NAG CL Interface
f08yxc (ztgevc)
1
Purpose
f08yxc computes some or all of the right and/or left generalized eigenvectors of a pair of complex upper triangular matrices .
2
Specification
void |
f08yxc (Nag_OrderType order,
Nag_SideType side,
Nag_HowManyType how_many,
const Nag_Boolean select[],
Integer n,
const Complex a[],
Integer pda,
const Complex b[],
Integer pdb,
Complex vl[],
Integer pdvl,
Complex vr[],
Integer pdvr,
Integer mm,
Integer *m,
NagError *fail) |
|
The function may be called by the names: f08yxc, nag_lapackeig_ztgevc or nag_ztgevc.
3
Description
f08yxc computes some or all of the right and/or left generalized eigenvectors of the matrix pair
which is assumed to be in upper triangular form. If the matrix pair
is not upper triangular then the function
f08xsc should be called before invoking
f08yxc.
The right generalized eigenvector
and the left generalized eigenvector
of
corresponding to a generalized eigenvalue
are defined by
and
If a generalized eigenvalue is determined as
, which is due to zero diagonal elements at the same locations in both
and
, a unit vector is returned as the corresponding eigenvector.
Note that the generalized eigenvalues are computed using
f08xsc but
f08yxc does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by
f08yxc.
If all eigenvectors are requested, the function may either return the matrices
and/or
of right or left eigenvectors of
, or the products
and/or
, where
and
are two matrices supplied by you. Usually,
and
are chosen as the unitary matrices returned by
f08xsc. Equivalently,
and
are the left and right Schur vectors of the matrix pair supplied to
f08xsc. In that case,
and
are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to
f08xsc.
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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London
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_SideType
Input
-
On entry: specifies the required sets of generalized eigenvectors.
- Only right eigenvectors are computed.
- Only left eigenvectors are computed.
- Both left and right eigenvectors are computed.
Constraint:
, or .
-
3:
– Nag_HowManyType
Input
-
On entry: specifies further details of the required generalized eigenvectors.
- All right and/or left eigenvectors are computed.
- All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays vr and/or vl.
- Selected right and/or left eigenvectors, defined by the array select, are computed.
Constraint:
, or .
-
4:
– const Nag_Boolean
Input
-
Note: the dimension,
dim, of the array
select
must be at least
- when ;
- otherwise select may be NULL.
On entry: specifies the eigenvectors to be computed if
. To select the generalized eigenvector corresponding to the
th generalized eigenvalue, the
th element of
select should be set to Nag_TRUE.
Constraint:
if , or , for .
-
5:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
6:
– const Complex
Input
-
Note: the dimension,
dim, of the array
a
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the matrix
must be in upper triangular form. Usually, this is the matrix
returned by
f08xsc.
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
8:
– const Complex
Input
-
Note: the dimension,
dim, of the array
b
must be at least
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the matrix
must be in upper triangular form with non-negative real diagonal elements. Usually, this is the matrix
returned by
f08xsc.
-
9:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
-
10:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
vl
must be at least
- when
or and
;
- when
or and
;
- otherwise vl may be NULL.
th element of the
th vector is stored in
- when ;
- when .
On entry: if
and
or
,
vl must be initialized to an
by
matrix
. Usually, this is the unitary matrix
of left Schur vectors returned by
f08xsc.
On exit: if
or
,
vl contains:
- if , the matrix of left eigenvectors of ;
- if , the matrix ;
- if , the left eigenvectors of specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vl, in the same order as their corresponding eigenvalues.
-
11:
– Integer
Input
-
On entry: the stride used in the array
vl.
Constraints:
- if ,
- if or , ;
- if , vl may be NULL;
- if ,
- if or ,
;
- if ,
vl may be NULL.
-
12:
– Complex
Input/Output
-
Note: the dimension,
dim, of the array
vr
must be at least
- when
or and
;
- when
or and
;
- otherwise vr may be NULL.
th element of the
th vector is stored in
- when ;
- when .
On entry: if
and
or
,
vr must be initialized to an
by
matrix
. Usually, this is the unitary matrix
of right Schur vectors returned by
f08xec.
On exit: if
or
,
vr contains:
- if , the matrix of right eigenvectors of ;
- if , the matrix ;
- if , the right eigenvectors of specified by select, stored consecutively in the rows or columns (depending on the value of order) of the array vr, in the same order as their corresponding eigenvalues.
-
13:
– Integer
Input
-
On entry: the stride used in the array
vr.
Constraints:
- if ,
- if or , ;
- if , vr may be NULL;
- if ,
- if or ,
;
- if ,
vr may be NULL.
-
14:
– Integer
Input
-
On entry: the number of columns in the arrays
vl and/or
vr.
Constraints:
- if or , ;
- if , mm must not be less than the number of requested eigenvectors.
-
15:
– Integer *
Output
-
On exit: the number of columns in the arrays
vl and/or
vr actually used to store the eigenvectors. If
or
,
m is set to
n. Each selected eigenvector occupies one column.
-
16:
– 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_CONSTRAINT
-
Constraint: if , or , for .
- NE_ENUM_INT_2
-
On entry,
,
and
.
Constraint: if
or
,
;
if
,
mm must not be less than the number of requested eigenvectors.
On entry, , and .
Constraint: if or ,
.
On entry, , and .
Constraint: if or , .
On entry, , and .
Constraint: if or ,
.
On entry, , and .
Constraint: if or , .
- 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: .
- 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
It is beyond the scope of this manual to summarise the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see
Anderson et al. (1999)) and Chapter 6 of
Stewart and Sun (1990).
8
Parallelism and Performance
f08yxc 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.
f08yxc is the sixth step in the solution of the complex generalized eigenvalue problem and is usually called after
f08xsc.
The real analogue of this function is
f08ykc.
10
Example
This example computes the
and
arguments, which defines the generalized eigenvalues and the corresponding left and right eigenvectors, of the matrix pair
given by
and
To compute generalized eigenvalues, it is required to call five functions:
f08wvc to balance the matrix,
f08asc to perform the
factorization of
,
f08auc to apply
to
,
f08wsc to reduce the matrix pair to the generalized Hessenberg form and
f08xsc to compute the eigenvalues via the
algorithm.
The computation of generalized eigenvectors is done by calling
f08yxc to compute the eigenvectors of the balanced matrix pair. The function
f08wwc is called to backward transform the eigenvectors to the user-supplied matrix pair. If both left and right eigenvectors are required then
f08wwc must be called twice.
10.1
Program Text
10.2
Program Data
10.3
Program Results