NAG Library Routine Document
F08TQF (ZHPGVD)
1 Purpose
F08TQF (ZHPGVD) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite eigenproblem, of the form
where
and
are Hermitian, stored in packed format, and
is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.
2 Specification
SUBROUTINE F08TQF ( |
ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO) |
INTEGER |
ITYPE, N, LDZ, LWORK, LRWORK, IWORK(max(1,LIWORK)), LIWORK, INFO |
REAL (KIND=nag_wp) |
W(N), RWORK(max(1,LRWORK)) |
COMPLEX (KIND=nag_wp) |
AP(*), BP(*), Z(LDZ,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBZ, UPLO |
|
The routine may be called by its
LAPACK
name zhpgvd.
3 Description
F08TQF (ZHPGVD) first performs a Cholesky factorization of the matrix
as
, when
or
, when
. The generalized problem is then reduced to a standard symmetric eigenvalue problem
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
, the eigenvectors are normalized so that the matrix of eigenvectors,
, satisfies
where
is the diagonal matrix whose diagonal elements are the eigenvalues. For the problem
we correspondingly have
and for
we have
4 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
5 Parameters
- 1: ITYPE – INTEGERInput
On entry: specifies the problem type to be solved.
- .
- .
- .
Constraint:
, or .
- 2: JOBZ – CHARACTER(1)Input
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 3: UPLO – CHARACTER(1)Input
On entry: if
, the upper triangles of
and
are stored.
If , the lower triangles of and are stored.
Constraint:
or .
- 4: N – INTEGERInput
On entry: , the order of the matrices and .
Constraint:
.
- 5: AP() – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
AP
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian matrix
, packed by columns.
More precisely,
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
On exit: the contents of
AP are destroyed.
- 6: BP() – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
BP
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian matrix
, packed by columns.
More precisely,
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
On exit: the triangular factor or from the Cholesky factorization or , in the same storage format as .
- 7: W(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the eigenvalues in ascending order.
- 8: Z(LDZ,) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
Z
must be at least
if
, and at least
otherwise.
On exit: if
,
Z contains the matrix
of eigenvectors. The eigenvectors are normalized as follows:
- if or , ;
- if , .
If
,
Z is not referenced.
- 9: LDZ – INTEGERInput
On entry: the first dimension of the array
Z as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
Constraints:
- if , ;
- otherwise .
- 10: WORK() – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
, the real part of
contains the minimum value of
LWORK required for optimal performance.
- 11: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If
, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Constraints:
- if , ;
- if and , ;
- if and , .
- 12: RWORK() – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
,
returns the optimal
LRWORK.
- 13: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If
, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Constraints:
- if , ;
- if and , ;
- if and , .
- 14: IWORK() – INTEGER arrayWorkspace
On exit: if
,
returns the optimal
LIWORK.
- 15: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which F08TQF (ZHPGVD) is called.
If
, a workspace query is assumed; the routine only calculates the optimal sizes of the
WORK,
RWORK and
IWORK arrays, returns these values as the first entries of the
WORK,
RWORK and
IWORK arrays, and no error message related to
LWORK,
LRWORK or
LIWORK is issued.
Constraints:
- if or , ;
- if and , .
- 16: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
F07GRF (ZPPTRF)
or
F08GQF (ZHPEVD)
returned an error code:
|
if ,
F08GQF (ZHPEVD)
failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero; |
|
if , for , then the leading minor of order of is not positive definite. The factorization of could not be completed and no eigenvalues or eigenvectors were computed. |
7 Accuracy
If
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
differ widely in magnitude the eigenvalues and eigenvectors may be less sensitive than the condition of
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.
The total number of floating point operations is proportional to .
The real analogue of this routine is
F08TCF (DSPGVD).
9 Example
This example finds all the eigenvalues and eigenvectors of the generalized Hermitian eigenproblem
, where
and
together with an estimate of the condition number of
, and approximate error bounds for the computed eigenvalues and eigenvectors.
The example program for
F08TNF (ZHPGV) illustrates solving a generalized Hermitian eigenproblem of the form
.
9.1 Program Text
Program Text (f08tqfe.f90)
9.2 Program Data
Program Data (f08tqfe.d)
9.3 Program Results
Program Results (f08tqfe.r)