NAG FL Interface
f08usf (zhbgst)
1
Purpose
f08usf reduces a complex Hermitian-definite generalized eigenproblem
to the standard form
, where
and
are band matrices,
is a complex Hermitian matrix, and
has been factorized by
f08utf.
2
Specification
Fortran Interface
Subroutine f08usf ( |
vect, uplo, n, ka, kb, ab, ldab, bb, ldbb, x, ldx, work, rwork, info) |
Integer, Intent (In) |
:: |
n, ka, kb, ldab, ldbb, ldx |
Integer, Intent (Out) |
:: |
info |
Real (Kind=nag_wp), Intent (Out) |
:: |
rwork(n) |
Complex (Kind=nag_wp), Intent (In) |
:: |
bb(ldbb,*) |
Complex (Kind=nag_wp), Intent (Inout) |
:: |
ab(ldab,*), x(ldx,*) |
Complex (Kind=nag_wp), Intent (Out) |
:: |
work(n) |
Character (1), Intent (In) |
:: |
vect, uplo |
|
C Header Interface
#include <nag.h>
void |
f08usf_ (const char *vect, const char *uplo, const Integer *n, const Integer *ka, const Integer *kb, Complex ab[], const Integer *ldab, const Complex bb[], const Integer *ldbb, Complex x[], const Integer *ldx, Complex work[], double rwork[], Integer *info, const Charlen length_vect, const Charlen length_uplo) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08usf_ (const char *vect, const char *uplo, const Integer &n, const Integer &ka, const Integer &kb, Complex ab[], const Integer &ldab, const Complex bb[], const Integer &ldbb, Complex x[], const Integer &ldx, Complex work[], double rwork[], Integer &info, const Charlen length_vect, const Charlen length_uplo) |
}
|
The routine may be called by the names f08usf, nagf_lapackeig_zhbgst or its LAPACK name zhbgst.
3
Description
To reduce the complex Hermitian-definite generalized eigenproblem
to the standard form
, where
,
and
are banded,
f08usf must be preceded by a call to
f08utf which computes the split Cholesky factorization of the positive definite matrix
:
. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This routine overwrites with , where and is a unitary matrix chosen (implicitly) to preserve the bandwidth of . The routine also has an option to allow the accumulation of , and then, if is an eigenvector of , is an eigenvector of the original system.
4
References
Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86
5
Arguments
-
1:
– Character(1)
Input
-
On entry: indicates whether
is to be returned.
- is not returned.
- is returned.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: indicates whether the upper or lower triangular part of
is stored.
- The upper triangular part of is stored.
- The lower triangular part of is stored.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
4:
– Integer
Input
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
-
5:
– Integer
Input
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
-
6:
– Complex (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
ab
must be at least
.
On entry: the upper or lower triangle of the
by
Hermitian band matrix
.
The matrix is stored in rows
to
, more precisely,
- if , the elements of the upper triangle of within the band must be stored with element in ;
- if , the elements of the lower triangle of within the band must be stored with element in
On exit: the upper or lower triangle of
ab is overwritten by the corresponding upper or lower triangle of
as specified by
uplo.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
ab as declared in the (sub)program from which
f08usf is called.
Constraint:
.
-
8:
– Complex (Kind=nag_wp) array
Input
-
Note: the second dimension of the array
bb
must be at least
.
On entry: the banded split Cholesky factor of
as specified by
uplo,
n and
kb and returned by
f08utf.
-
9:
– Integer
Input
-
On entry: the first dimension of the array
bb as declared in the (sub)program from which
f08usf is called.
Constraint:
.
-
10:
– Complex (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
x
must be at least
if
and at least
if
.
On exit: the
by
matrix
, if
.
If
,
x is not referenced.
-
11:
– Integer
Input
-
On entry: the first dimension of the array
x as declared in the (sub)program from which
f08usf is called.
Constraints:
- if , ;
- if , .
-
12:
– Complex (Kind=nag_wp) array
Workspace
-
-
13:
– Real (Kind=nag_wp) array
Workspace
-
-
14:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
7
Accuracy
Forming the reduced matrix is a stable procedure. However it involves implicit multiplication by . When f08usf is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if is ill-conditioned with respect to inversion.
8
Parallelism and Performance
f08usf 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 routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of real floating-point operations is approximately , when , assuming ; there are an additional operations when .
The real analogue of this routine is
f08uef.
10
Example
This example computes all the eigenvalues of
, where
and
Here
is Hermitian,
is Hermitian positive definite, and
and
are treated as band matrices.
must first be factorized by
f08utf. The program calls
f08usf to reduce the problem to the standard form
, then
f08hsf to reduce
to tridiagonal form, and
f08jff to compute the eigenvalues.
10.1
Program Text
10.2
Program Data
10.3
Program Results