0i,  0.5 + 0i];
bb = [0,  1.08 - 1.73i,  -0.04 + 0.29i,  -0.33 + 2.24i;
      9.89 + 0i,  1.69 + 0i,  2.65 + 0i,  2.17 + 0i];
[abOut, bbOut, w, z, info] = f08uq(jobz, uplo, ka, kb, ab, bb)
 

abOut =

   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.8302 + 0.1482i  -0.8561 + 1.5855i
   0.0000 + 0.0000i   1.5456 + 0.0000i   2.7633 + 0.0000i   2.4855 + 0.0000i
  -0.1143 + 0.0000i  -1.7545 + 0.0000i  -3.7946 + 0.0000i  -1.0557 + 0.0000i


bbOut =

   0.0000 + 0.0000i   0.3434 - 0.5501i  -0.0746 + 0.5408i  -0.2240 + 1.5206i
   3.1448 + 0.0000i   0.9856 + 0.0000i   0.5362 + 0.0000i   1.4731 + 0.0000i


w =

   -6.6089
   -2.0416
    0.1603
    1.7712


z =

   0.0000e+00 +6.9239e-310i


info =

                    0



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

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