NAG FL Interface
f08wbf (dggevx)
1
Purpose
f08wbf computes for a pair of by real nonsymmetric matrices the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.
2
Specification
Fortran Interface
Subroutine f08wbf ( |
balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info) |
Integer, Intent (In) |
:: |
n, lda, ldb, ldvl, ldvr, lwork |
Integer, Intent (Inout) |
:: |
iwork(*) |
Integer, Intent (Out) |
:: |
ilo, ihi, info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
a(lda,*), b(ldb,*), vl(ldvl,*), vr(ldvr,*), rconde(*), rcondv(*) |
Real (Kind=nag_wp), Intent (Out) |
:: |
alphar(n), alphai(n), beta(n), lscale(n), rscale(n), abnrm, bbnrm, work(max(1,lwork)) |
Logical, Intent (Inout) |
:: |
bwork(*) |
Character (1), Intent (In) |
:: |
balanc, jobvl, jobvr, sense |
|
C Header Interface
#include <nag.h>
void |
f08wbf_ (const char *balanc, const char *jobvl, const char *jobvr, const char *sense, const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, double alphar[], double alphai[], double beta[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], double work[], const Integer *lwork, Integer iwork[], logical bwork[], Integer *info, const Charlen length_balanc, const Charlen length_jobvl, const Charlen length_jobvr, const Charlen length_sense) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08wbf_ (const char *balanc, const char *jobvl, const char *jobvr, const char *sense, const Integer &n, double a[], const Integer &lda, double b[], const Integer &ldb, double alphar[], double alphai[], double beta[], double vl[], const Integer &ldvl, double vr[], const Integer &ldvr, Integer &ilo, Integer &ihi, double lscale[], double rscale[], double &abnrm, double &bbnrm, double rconde[], double rcondv[], double work[], const Integer &lwork, Integer iwork[], logical bwork[], Integer &info, const Charlen length_balanc, const Charlen length_jobvl, const Charlen length_jobvr, const Charlen length_sense) |
}
|
The routine may be called by the names f08wbf, nagf_lapackeig_dggevx or its LAPACK name dggevx.
3
Description
A generalized eigenvalue for a pair of matrices is a scalar or a ratio , such that is singular. It is usually represented as the pair , as there is a reasonable interpretation for , and even for both being zero.
The right eigenvector
corresponding to the eigenvalue
of
satisfies
The left eigenvector
corresponding to the eigenvalue
of
satisfies
where
is the conjugate-transpose of
.
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem
, where
and
are real, square matrices, are determined using the
algorithm. The
algorithm consists of four stages:
-
1. is reduced to upper Hessenberg form and at the same time is reduced to upper triangular form.
-
2. is further reduced to quasi-triangular form while the triangular form of is maintained. This is the real generalized Schur form of the pair .
-
3.The quasi-triangular form of is reduced to triangular form and the eigenvalues extracted. This routine does not actually produce the eigenvalues , but instead returns and such that
The division by becomes your responsibility, since may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with and complex conjugates, even though and are not conjugate.
-
4.If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
For details of the balancing option, see
Section 3 in
f08whf.
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
Wilkinson J H (1979) Kronecker's canonical form and the algorithm Linear Algebra Appl. 28 285–303
5
Arguments
-
1:
– Character(1)
Input
-
On entry: specifies the balance option to be performed.
- Do not diagonally scale or permute.
- Permute only.
- Scale only.
- Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, is recommended.
Constraint:
, , or .
-
2:
– Character(1)
Input
-
On entry: if
, do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
-
3:
– Character(1)
Input
-
On entry: if
, do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
-
4:
– Character(1)
Input
-
On entry: determines which reciprocal condition numbers are computed.
- None are computed.
- Computed for eigenvalues only.
- Computed for eigenvectors only.
- Computed for eigenvalues and eigenvectors.
Constraint:
, , or .
-
5:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
6:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
a
must be at least
.
On entry: the matrix in the pair .
On exit:
a has been overwritten. If
or
or both, then
contains the first part of the real Schur form of the ‘balanced’ versions of the input
and
.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
a as declared in the (sub)program from which
f08wbf is called.
Constraint:
.
-
8:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the second dimension of the array
b
must be at least
.
On entry: the matrix in the pair .
On exit:
b has been overwritten.
-
9:
– Integer
Input
-
On entry: the first dimension of the array
b as declared in the (sub)program from which
f08wbf is called.
Constraint:
.
-
10:
– Real (Kind=nag_wp) array
Output
-
On exit: the element contains the real part of .
-
11:
– Real (Kind=nag_wp) array
Output
-
On exit: the element contains the imaginary part of .
-
12:
– Real (Kind=nag_wp) array
Output
-
On exit:
, for
, will be the generalized eigenvalues.
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, will always be less than and usually comparable with in magnitude, and will always be less than and usually comparable with .
-
13:
– Real (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
vl
must be at least
if
, and at least
otherwise.
On exit: if
, the left generalized eigenvectors
are stored one after another in the columns of
vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have
.
If
,
vl is not referenced.
-
14:
– Integer
Input
-
On entry: the first dimension of the array
vl as declared in the (sub)program from which
f08wbf is called.
Constraints:
- if , ;
- otherwise .
-
15:
– Real (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
vr
must be at least
if
, and at least
otherwise.
On exit: if
, the right generalized eigenvectors
are stored one after another in the columns of
vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have
.
If
,
vr is not referenced.
-
16:
– Integer
Input
-
On entry: the first dimension of the array
vr as declared in the (sub)program from which
f08wbf is called.
Constraints:
- if , ;
- otherwise .
-
17:
– Integer
Output
-
18:
– Integer
Output
-
On exit:
ilo and
ihi are integer values such that
and
if
and
or
.
If or , and .
-
19:
– Real (Kind=nag_wp) array
Output
-
On exit: details of the permutations and scaling factors applied to the left side of
and
.
If
is the index of the row interchanged with row
, and
is the scaling factor applied to row
, then:
- , for ;
- , for ;
- , for .
The order in which the interchanges are made is
n to
, then
to
.
-
20:
– Real (Kind=nag_wp) array
Output
-
On exit: details of the permutations and scaling factors applied to the right side of
and
.
If
is the index of the column interchanged with column
, and
is the scaling factor applied to column
, then:
- , for ;
- if
, for ;
- if
, for .
The order in which the interchanges are made is
n to
, then
to
.
-
21:
– Real (Kind=nag_wp)
Output
-
On exit: the -norm of the balanced matrix .
-
22:
– Real (Kind=nag_wp)
Output
-
On exit: the -norm of the balanced matrix .
-
23:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
rconde
must be at least
.
On exit: if
or
, the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of
rconde are set to the same value. Thus
,
, and the
th columns of
vl and
vr all correspond to the
th eigenpair.
If
,
rconde is not referenced.
-
24:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
rcondv
must be at least
.
On exit: if
or
, the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of
rcondv are set to the same value.
If
,
rcondv is not referenced.
-
25:
– Real (Kind=nag_wp) array
Workspace
-
On exit: if
,
contains the minimum value of
lwork required for optimal performance.
-
26:
– Integer
Input
-
On entry: the dimension of the array
work as declared in the (sub)program from which
f08wbf is called.
If
, a workspace query is assumed; the routine only calculates the optimal size of the
work array, returns this value as the first entry of the
work array, and no error message related to
lwork is issued.
Suggested value:
for optimal performance,
lwork must generally be larger than the minimum; increase workspace by, say,
, where
is the optimal
block size.
Constraints:
- if ,
- if or and and , ;
- otherwise ;
- if , ;
- if or , .
-
27:
– Integer array
Workspace
-
Note: the dimension of the array
iwork
must be at least
.
If
,
iwork is not referenced.
-
28:
– Logical array
Workspace
-
Note: the dimension of the array
bwork
must be at least
.
If
,
bwork is not referenced.
-
29:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
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.
-
A failure occurred in
f08ykf while computing generalized eigenvectors.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for nearby matrices
and
, where
and
is the
machine precision.
An approximate error bound on the chordal distance between the
th computed generalized eigenvalue
and the corresponding exact eigenvalue
is
An approximate error bound for the angle between the
th computed eigenvector
or
is given by
For further explanation of the reciprocal condition numbers
rconde and
rcondv, see Section 4.11 of
Anderson et al. (1999).
Note: interpretation of results obtained with the
algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in
Wilkinson (1979), in relation to the significance of small values of
and
. It should be noted that if
and
are
both small for any
, it may be that no reliance can be placed on
any of the computed eigenvalues
. You are recommended to study
Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.
8
Parallelism and Performance
f08wbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wbf 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 routine. 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 routine is
f08wpf.
10
Example
This example finds all the eigenvalues and right eigenvectors of the matrix pair
,
where
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.
Note that the block size (NB) of assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
10.1
Program Text
10.2
Program Data
10.3
Program Results