hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_zhbgvx (f08up)

Purpose

nag_lapack_zhbgvx (f08up) computes selected 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. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

Syntax

[ab, bb, q, m, w, z, jfail, info] = f08up(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol, 'n', n)
[ab, bb, q, m, w, z, jfail, info] = nag_lapack_zhbgvx(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol, '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 required eigenvalues and eigenvectors, and the eigenvectors are then backtransformed to the eigenvectors of the original problem.
The eigenvectors are normalized so that
zH A z = λ   and   zH B z = 1 .
zH A z = λ   and   zH B z = 1 .

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
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
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:     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 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'.
4:     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.
5:     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.
6:     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).
7:     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 positive definite 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).
8:     vl – double scalar
9:     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.
10:   il – int64int32nag_int scalar
11:   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 .
12:   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 CC 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 = 1tonINFO=1ton, 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 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 ldq ldz work rwork iwork

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:     q(ldq, : :) – complex array
The first dimension, ldq, of the array q will be
  • if jobz = 'V'jobz='V', ldq max (1,n) ldq max(1,n) ;
  • otherwise ldq1ldq1.
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', the nn by nn matrix, QQ used in the reduction of the standard form, i.e., Cx = λxCx=λx, from symmetric banded to tridiagonal form.
If jobz = 'N'jobz='N', q is not referenced.
4:     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.
5:     w(n) – double array
The eigenvalues in ascending order.
6:     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.
7:     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 = 1tonINFO=1ton, jfail contains the indices of the eigenvectors that failed to converge.
If jobz = 'N'jobz='N', jfail is not referenced.
8:     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: ka, 6: kb, 7: ab, 8: ldab, 9: bb, 10: ldbb, 11: q, 12: ldq, 13: vl, 14: vu, 15: il, 16: iu, 17: abstol, 18: m, 19: w, 20: z, 21: ldz, 22: work, 23: rwork, 24: iwork, 25: jfail, 26: 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 = 1tonINFO=1ton
If info = iinfo=i, then ii eigenvectors failed to converge. Their indices are stored in array jfail. Please see abstol.
  INFO > NINFO>N
nag_lapack_dpbstf (f08uf) returned an error code; i.e., if info = n + iinfo=n+i, for 1in1in, then the leading minor of order ii of 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 range = 'A'range='A', and assuming that nka nka , is approximately proportional to n2 ka n2 ka  if jobz = 'N'jobz='N'. Otherwise the number of floating point operations depends upon the number of eigenvectors computed.
The real analogue of this function is nag_lapack_dsbgvx (f08ub).

Example

function nag_lapack_zhbgvx_example
jobz = 'Vectors';
range = 'Values in range';
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];
vl = 0;
vu = 2;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[abOut, bbOut, q, m, w, z, jfail, info] = ...
    nag_lapack_zhbgvx(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol)
 

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


q =

   0.3180 + 0.0000i  -0.1735 - 0.0251i  -0.0358 + 0.0788i   0.0357 - 0.0653i
   0.0000 + 0.0000i   0.3424 + 0.7783i   0.4161 - 0.0551i  -0.3604 + 0.0208i
   0.0000 + 0.0000i  -1.7389 + 0.2747i   0.6388 - 0.6501i  -0.5894 + 0.5169i
   0.0000 + 0.0000i  -0.5480 - 1.7533i   1.0400 + 0.2112i  -0.2779 - 0.9114i


m =

                    2


w =

    0.1603
    1.7712
         0
         0


z =

   0.1908 + 0.0137i   0.0494 - 0.0045i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.1413 + 0.1012i   0.2505 + 0.4427i   0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.0437 - 0.0905i  -0.9705 + 0.0679i   0.0000 + 0.0000i   0.0000 + 0.0000i
  -0.2135 + 0.2880i   0.0606 - 1.3227i   0.0000 + 0.0000i   0.0000 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0


function f08up_example
jobz = 'Vectors';
range = 'Values in range';
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];
vl = 0;
vu = 2;
il = int64(0);
iu = int64(8185080);
abstol = 0;
[abOut, bbOut, q, m, w, z, jfail, info] = ...
    f08up(jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol)
 

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


q =

   0.3180 + 0.0000i  -0.1735 - 0.0251i  -0.0358 + 0.0788i   0.0357 - 0.0653i
   0.0000 + 0.0000i   0.3424 + 0.7783i   0.4161 - 0.0551i  -0.3604 + 0.0208i
   0.0000 + 0.0000i  -1.7389 + 0.2747i   0.6388 - 0.6501i  -0.5894 + 0.5169i
   0.0000 + 0.0000i  -0.5480 - 1.7533i   1.0400 + 0.2112i  -0.2779 - 0.9114i


m =

                    2


w =

    0.1603
    1.7712
         0
         0


z =

   0.1908 + 0.0137i   0.0494 - 0.0045i      NaN +    NaNi   0.0000 + 0.0000i
   0.1413 + 0.1012i   0.2505 + 0.4427i      NaN +    NaNi   0.0000 + 0.0000i
  -0.0437 - 0.0905i  -0.9705 + 0.0679i      NaN +    NaNi   0.0000 + 0.0000i
  -0.2135 + 0.2880i   0.0606 - 1.3227i   0.0000 + 0.0000i   0.0000 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013