NAG Library Function Document
nag_zgges (f08xnc)
1
Purpose
nag_zgges (f08xnc) computes the generalized eigenvalues, the generalized Schur form
and, optionally, the left and/or right generalized Schur vectors for a pair of
by
complex nonsymmetric matrices
.
nag_zgges (f08xnc) is marked as
deprecated by LAPACK; the replacement routine is
nag_zgges3 (f08xqc) which makes better use of level 3 BLAS.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_zgges (Nag_OrderType order,
Nag_LeftVecsType jobvsl,
Nag_RightVecsType jobvsr,
Nag_SortEigValsType sort,
Nag_Boolean |
(*selctg)(Complex a,
Complex b),
|
|
Integer n,
Complex a[],
Integer pda,
Complex b[],
Integer pdb,
Integer *sdim,
Complex alpha[],
Complex beta[],
Complex vsl[],
Integer pdvsl,
Complex vsr[],
Integer pdvsr,
NagError *fail) |
|
3
Description
The generalized Schur factorization for a pair of complex matrices
is given by
where
and
are unitary,
and
are upper triangular. 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, nag_zgges (f08xnc) 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.
nag_zgges (f08xnc) computes to have real non-negative diagonal entries. 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
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
- 1:
– Nag_OrderTypeInput
-
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_LeftVecsTypeInput
-
On entry: if
, do not compute the left Schur vectors.
If , compute the left Schur vectors.
Constraint:
or .
- 3:
– Nag_RightVecsTypeInput
-
On entry: if
, do not compute the right Schur vectors.
If , compute the right Schur vectors.
Constraint:
or .
- 4:
– Nag_SortEigValsTypeInput
-
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 userExternal 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
nag_zgges (f08xnc), and may be specified as NULLFN.
The specification of
selctg is:
Nag_Boolean |
selctg (Complex a,
Complex b)
|
|
- 1:
– ComplexInput
- 2:
– ComplexInput
-
On entry: an eigenvalue
is selected if
is Nag_TRUE.
Note that in the ill-conditioned case, a selected generalized eigenvalue may no longer satisfy
after ordering.
NE_SCHUR_REORDER_SELECT in this case.
- 6:
– IntegerInput
-
On entry: , the order of the matrices and .
Constraint:
.
- 7:
– ComplexInput/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:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 9:
– ComplexInput/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:
– IntegerInput
-
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.
- 12:
– ComplexOutput
-
On exit: see the description of
beta.
- 13:
– ComplexOutput
-
On exit:
, for
, will be the generalized eigenvalues.
, for
and
, for
, are the diagonals of the complex Schur form
output by
nag_zgges (f08xnc). The
will be non-negative real.
Note: the quotients
may easily overflow or underflow, and
may even be zero. Thus, you should avoid naively computing the ratio
. However,
alpha will always be less than and usually comparable with
in magnitude, and
beta will always be less than and usually comparable with
.
- 14:
– ComplexOutput
-
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.
- 15:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vsl.
Constraints:
- if , ;
- otherwise .
- 16:
– ComplexOutput
-
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.
- 17:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vsr.
Constraints:
- if , ;
- otherwise .
- 18:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_ITERATION_QZ
-
The iteration did not converge and the matrix pair is not in the generalized Schur form.
The computed and should be correct for .
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 2.7.5 in How to Use the NAG Library and its Documentation 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
nag_zgges (f08xnc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgges (f08xnc) 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 real analogue of this function is
nag_dgges (f08xac).
10
Example
This example finds the generalized Schur factorization of the matrix pair
, where
and
10.1
Program Text
Program Text (f08xnce.c)
10.2
Program Data
Program Data (f08xnce.d)
10.3
Program Results
Program Results (f08xnce.r)