NAG Library Function Document
nag_dggev3 (f08wcc)
1
Purpose
nag_dggev3 (f08wcc) computes for a pair of by real nonsymmetric matrices the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the algorithm.
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dggev3 (Nag_OrderType order,
Nag_LeftVecsType jobvl,
Nag_RightVecsType jobvr,
Integer n,
double a[],
Integer pda,
double b[],
Integer pdb,
double alphar[],
double alphai[],
double beta[],
double vl[],
Integer pdvl,
double vr[],
Integer pdvr,
NagError *fail) |
|
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. |
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
http://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_OrderTypeInput
-
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_LeftVecsTypeInput
-
On entry: if
, do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
- 3:
– Nag_RightVecsTypeInput
-
On entry: if
, do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
- 4:
– IntegerInput
-
On entry: , the order of the matrices and .
Constraint:
.
- 5:
– doubleInput/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 matrix in the pair .
On exit:
a has been overwritten.
- 6:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraint:
.
- 7:
– doubleInput/Output
-
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 matrix in the pair .
On exit:
b has been overwritten.
- 8:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraint:
.
- 9:
– doubleOutput
-
On exit: the element contains the real part of .
- 10:
– doubleOutput
-
On exit: the element contains the imaginary part of .
- 11:
– doubleOutput
-
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 .
- 12:
– doubleOutput
-
Note: the dimension,
dim, of the array
vl
must be at least
- when
;
- otherwise.
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
, the left eigenvectors
are stored one after another in the columns of
vl, in the same order as the corresponding eigenvalues.
If the th eigenvalue is real, then , the th column of .
If the th and th eigenvalues form a complex conjugate pair, then and . Each eigenvector will be scaled so the largest component has .
If
,
vl is not referenced.
- 13:
– IntegerInput
-
On entry: the stride used in the array
vl.
Constraints:
- if , ;
- otherwise .
- 14:
– doubleOutput
-
Note: the dimension,
dim, of the array
vr
must be at least
- when
;
- otherwise.
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: if
, the right eigenvectors
are stored one after another in the columns of
vr, in the same order as the corresponding eigenvalues.
If the th eigenvalue is real, then , the th column of .
If the th and th eigenvalues form a complex conjugate pair, then and . Each eigenvector will be scaled so the largest component has .
If
,
vr is not referenced.
- 15:
– IntegerInput
-
On entry: the stride used in the array
vr.
Constraints:
- if , ;
- otherwise .
- 16:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_EIGENVECTORS
-
A failure occurred in
nag_dtgevc (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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The computed eigenvalues and eigenvectors are exact for nearby matrices
and
, where
and
is the
machine precision. See Section 4.11 of
Anderson et al. (1999) for further details.
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
nag_dggev3 (f08wcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggev3 (f08wcc) 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
nag_zggev3 (f08wqc).
10
Example
This example finds all the eigenvalues and right eigenvectors of the matrix pair
,
where
10.1
Program Text
Program Text (f08wcce.c)
10.2
Program Data
Program Data (f08wcce.d)
10.3
Program Results
Program Results (f08wcce.r)