NAG CL Interface
f08uec (dsbgst)
1
Purpose
f08uec reduces a real symmetric-definite generalized eigenproblem
to the standard form
, where
and
are band matrices,
is a real symmetric matrix, and
has been factorized by
f08ufc.
2
Specification
void |
f08uec (Nag_OrderType order,
Nag_VectType vect,
Nag_UploType uplo,
Integer n,
Integer ka,
Integer kb,
double ab[],
Integer pdab,
const double bb[],
Integer pdbb,
double x[],
Integer pdx,
NagError *fail) |
|
The function may be called by the names: f08uec, nag_lapackeig_dsbgst or nag_dsbgst.
3
Description
To reduce the real symmetric-definite generalized eigenproblem
to the standard form
, where
,
and
are banded,
f08uec must be preceded by a call to
f08ufc 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 function overwrites with , where and is a orthogonal matrix chosen (implicitly) to preserve the bandwidth of . The function 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:
– Nag_OrderType
Input
-
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_VectType
Input
-
On entry: indicates whether
is to be returned.
- is not returned.
- is returned.
Constraint:
or .
-
3:
– Nag_UploType
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 .
-
4:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
5:
– Integer
Input
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
-
6:
– Integer
Input
-
On entry: if
, the number of superdiagonals,
, of the matrix
.
If , the number of subdiagonals, , of the matrix .
Constraint:
.
-
7:
– double
Input/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the upper or lower triangle of the
by
symmetric 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 upper or lower triangle of
ab is overwritten by the corresponding upper or lower triangle of
as specified by
uplo.
-
8:
– Integer
Input
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
-
9:
– const double
Input
-
Note: the dimension,
dim, 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
f08ufc.
-
10:
– Integer
Input
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix in the array
bb.
Constraint:
.
-
11:
– double
Output
-
Note: the dimension,
dim, of the array
x
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: the
by
matrix
, if
.
If
,
x is not referenced.
-
12:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if , ;
- if , .
-
13:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
if , .
- 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
Forming the reduced matrix is a stable procedure. However it involves implicit multiplication by . When f08uec 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
f08uec 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 approximately , when , assuming ; there are an additional operations when .
The complex analogue of this function is
f08usc.
10
Example
This example computes all the eigenvalues of
, where
Here
is symmetric,
is symmetric positive definite, and
and
are treated as band matrices.
must first be factorized by
f08ufc. The program calls
f08uec to reduce the problem to the standard form
, then
f08hec to reduce
to tridiagonal form, and
f08jfc to compute the eigenvalues.
10.1
Program Text
10.2
Program Data
10.3
Program Results