NAG CL Interface
f08pac (dgees)
1
Purpose
f08pac computes the eigenvalues, the real Schur form , and, optionally, the matrix of Schur vectors for an by real nonsymmetric matrix .
2
Specification
void |
f08pac (Nag_OrderType order,
Nag_JobType jobvs,
Nag_SortEigValsType sort,
Integer n,
double a[],
Integer pda,
Integer *sdim,
double wr[],
double wi[],
double vs[],
Integer pdvs,
NagError *fail) |
|
The function may be called by the names: f08pac, nag_lapackeig_dgees or nag_dgees.
3
Description
The real Schur factorization of
is given by
where
, the matrix of Schur vectors, is orthogonal and
is the real Schur form. A matrix is in real Schur form if it is upper quasi-triangular with
by
and
by
blocks.
by
blocks will be standardized in the form
where
. The eigenvalues of such a block are
.
Optionally, f08pac also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left. The leading columns of form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.
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 (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:
– Nag_JobType
Input
-
On entry: if
, Schur vectors are not computed.
If , Schur vectors are computed.
Constraint:
or .
-
3:
– Nag_SortEigValsType
Input
-
On entry: specifies whether or not to order the eigenvalues on the diagonal of the Schur form.
- Eigenvalues are not ordered.
- Eigenvalues are ordered (see select).
Constraint:
or .
-
4:
– function, supplied by the user
External Function
-
If
,
select is used to select eigenvalues to sort to the top left of the Schur form.
If
,
select is not referenced and
f08pac may be specified as NULLFN.
An eigenvalue is selected if is Nag_TRUE. If either one of a complex conjugate pair of eigenvalues is selected, then both are. Note that a selected complex eigenvalue may no longer satisfy after ordering, since ordering may change the value of complex eigenvalues (especially if the eigenvalue is ill-conditioned); in this case is set to .
The specification of
select is:
Nag_Boolean |
select (double wr,
double wi)
|
|
-
1:
– double
Input
-
2:
– double
Input
-
On entry: the real and imaginary parts of the eigenvalue.
-
5:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
6:
– 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 by matrix .
On exit:
a is overwritten by its real Schur form
.
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
8:
– Integer *
Output
-
On exit: if
,
.
If
,
number of eigenvalues (after sorting) for which
select is Nag_TRUE. (Complex conjugate pairs for which
select is Nag_TRUE for either eigenvalue count as
.)
-
9:
– double
Output
-
Note: the dimension,
dim, of the array
wr
must be at least
.
On exit: see the description of
wi.
-
10:
– double
Output
-
Note: the dimension,
dim, of the array
wi
must be at least
.
On exit:
wr and
wi contain the real and imaginary parts, respectively, of the computed eigenvalues in the same order that they appear on the diagonal of the output Schur form
. Complex conjugate pairs of eigenvalues will appear consecutively with the eigenvalue having the positive imaginary part first.
-
11:
– double
Output
-
Note: the dimension,
dim, of the array
vs
must be at least
- when
;
- otherwise.
th element of the
th vector is stored in
- when ;
- when .
On exit: if
,
vs contains the orthogonal matrix
of Schur vectors.
If
,
vs is not referenced.
-
12:
– Integer
Input
-
On entry: the stride used in the array
vs.
Constraints:
- if , ;
- otherwise .
-
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_CONVERGENCE
-
The algorithm failed to compute all the eigenvalues.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
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.
- 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 Schur form no longer satisfy . This could also be caused by underflow due to scaling.
7
Accuracy
The computed Schur factorization satisfies
where
and
is the
machine precision. See Section 4.8 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f08pac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08pac 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
f08pnc.
10
Example
This example finds the Schur factorization of the matrix
such that the real positive eigenvalues of
are the top left diagonal elements of the Schur form,
.
10.1
Program Text
10.2
Program Data
10.3
Program Results