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NAG Toolbox: nag_lapack_zhegvd (f08sq)

Purpose

nag_lapack_zhegvd (f08sq) computes all the eigenvalues and, optionally, the 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 and BB is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.

Syntax

[a, b, w, info] = f08sq(itype, jobz, uplo, a, b, 'n', n)
[a, b, w, info] = nag_lapack_zhegvd(itype, jobz, uplo, a, b, 'n', n)

Description

nag_lapack_zhegvd (f08sq) 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 eigenvalues and, optionally, the 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
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:     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:     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:     b(ldb, : :) – complex array
The first dimension of the array b 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 BB.
  • If uplo = 'U'uplo='U', the upper triangular part of bb 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 bb must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, b and the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
nn, the order of the matrices AA and BB.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda ldb work lwork rwork lrwork iwork liwork

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).
If jobz = 'V'jobz='V', a contains the matrix ZZ of eigenvectors. The eigenvectors are normalized as follows:
  • if itype = 1itype=1 or 22, ZHBZ = IZHBZ=I;
  • if itype = 3itype=3, ZHB1Z = IZHB-1Z=I.
If jobz = 'N'jobz='N', the upper triangle (if uplo = 'U'uplo='U') or the lower triangle (if uplo = 'L'uplo='L') of a, including the diagonal, is overwritten.
2:     b(ldb, : :) – complex array
The first dimension of the array b will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldbmax (1,n)ldbmax(1,n).
The triangular factor UU or LL from the Cholesky factorization B = UHUB=UHU or B = LLHB=LLH.
3:     w(n) – double array
The eigenvalues in ascending order.
4:     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: itype, 2: jobz, 3: uplo, 4: n, 5: a, 6: lda, 7: b, 8: ldb, 9: w, 10: work, 11: lwork, 12: rwork, 13: lrwork, 14: iwork, 15: liwork, 16: 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 = 1tonINFO=1ton
If info = iinfo=i, nag_lapack_zheevd (f08fq) failed to converge; ii ii off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
  INFO > NINFO>N
nag_lapack_zpotrf (f07fr) 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.
The example program below illustrates the computation of approximate 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_dsygvd (f08sc).

Example

function nag_lapack_zhegvd_example
itype = int64(2);
jobz = 'Vectors';
uplo = 'Upper';
a = [-7.36,  0.77 - 0.43i,  -0.64 - 0.92i,  3.01 - 6.97i;
      0 + 0i,  3.49 + 0i,  2.19 + 4.45i,  1.9 + 3.73i;
      0 + 0i,  0 + 0i,  0.12 + 0i,  2.88 - 3.17i;
      0 + 0i,  0 + 0i,  0 + 0i,  -2.54 + 0i];
b = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
      0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
      0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
      0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[aOut, bOut, w, info] = nag_lapack_zhegvd(itype, jobz, uplo, a, b)
 

aOut =

   0.2729 - 0.2791i   0.1219 - 0.1054i   0.9130 - 2.1011i  -0.1738 - 0.1110i
  -0.1187 + 0.1377i   0.3886 + 0.0797i  -0.8539 + 0.1787i   0.3785 + 0.0873i
   0.0548 - 0.0077i  -0.3047 + 0.4414i  -0.9232 + 0.9837i   0.1111 - 0.1894i
  -0.3177 + 0.0000i   0.2285 + 0.0000i   1.4319 + 0.0000i   0.3000 + 0.0000i


bOut =

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


w =

  -61.7321
   -6.6195
    0.0725
   43.1883


info =

                    0


function f08sq_example
itype = int64(2);
jobz = 'Vectors';
uplo = 'Upper';
a = [-7.36,  0.77 - 0.43i,  -0.64 - 0.92i,  3.01 - 6.97i;
      0 + 0i,  3.49 + 0i,  2.19 + 4.45i,  1.9 + 3.73i;
      0 + 0i,  0 + 0i,  0.12 + 0i,  2.88 - 3.17i;
      0 + 0i,  0 + 0i,  0 + 0i,  -2.54 + 0i];
b = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
      0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
      0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
      0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[aOut, bOut, w, info] = f08sq(itype, jobz, uplo, a, b)
 

aOut =

   0.2729 - 0.2791i   0.1219 - 0.1054i   0.9130 - 2.1011i  -0.1738 - 0.1110i
  -0.1187 + 0.1377i   0.3886 + 0.0797i  -0.8539 + 0.1787i   0.3785 + 0.0873i
   0.0548 - 0.0077i  -0.3047 + 0.4414i  -0.9232 + 0.9837i   0.1111 - 0.1894i
  -0.3177 + 0.0000i   0.2285 + 0.0000i   1.4319 + 0.0000i   0.3000 + 0.0000i


bOut =

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


w =

  -61.7321
   -6.6195
    0.0725
   43.1883


info =

                    0



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