NAG CL Interface
f08tbc (dspgvx)
1
Purpose
f08tbc computes selected eigenvalues and, optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form
where
and
are symmetric, stored in packed storage, and
is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.
2
Specification
void |
f08tbc (Nag_OrderType order,
Integer itype,
Nag_JobType job,
Nag_RangeType range,
Nag_UploType uplo,
Integer n,
double ap[],
double bp[],
double vl,
double vu,
Integer il,
Integer iu,
double abstol,
Integer *m,
double w[],
double z[],
Integer pdz,
Integer jfail[],
NagError *fail) |
|
The function may be called by the names: f08tbc, nag_lapackeig_dspgvx or nag_dspgvx.
3
Description
f08tbc 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 desired eigenvalues and 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
https://www.netlib.org/lapack/lug
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices SIAM J. Sci. Statist. Comput. 11 873–912
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
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:
– Integer
Input
-
On entry: specifies the problem type to be solved.
- .
- .
- .
Constraint:
, or .
-
3:
– Nag_JobType
Input
-
On entry: indicates whether eigenvectors are computed.
- Only eigenvalues are computed.
- Eigenvalues and eigenvectors are computed.
Constraint:
or .
-
4:
– Nag_RangeType
Input
-
On entry: if
, all eigenvalues will be found.
If , all eigenvalues in the half-open interval will be found.
If
, the
ilth to
iuth eigenvalues will be found.
Constraint:
, or .
-
5:
– Nag_UploType
Input
-
On entry: if
, the upper triangles of
and
are stored.
If , the lower triangles of and are stored.
Constraint:
or .
-
6:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
7:
– double
Input/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the upper or lower triangle of the
by
symmetric matrix
, packed by rows or columns.
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for .
On exit: the contents of
ap are destroyed.
-
8:
– double
Input/Output
-
Note: the dimension,
dim, of the array
bp
must be at least
.
On entry: the upper or lower triangle of the
by
symmetric matrix
, packed by rows or columns.
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for .
On exit: the triangular factor or from the Cholesky factorization or , in the same storage format as .
-
9:
– double
Input
-
10:
– double
Input
-
On entry: if
, the lower and upper bounds of the interval to be searched for eigenvalues.
If
or
,
vl and
vu are not referenced.
Constraint:
if , .
-
11:
– Integer
Input
-
12:
– Integer
Input
-
On entry: if
,
il and
iu specify the indices (in ascending order) of the smallest and largest eigenvalues to be returned, respectively.
If
or
,
il and
iu are not referenced.
Constraints:
- if and , and ;
- if and , .
-
13:
– double
Input
-
On entry: the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval
of width less than or equal to
where
is the
machine precision. If
abstol is less than or equal to zero, then
will be used in its place, where
is the tridiagonal matrix obtained by reducing
to tridiagonal form. Eigenvalues will be computed most accurately when
abstol is set to twice the underflow threshold
, not zero. If this function returns with
NE_CONVERGENCE, indicating that some eigenvectors did not converge, try setting
abstol to
. See
Demmel and Kahan (1990).
-
14:
– Integer *
Output
-
On exit: the total number of eigenvalues found.
.
If , .
If , .
-
15:
– double
Output
-
On exit: the first
m elements contain the selected eigenvalues in ascending order.
-
16:
– double
Output
-
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
, then
- if NE_NOERROR, the first m columns of contain the orthonormal eigenvectors of the matrix corresponding to the selected eigenvalues, with the th column of holding the eigenvector associated with . The eigenvectors are normalized as follows:
- if or , ;
- if , ;
- if an eigenvector fails to converge ( NE_CONVERGENCE), then that column of contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in jfail.
If
,
z is not referenced.
-
17:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
z.
Constraints:
- if , ;
- otherwise .
-
18:
– Integer
Output
-
Note: the dimension,
dim, of the array
jfail
must be at least
.
On exit: if
, then
- if NE_NOERROR, the first m elements of jfail are zero;
- if NE_CONVERGENCE, the first elements of jfail contains the indices of the eigenvectors that failed to converge.
If
,
jfail is not referenced.
-
19:
– 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_CONVERGENCE
-
The algorithm failed to converge;
eigenvectors did not converge. Their indices are stored in array
jfail.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: if and , and ;
if and , .
- NE_ENUM_REAL_2
-
On entry, , and .
Constraint: if , .
- NE_INT
-
On entry, .
Constraint: , or .
On entry, .
Constraint: .
On entry, .
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_MAT_NOT_POS_DEF
-
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.
- 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
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
f08tbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08tbc 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 .
The complex analogue of this function is
f08tpc.
10
Example
This example finds the eigenvalues in the half-open interval
, and corresponding eigenvectors, of the generalized symmetric eigenproblem
, where
The example program for
f08tcc illustrates solving a generalized symmetric eigenproblem of the form
.
10.1
Program Text
10.2
Program Data
10.3
Program Results