NAG CL Interface
f08xcc (dgges3)
1
Purpose
f08xcc computes the generalized eigenvalues, the generalized real Schur form and, optionally, the left and/or right generalized Schur vectors for a pair of by real nonsymmetric matrices .
2
Specification
void |
f08xcc (Nag_OrderType order,
Nag_LeftVecsType jobvsl,
Nag_RightVecsType jobvsr,
Nag_SortEigValsType sort,
Integer n,
double a[],
Integer pda,
double b[],
Integer pdb,
Integer *sdim,
double alphar[],
double alphai[],
double beta[],
double vsl[],
Integer pdvsl,
double vsr[],
Integer pdvsr,
NagError *fail) |
|
The function may be called by the names: f08xcc, nag_lapackeig_dgges3 or nag_dgges3.
3
Description
The generalized Schur factorization for a pair of real matrices
is given by
where
and
are orthogonal,
is upper triangular and
is upper quasi-triangular with
by
and
by
diagonal blocks. The generalized eigenvalues,
, of
are computed from the diagonals of
and
and satisfy
where
is the corresponding generalized eigenvector.
is actually returned as the pair
such that
since
, or even both
and
can be zero. The columns of
and
are the left and right generalized Schur vectors of
.
Optionally, f08xcc can order the generalized eigenvalues on the diagonals of so that selected eigenvalues are at the top left. The leading columns of and then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xcc computes to have non-negative diagonal elements, and the by blocks of correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the algorithm.
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
Golub G H and Van Loan C F (2012) Matrix Computations (4th 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:
– Nag_LeftVecsType
Input
-
On entry: if
, do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
-
3:
– Nag_RightVecsType
Input
-
On entry: if
, do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
-
4:
– Nag_SortEigValsType
Input
-
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
- Eigenvalues are not ordered.
- Eigenvalues are ordered (see selctg).
Constraint:
or .
-
5:
– function, supplied by the user
External Function
-
If
,
selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.
If
,
selctg is not referenced by
f08xcc, and may be specified as NULLFN.
The specification of
selctg is:
Nag_Boolean |
selctg (double ar,
double ai,
double b)
|
|
-
1:
– double
Input
-
2:
– double
Input
-
3:
– double
Input
-
On entry: an eigenvalue
is selected if
. If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy
after ordering.
NE_SCHUR_REORDER_SELECT in this case.
-
6:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
7:
– double
Input/Output
-
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 first of the pair of matrices, .
On exit:
a has been overwritten by its generalized Schur form
.
-
8:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
9:
– double
Input/Output
-
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 second of the pair of matrices, .
On exit:
b has been overwritten by its generalized Schur form
.
-
10:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
-
11:
– Integer *
Output
-
On exit: if
,
.
If
,
number of eigenvalues (after sorting) for which
selctg is Nag_TRUE. (Complex conjugate pairs for which
selctg is Nag_TRUE for either eigenvalue count as
.)
-
12:
– double
Output
-
On exit: see the description of
beta.
-
13:
– double
Output
-
On exit: see the description of
beta.
-
14:
– double
Output
-
On exit:
, for
, will be the generalized eigenvalues.
, and
, for
, are the diagonals of the complex Schur form
that would result if the
by
diagonal blocks of the real Schur form of
were further reduced to triangular form using
by
complex unitary transformations.
If is zero, then the th eigenvalue is real; if positive, then the th and st eigenvalues are a complex conjugate pair, with negative.
Note: the quotients
and
may easily overflow or underflow, and
may even be zero. Thus, you should avoid naively computing the ratio
. However,
alphar and
alphai will always be less than and usually comparable with
in magnitude, and
beta will always be less than and usually comparable with
.
-
15:
– double
Output
-
Note: the dimension,
dim, of the array
vsl
must be at least
- when
;
- otherwise.
The
th element of the matrix is stored in
- when ;
- when .
On exit: if
,
vsl will contain the left Schur vectors,
.
If
,
vsl is not referenced.
-
16:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vsl.
Constraints:
- if , ;
- otherwise .
-
17:
– double
Output
-
Note: the dimension,
dim, of the array
vsr
must be at least
- when
;
- otherwise.
The
th element of the matrix is stored in
- when ;
- when .
On exit: if
,
vsr will contain the right Schur vectors,
.
If
,
vsr is not referenced.
-
18:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vsr.
Constraints:
- if , ;
- otherwise .
-
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_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
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: .
- 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_ITERATION_QZ
-
The iteration failed. No eigenvectors have been calculated but , and should be correct from element .
The
iteration failed with an unexpected error, please contact
NAG.
- 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.
- NE_SCHUR_REORDER
-
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).
- NE_SCHUR_REORDER_SELECT
-
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy . This could also be caused by underflow due to scaling.
7
Accuracy
The computed generalized Schur factorization satisfies
where
and
is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f08xcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xcc 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
f08xqc.
10
Example
This example finds the generalized Schur factorization of the matrix pair
, where
such that the real positive eigenvalues of
correspond to the top left diagonal elements of the generalized Schur form,
.
10.1
Program Text
10.2
Program Data
10.3
Program Results