NAG CL Interface
f08wbc (dggevx)
1
Purpose
f08wbc 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
void |
f08wbc (Nag_OrderType order,
Nag_BalanceType balanc,
Nag_LeftVecsType jobvl,
Nag_RightVecsType jobvr,
Nag_RCondType sense,
Integer n,
double a[],
Integer pda,
double b[],
Integer pdb,
double alphar[],
double alphai[],
double beta[],
double vl[],
Integer pdvl,
double vr[],
Integer pdvr,
Integer *ilo,
Integer *ihi,
double lscale[],
double rscale[],
double *abnrm,
double *bbnrm,
double rconde[],
double rcondv[],
NagError *fail) |
|
The function may be called by the names: f08wbc, nag_lapackeig_dggevx or nag_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 function 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
f08whc.
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:
– 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_BalanceType
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 .
-
3:
– Nag_LeftVecsType
Input
-
On entry: if
, do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
-
4:
– Nag_RightVecsType
Input
-
On entry: if
, do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
-
5:
– Nag_RCondType
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 .
-
6:
– Integer
Input
-
On entry: , the order of the matrices and .
Constraint:
.
-
7:
– double
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
.
where
appears in this document, it refers to the array element
- when ;
- when .
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
.
-
8:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
-
9:
– double
Input/Output
-
Note: the dimension,
dim, of the array
b
must be at least
.
where
appears in this document, it refers to the array element
- when ;
- when .
On entry: the matrix in the pair .
On exit:
b has been overwritten.
-
10:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
-
11:
– double
Output
-
On exit: the element contains the real part of .
-
12:
– double
Output
-
On exit: the element contains the imaginary part of .
-
13:
– double
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 .
-
14:
– double
Output
-
Note: the dimension,
dim, of the array
vl
must be at least
- when
;
- otherwise.
The
th element of the matrix is stored in
- when ;
- when .
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.
-
15:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vl.
Constraints:
- if , ;
- otherwise .
-
16:
– double
Output
-
Note: the dimension,
dim, of the array
vr
must be at least
- when
;
- otherwise.
The
th element of the matrix is stored in
- when ;
- when .
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.
-
17:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vr.
Constraints:
- if , ;
- otherwise .
-
18:
– Integer *
Output
-
19:
– Integer *
Output
-
On exit:
ilo and
ihi are integer values such that
and
if
and
or
.
If or , and .
-
20:
– double
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
.
-
21:
– double
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
.
-
22:
– double *
Output
-
On exit: the -norm of the balanced matrix .
-
23:
– double *
Output
-
On exit: the -norm of the balanced matrix .
-
24:
– double
Output
-
Note: the dimension,
dim, 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.
-
25:
– double
Output
-
Note: the dimension,
dim, 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.
-
26:
– 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_EIGENVECTORS
-
A failure occurred in
f08ykc while computing generalized eigenvectors.
- 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_ITERATION_QZ
-
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.
- 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
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
f08wbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wbc 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
f08wpc.
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.
10.1
Program Text
10.2
Program Data
10.3
Program Results