NAG Library Function Document
nag_zhbgvd (f08uqc)
1
Purpose
nag_zhbgvd (f08uqc) computes all the eigenvalues and, optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form
where
and
are Hermitian and banded, and
is also positive definite. If eigenvectors are desired, it uses a divide-and-conquer algorithm.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zhbgvd (Nag_OrderType order,
Nag_JobType job,
Nag_UploType uplo,
Integer n,
Integer ka,
Integer kb,
Complex ab[],
Integer pdab,
Complex bb[],
Integer pdbb,
double w[],
Complex z[],
Integer pdz,
NagError *fail) |
|
3
Description
The generalized Hermitian-definite band problem
is first reduced to a standard band Hermitian problem
where
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,
, satisfies
where
is the diagonal matrix whose diagonal elements are the eigenvalues.
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
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
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_JobTypeInput
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
- 3:
– Nag_UploTypeInput
-
On entry: if
, the upper triangles of
and
are stored.
If , the lower triangles of and are stored.
Constraint:
or .
- 4:
– IntegerInput
-
On entry: , the order of the matrices and .
Constraint:
.
- 5:
– IntegerInput
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
- 7:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian band matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of
, depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and .
On exit: the contents of
ab are overwritten.
- 8:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 9:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
bb
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian band matrix
.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements of
, depends on the
order and
uplo arguments as follows:
- if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and ; - if and ,
is stored in , for and .
On exit: the factor
from the split Cholesky factorization
, as returned by
nag_zpbstf (f08utc).
- 10:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
bb.
Constraint:
.
- 11:
– doubleOutput
-
On exit: the eigenvalues in ascending order.
- 12:
– ComplexOutput
-
Note: the dimension,
dim, of the array
z
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
z contains the matrix
of eigenvectors, with the
th column of
holding the eigenvector associated with
. The eigenvectors are normalized so that
.
If
,
z is not referenced.
- 13:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
- 14:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The algorithm failed to converge; off-diagonal elements of an intermediate tridiagonal form did not converge to zero.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_MAT_NOT_POS_DEF
-
If
, for
, then
nag_zpbstf (f08utc) returned
:
is not positive definite. The factorization of
could not be completed and no eigenvalues or eigenvectors were computed.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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.
8
Parallelism and Performance
nag_zhbgvd (f08uqc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zhbgvd (f08uqc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is proportional to if and, assuming that , is approximately proportional to otherwise.
The real analogue of this function is
nag_dsbgvd (f08ucc).
10
Example
This example finds all the eigenvalues of the generalized band Hermitian eigenproblem
, where
and
10.1
Program Text
Program Text (f08uqce.c)
10.2
Program Data
Program Data (f08uqce.d)
10.3
Program Results
Program Results (f08uqce.r)