NAG Library Routine Document
F08WBF (DGGEVX)
1 Purpose
F08WBF (DGGEVX) 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
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 |
N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, IWORK(*), INFO |
REAL (KIND=nag_wp) |
A(LDA,*), B(LDB,*), ALPHAR(N), ALPHAI(N), BETA(N), VL(LDVL,*), VR(LDVR,*), LSCALE(N), RSCALE(N), ABNRM, BBNRM, RCONDE(*), RCONDV(*), WORK(max(1,LWORK)) |
LOGICAL |
BWORK(*) |
CHARACTER(1) |
BALANC, JOBVL, JOBVR, SENSE |
|
The routine may be called by 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:
- is reduced to upper Hessenberg form and at the same time is reduced to upper triangular form.
- is further reduced to quasi-triangular form while the triangular form of is maintained. This is the real generalized Schur form of the pair .
- 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.
- 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 (DGGBAL).
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 (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the algorithm Linear Algebra Appl. 28 285–303
5 Parameters
- 1: BALANC – 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: JOBVL – CHARACTER(1)Input
On entry: if
, do not compute the left generalized eigenvectors.
If , compute the left generalized eigenvectors.
Constraint:
or .
- 3: JOBVR – CHARACTER(1)Input
On entry: if
, do not compute the right generalized eigenvectors.
If , compute the right generalized eigenvectors.
Constraint:
or .
- 4: SENSE – 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: N – INTEGERInput
On entry: , the order of the matrices and .
Constraint:
.
- 6: A(LDA,) – REAL (KIND=nag_wp) arrayInput/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: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08WBF (DGGEVX) is called.
Constraint:
.
- 8: B(LDB,) – REAL (KIND=nag_wp) arrayInput/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: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08WBF (DGGEVX) is called.
Constraint:
.
- 10: ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element contains the real part of .
- 11: ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element contains the imaginary part of .
- 12: BETA(N) – REAL (KIND=nag_wp) arrayOutput
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: VL(LDVL,) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VL
must be at least
if
, and at least
otherwise.
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.
- 14: LDVL – INTEGERInput
On entry: the first dimension of the array
VL as declared in the (sub)program from which F08WBF (DGGEVX) is called.
Constraints:
- if , ;
- otherwise .
- 15: VR(LDVR,) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VR
must be at least
if
, and at least
otherwise.
On exit: if
, the right eigenvectors
are stored one after another in the columns of
VR, in the same order as their 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.
- 16: LDVR – INTEGERInput
On entry: the first dimension of the array
VR as declared in the (sub)program from which F08WBF (DGGEVX) is called.
Constraints:
- if , ;
- otherwise .
- 17: ILO – INTEGEROutput
- 18: IHI – INTEGEROutput
On exit:
ILO and
IHI are integer values such that
and
if
and
or
.
If or , and .
- 19: LSCALE(N) – REAL (KIND=nag_wp) arrayOutput
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: RSCALE(N) – REAL (KIND=nag_wp) arrayOutput
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: ABNRM – REAL (KIND=nag_wp)Output
On exit: the -norm of the balanced matrix .
- 22: BBNRM – REAL (KIND=nag_wp)Output
On exit: the -norm of the balanced matrix .
- 23: RCONDE() – REAL (KIND=nag_wp) arrayOutput
-
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: RCONDV() – REAL (KIND=nag_wp) arrayOutput
-
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: WORK() – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
,
contains the minimum value of
LWORK required for optimal performance.
- 26: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08WBF (DGGEVX) 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: IWORK() – INTEGER arrayWorkspace
-
Note: the dimension of the array
IWORK
must be at least
.
If
,
IWORK is not referenced.
- 28: BWORK() – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
.
If
,
BWORK is not referenced.
- 29: INFO – INTEGEROutput
On exit:
unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
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 for .
Unexpected error returned from
F08XEF (DHGEQZ).
Error returned from
F08YKF (DTGEVC).
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a 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.
The total number of floating point operations is proportional to .
The complex analogue of this routine is
F08WPF (ZGGEVX).
9 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.
9.1 Program Text
Program Text (f08wbfe.f90)
9.2 Program Data
Program Data (f08wbfe.d)
9.3 Program Results
Program Results (f08wbfe.r)