NAG CL Interface
f08ylc (dtgsna)
1
Purpose
f08ylc estimates condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair in generalized real Schur form.
2
Specification
void |
f08ylc (Nag_OrderType order,
Nag_JobType job,
Nag_HowManyType howmny,
const Nag_Boolean select[],
Integer n,
const double a[],
Integer pda,
const double b[],
Integer pdb,
const double vl[],
Integer pdvl,
const double vr[],
Integer pdvr,
double s[],
double dif[],
Integer mm,
Integer *m,
NagError *fail) |
|
The function may be called by the names: f08ylc, nag_lapackeig_dtgsna or nag_dtgsna.
3
Description
f08ylc estimates condition numbers for specified eigenvalues and/or right eigenvectors of an by matrix pair in real generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair
are in real generalized Schur form if
is block upper triangular with
by
and
by
diagonal blocks and
is upper triangular as returned, for example, by
f08xac or
f08xbc, or
f08xec with
. The diagonal elements, or blocks, define the generalized eigenvalues
, for
, of the pair
and the eigenvalues are given by
so that
where
is the corresponding (right) eigenvector.
If
and
are the result of a generalized Schur factorization of a matrix pair
then the eigenvalues and condition numbers of the pair
are the same as those of the pair
.
Let
be a simple generalized eigenvalue of
. Then the reciprocal of the condition number of the eigenvalue
is defined as
where
and
are the right and left eigenvectors of
corresponding to
. If both
and
are zero, then
is singular and
is returned.
The definition of the reciprocal of the estimated condition number of the right eigenvector and the left eigenvector corresponding to the simple eigenvalue depends upon whether is a real eigenvalue, or one of a complex conjugate pair.
If the eigenvalue
is real and
and
are orthogonal transformations such that
where
and
are
by
matrices, then the reciprocal condition number is given by
where
denotes the smallest singular value of the
by
matrix
and
is the Kronecker product.
If
is part of a complex conjugate pair and
and
are orthogonal transformations such that
where
and
are two by two matrices,
and
are
by
matrices, and
corresponds to the complex conjugate eigenvalue pair
,
, then there exist unitary matrices
and
such that
The eigenvalues are given by
and
. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by
where
denotes the real part of
,
,
is the complex two by two matrix
and
is an upper bound on
; i.e., an upper bound on
, where
is the
by
matrix
See Sections 2.4.8 and 4.11 of
Anderson et al. (1999) and
Kågström and Poromaa (1996) for further details and information.
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
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103
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: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
- Condition numbers for eigenvalues only are computed.
- Condition numbers for eigenvectors only are computed.
- Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint:
, or .
-
3:
– Nag_HowManyType
Input
-
On entry: indicates how many condition numbers are to be computed.
- Condition numbers for all eigenpairs are computed.
- Condition numbers for selected eigenpairs (as specified by 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 eigenpairs for which condition numbers are to be computed if
. To select condition numbers for the eigenpair corresponding to the real eigenvalue
,
must be set Nag_TRUE. To select condition numbers corresponding to a complex conjugate pair of eigenvalues
and
,
and/or
must be set to Nag_TRUE.
If
,
select is not referenced and may be
NULL.
-
5:
– Integer
Input
-
On entry: , the order of the matrix pair .
Constraint:
.
-
6:
– const double
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 upper quasi-triangular matrix .
-
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 double
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 upper triangular matrix .
-
9:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
-
10:
– const double
Input
-
Note: the dimension,
dim, of the array
vl
must be at least
- when
or and
;
- when
or and
;
- otherwise vl may be NULL.
The
th element of the matrix is stored in
- when ;
- when .
On entry: if
or
,
vl must contain left eigenvectors of
, corresponding to the eigenpairs specified by
howmny and
select. The eigenvectors must be stored in consecutive columns of
vl, as returned by
f08wac or
f08ykc.
If
,
vl is not referenced and may be
NULL.
-
11:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vl.
Constraints:
- if ,
- if or , ;
- otherwise ;
- if ,
- if or ,
;
- otherwise vl may be NULL.
-
12:
– const double
Input
-
Note: the dimension,
dim, of the array
vr
must be at least
- when
or and
;
- when
or and
;
- otherwise vr may be NULL.
The
th element of the matrix is stored in
- when ;
- when .
On entry: if
or
,
vr must contain right eigenvectors of
, corresponding to the eigenpairs specified by
howmny and
select. The eigenvectors must be stored in consecutive columns of
vr, as returned by
f08wac or
f08ykc.
If
,
vr is not referenced and may be
NULL.
-
13:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vr.
Constraints:
- if ,
- if or , ;
- otherwise ;
- if ,
- if or ,
;
- otherwise vr may be NULL.
-
14:
– double
Output
-
Note: the dimension,
dim, of the array
s
must be at least
- when or ;
- otherwise s may be NULL.
On exit: if
or
, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of
s are set to the same value. Thus
,
, and the
th columns of
and
all correspond to the same eigenpair (but not in general the
th eigenpair, unless all eigenpairs are selected).
If
,
s is not referenced and may be
NULL.
-
15:
– double
Output
-
Note: the dimension,
dim, of the array
dif
must be at least
- when or ;
- otherwise dif may be NULL.
On exit: if
or
, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of
dif are set to the same value. If the eigenvalues cannot be reordered to compute
,
is set to
; this can only occur when the true value would be very small anyway.
If
,
dif is not referenced and may be
NULL.
-
16:
– Integer
Input
-
On entry: the number of elements in the arrays
s and
dif.
Constraints:
- if , ;
- otherwise .
-
17:
– Integer *
Output
-
On exit:
m, the number of elements of the arrays
s and
dif used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If
,
m is set to
n.
-
18:
– 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 or ,
.
On entry, , and .
Constraint: if or , .
On entry, , and .
Constraint: if or ,
.
On entry, , and .
Constraint: if or , .
- NE_ENUM_INT_3
-
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_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
None.
8
Parallelism and Performance
f08ylc 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.
An approximate asymptotic error bound on the chordal distance between the computed eigenvalue
and the corresponding exact eigenvalue
is
where
is the
machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors
or
corresponding to the right and left eigenvectors
and
is given by
The complex analogue of this function is
f08yyc.
10
Example
This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair
given by
The eigenvalues and eigenvectors are computed by calling
f08ykc.
10.1
Program Text
10.2
Program Data
10.3
Program Results