NAG CL Interface
f08ppc (zgeesx)
1
Purpose
f08ppc computes the eigenvalues, the Schur form , and, optionally, the matrix of Schur vectors for an by complex nonsymmetric matrix .
2
Specification
void |
f08ppc (Nag_OrderType order,
Nag_JobType jobvs,
Nag_SortEigValsType sort,
Nag_RCondType sense,
Integer n,
Complex a[],
Integer pda,
Integer *sdim,
Complex w[],
Complex vs[],
Integer pdvs,
double *rconde,
double *rcondv,
NagError *fail) |
|
The function may be called by the names: f08ppc, nag_lapackeig_zgeesx or nag_zgeesx.
3
Description
The Schur factorization of
is given by
where
, the matrix of Schur vectors, is unitary and
is the Schur form. A complex matrix is in Schur form if it is upper triangular.
Optionally,
f08ppc also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (
rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (
rcondv). The leading columns of
form an orthonormal basis for this invariant subspace.
For further explanation of the reciprocal condition numbers
rconde and
rcondv, see Section 4.8 of
Anderson et al. (1999) (where these quantities are called
and
respectively).
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
f08ppc may be specified as NULLFN.
An eigenvalue is selected if is Nag_TRUE.
The specification of
select is:
Nag_Boolean |
select (Complex w)
|
|
-
1:
– Complex
Input
-
On entry: the real and imaginary parts of the eigenvalue.
-
5:
– Nag_RCondType
Input
-
On entry: determines which reciprocal condition numbers are computed.
- None are computed.
- Computed for average of selected eigenvalues only.
- Computed for selected right invariant subspace only.
- Computed for both.
If , or , .
Constraint:
, , or .
-
6:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
7:
– Complex
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 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:
– Integer *
Output
-
On exit: if
,
.
If
,
number of eigenvalues for which
select is Nag_TRUE.
-
10:
– Complex
Output
-
Note: the dimension,
dim, of the array
w
must be at least
.
On exit: contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form .
-
11:
– Complex
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 unitary 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:
– double *
Output
-
On exit: if
or
, contains the reciprocal condition number for the average of the selected eigenvalues.
If
or
,
rconde is not referenced.
-
14:
– double *
Output
-
On exit: if
or
,
rcondv contains the reciprocal condition number for the selected right invariant subspace.
If
or
,
rcondv is not referenced.
-
15:
– 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
f08ppc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08ppc 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
f08pbc.
10
Example
This example finds the Schur factorization of the matrix
such that the eigenvalues of
with positive real part of are the top left diagonal elements of the Schur form,
. Estimates of the condition numbers for the selected eigenvalue cluster and corresponding invariant subspace are also returned.
10.1
Program Text
10.2
Program Data
10.3
Program Results