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NAG Toolbox

NAG Toolbox: nag_lapack_zhpgvx (f08tp)

Purpose

nag_lapack_zhpgvx (f08tp) computes selected eigenvalues and, optionally, eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
Az = λBz ,   ABz = λz   or   BAz = λz ,
Az=λBz ,   ABz=λz   or   BAz=λz ,
where AA and BB are Hermitian, stored in packed format, and BB is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Syntax

[ap, bp, m, w, z, jfail, info] = f08tp(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol)
[ap, bp, m, w, z, jfail, info] = nag_lapack_zhpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol)

Description

nag_lapack_zhpgvx (f08tp) first performs a Cholesky factorization of the matrix BB as B = UHU B=UHU , when uplo = 'U'uplo='U' or B = LLH B=LLH , when uplo = 'L'uplo='L'. The generalized problem is then reduced to a standard symmetric eigenvalue problem
Cx = λx ,
Cx=λx ,
which is solved for the desired eigenvalues and eigenvectors; the eigenvectors are then backtransformed to give the eigenvectors of the original problem.
For the problem Az = λBz Az=λBz , 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. For the problem A B z = λ z A B z = λ z  we correspondingly have
Z1 A ZH = Λ   and   ZH B Z = I ,
Z-1 A Z-H = Λ   and   ZH B Z = I ,
and for B A z = λ z B A z = λ z  we have
ZH A Z = Λ   and   ZH B1 Z = I .
ZH A Z = Λ   and   ZH B-1 Z = I .

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:     itype – int64int32nag_int scalar
Specifies the problem type to be solved.
itype = 1itype=1
Az = λBzAz=λBz.
itype = 2itype=2
ABz = λzABz=λz.
itype = 3itype=3
BAz = λzBAz=λz.
Constraint: itype = 1itype=1, 22 or 33.
2:     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'.
3:     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'.
4:     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'.
5:     n – int64int32nag_int scalar
nn, the order of the matrices AA and BB.
Constraint: n0n0.
6:     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.
7:     bp( : :) – complex array
Note: the dimension of the array bp 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 BB, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of BB must be stored with element BijBij in bp(i + j(j1) / 2)bpi+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of BB must be stored with element BijBij in bp(i + (2nj)(j1) / 2)bpi+(2n-j)(j-1)/2 for ijij.
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

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).
The contents of ap are destroyed.
2:     bp( : :) – complex array
Note: the dimension of the array bp must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The triangular factor UU or LL from the Cholesky factorization B = UHUB=UHU or B = LLHB=LLH, in the same storage format as BB.
3:     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.
4:     w(n) – double array
The first m elements contain the selected eigenvalues in ascending order.
5:     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. The eigenvectors are normalized as follows:
    • if itype = 1itype=1 or 22, ZHBZ = IZHBZ=I;
    • if itype = 3itype=3, ZHB1Z = IZHB-1Z=I;
  • if an eigenvector fails to converge (INFO = 1tonINFO=1ton), 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.
6:     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.
7:     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: itype, 2: jobz, 3: range, 4: uplo, 5: n, 6: ap, 7: bp, 8: vl, 9: vu, 10: il, 11: iu, 12: abstol, 13: m, 14: w, 15: z, 16: ldz, 17: work, 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 = 1tonINFO=1ton
If info = iinfo=i, nag_lapack_zhpevx (f08gp) failed to converge; ii eigenvectors failed to converge. Their indices are stored in array jfail.
  INFO > NINFO>N
nag_lapack_zpptrf (f07gr) 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 .
The real analogue of this function is nag_lapack_dspgvx (f08tb).

Example

function nag_lapack_zhpgvx_example
itype = int64(1);
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
n = int64(4);
ap = [-7.36;
      0.77 - 0.43i;
      3.49 + 0i;
      -0.64 - 0.92i;
      2.19 + 4.45i;
      0.12 + 0i;
      3.01 - 6.97i;
      1.9 + 3.73i;
      2.88 - 3.17i;
      -2.54 + 0i];
bp = [3.23;
      1.51 - 1.92i;
      3.58 + 0i;
      1.9 + 0.84i;
      -0.23 + 1.11i;
      4.09 + 0i;
      0.42 + 2.5i;
      -1.18 + 1.37i;
      2.33 - 0.14i;
      4.29 + 0i];
vl = -3;
vu = 3;
il = int64(10581250);
iu = int64(-1233178000);
abstol = 0;
[apOut, bpOut, m, w, z, jfail, info] = ...
    nag_lapack_zhpgvx(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol)
 

apOut =

  -1.2636 + 0.0000i
  -2.3214 + 0.0000i
  -1.8095 + 0.0000i
  -0.5211 - 0.0656i
  -2.7959 + 0.0000i
  -0.7025 + 0.0000i
  -0.0802 + 0.4016i
  -0.1903 + 0.1121i
  -3.8021 + 0.0000i
  -0.7133 + 0.0000i


bpOut =

   1.7972 + 0.0000i
   0.8402 - 1.0683i
   1.3164 + 0.0000i
   1.0572 + 0.4674i
  -0.4702 - 0.3131i
   1.5604 + 0.0000i
   0.2337 + 1.3910i
   0.0834 - 0.0368i
   0.9360 - 0.9900i
   0.6603 + 0.0000i


m =

                    2


w =

   -2.9936
    0.5047
         0
         0


z =

  -0.3504 + 0.6060i   0.2835 - 0.5806i
  -0.0993 + 0.0631i  -0.3769 - 0.3194i
   0.6851 - 0.5987i  -0.3338 - 0.0134i
  -0.8127 + 0.0000i   0.6663 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0


function f08tp_example
itype = int64(1);
jobz = 'Vectors';
range = 'Values in range';
uplo = 'U';
n = int64(4);
ap = [-7.36;
      0.77 - 0.43i;
      3.49 + 0i;
      -0.64 - 0.92i;
      2.19 + 4.45i;
      0.12 + 0i;
      3.01 - 6.97i;
      1.9 + 3.73i;
      2.88 - 3.17i;
      -2.54 + 0i];
bp = [3.23;
      1.51 - 1.92i;
      3.58 + 0i;
      1.9 + 0.84i;
      -0.23 + 1.11i;
      4.09 + 0i;
      0.42 + 2.5i;
      -1.18 + 1.37i;
      2.33 - 0.14i;
      4.29 + 0i];
vl = -3;
vu = 3;
il = int64(10581250);
iu = int64(-1233178000);
abstol = 0;
[apOut, bpOut, m, w, z, jfail, info] = ...
    f08tp(itype, jobz, range, uplo, n, ap, bp, vl, vu, il, iu, abstol)
 

apOut =

  -1.2636 + 0.0000i
  -2.3214 + 0.0000i
  -1.8095 + 0.0000i
  -0.5211 - 0.0656i
  -2.7959 + 0.0000i
  -0.7025 + 0.0000i
  -0.0802 + 0.4016i
  -0.1903 + 0.1121i
  -3.8021 + 0.0000i
  -0.7133 + 0.0000i


bpOut =

   1.7972 + 0.0000i
   0.8402 - 1.0683i
   1.3164 + 0.0000i
   1.0572 + 0.4674i
  -0.4702 - 0.3131i
   1.5604 + 0.0000i
   0.2337 + 1.3910i
   0.0834 - 0.0368i
   0.9360 - 0.9900i
   0.6603 + 0.0000i


m =

                    2


w =

   -2.9936
    0.5047
         0
         0


z =

  -0.3504 + 0.6060i   0.2835 - 0.5806i
  -0.0993 + 0.0631i  -0.3769 - 0.3194i
   0.6851 - 0.5987i  -0.3338 - 0.0134i
  -0.8127 + 0.0000i   0.6663 + 0.0000i


jfail =

                    0
                    0
                    0
                    0


info =

                    0



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