NAG Library Routine Document
F08WBF (DGGEVX)
1 Purpose
F08WBF (DGGEVX) computes for a pair of n by n real nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ 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 A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.
The right eigenvector
vj corresponding to the eigenvalue
λj of
A,B satisfies
The left eigenvector
uj corresponding to the eigenvalue
λj of
A,B satisfies
where
ujH is the conjugate-transpose of
uj.
All the eigenvalues and, if required, all the eigenvectors of the generalized eigenproblem
Ax=λBx, where
A and
B are real, square matrices, are determined using the
QZ algorithm. The
QZ algorithm consists of four stages:
- A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
- A is further reduced to quasi-triangular form while the triangular form of B is maintained. This is the real generalized Schur form of the pair
A,B
.
- The quasi-triangular form of A is reduced to triangular form and the eigenvalues extracted. This routine does not actually produce the eigenvalues λj, but instead returns αj and βj such that
The division by βj becomes your responsibility, since βj may be zero, indicating an infinite eigenvalue. Pairs of complex eigenvalues occur with αj/βj and αj+1/βj+1 complex conjugates, even though αj and αj+1 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
QZ algorithm
Linear Algebra Appl. 28 285–303
5 Parameters
- 1: BALANC – CHARACTER(1)Input
On entry: specifies the balance option to be performed.
- BALANC='N'
- Do not diagonally scale or permute.
- BALANC='P'
- Permute only.
- BALANC='S'
- Scale only.
- BALANC='B'
- 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, BALANC='B' is recommended.
Constraint:
BALANC='N', 'P', 'S' or 'B'.
- 2: JOBVL – CHARACTER(1)Input
On entry: if
JOBVL='N', do not compute the left generalized eigenvectors.
If JOBVL='V', compute the left generalized eigenvectors.
Constraint:
JOBVL='N' or 'V'.
- 3: JOBVR – CHARACTER(1)Input
On entry: if
JOBVR='N', do not compute the right generalized eigenvectors.
If JOBVR='V', compute the right generalized eigenvectors.
Constraint:
JOBVR='N' or 'V'.
- 4: SENSE – CHARACTER(1)Input
On entry: determines which reciprocal condition numbers are computed.
- SENSE='N'
- None are computed.
- SENSE='E'
- Computed for eigenvalues only.
- SENSE='V'
- Computed for eigenvectors only.
- SENSE='B'
- Computed for eigenvalues and eigenvectors.
Constraint:
SENSE='N', 'E', 'V' or 'B'.
- 5: N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint:
N≥0.
- 6: A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the matrix A in the pair A,B.
On exit:
A has been overwritten. If
JOBVL='V' or
JOBVR='V' or both, then
A contains the first part of the real Schur form of the ‘balanced’ versions of the input
A and
B.
- 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:
LDA≥max1,N.
- 8: B(LDB,*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the matrix B in the pair A,B.
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:
LDB≥max1,N.
- 10: ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element ALPHARj contains the real part of αj.
- 11: ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element ALPHAIj contains the imaginary part of αj.
- 12: BETA(N) – REAL (KIND=nag_wp) arrayOutput
On exit:
ALPHARj+ALPHAIj×i/BETAj, for
j=1,2,…,N, will be the generalized eigenvalues.
If ALPHAIj is zero, then the jth eigenvalue is real; if positive, then the jth and j+1st eigenvalues are a complex conjugate pair, with ALPHAIj+1 negative.
Note: the quotients ALPHARj/BETAj and ALPHAIj/BETAj may easily overflow or underflow, and BETAj may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with A2 in magnitude, and maxβj will always be less than and usually comparable with B2.
- 13: VL(LDVL,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VL
must be at least
max1,N if
JOBVL='V', and at least
1 otherwise.
On exit: if
JOBVL='V', the left eigenvectors
uj are stored one after another in the columns of
VL, in the same order as the corresponding eigenvalues.
If the jth eigenvalue is real, then uj=VL:,j, the jth column of VL.
If the jth and j+1th eigenvalues form a complex conjugate pair, then uj=VL:,j+i×VL:,j+1 and uj+1=VL:,j-i×VL:,j+1. Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If
JOBVL='N',
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 JOBVL='V', LDVL≥ max1,N ;
- otherwise LDVL≥1.
- 15: VR(LDVR,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VR
must be at least
max1,N if
JOBVR='V', and at least
1 otherwise.
On exit: if
JOBVR='V', the right eigenvectors
vj are stored one after another in the columns of
VR, in the same order as their eigenvalues.
If the jth eigenvalue is real, then vj=VR:,j, the jth column of VR.
If the jth and j+1th eigenvalues form a complex conjugate pair, then vj=VR:,j+i×VR:,j+1 and vj+1 = VR:,j -i×VR:,j+1.
Each eigenvector will be scaled so the largest component has real part+imag. part=1.
If
JOBVR='N',
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 JOBVR='V', LDVR≥ max1,N ;
- otherwise LDVR≥1.
- 17: ILO – INTEGEROutput
- 18: IHI – INTEGEROutput
On exit:
ILO and
IHI are integer values such that
Aij=0 and
Bij=0 if
i>j and
j=1,2,…,ILO-1 or
i=IHI+1,…,N.
If BALANC='N' or 'S', ILO=1 and IHI=N.
- 19: LSCALE(N) – REAL (KIND=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the left side of
A and
B.
If
plj is the index of the row interchanged with row
j, and
dlj is the scaling factor applied to row
j, then:
- LSCALEj = plj , for j=1,2,…,ILO-1;
- LSCALE = dlj , for j=ILO,…,IHI;
- LSCALE = plj , for j=IHI+1,…,N.
The order in which the interchanges are made is
N to
IHI+1, then
1 to
ILO-1.
- 20: RSCALE(N) – REAL (KIND=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the right side of
A and
B.
If
prj is the index of the column interchanged with column
j, and
drj is the scaling factor applied to column
j, then:
- RSCALEj=prj, for j=1,2,…,ILO-1;
- if
RSCALE=drj, for j=ILO,…,IHI;
- if
RSCALE=prj, for j=IHI+1,…,N.
The order in which the interchanges are made is
N to
IHI+1, then
1 to
ILO-1.
- 21: ABNRM – REAL (KIND=nag_wp)Output
On exit: the 1-norm of the balanced matrix A.
- 22: BBNRM – REAL (KIND=nag_wp)Output
On exit: the 1-norm of the balanced matrix B.
- 23: RCONDE(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDE
must be at least
max1,N.
On exit: if
SENSE='E' or
'B', 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
RCONDEj,
RCONDVj, and the
jth columns of
VL and
VR all correspond to the
jth eigenpair.
If
SENSE='V',
RCONDE is not referenced.
- 24: RCONDV(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDV
must be at least
max1,N.
On exit: if
SENSE='V' or
'B', 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
SENSE='E',
RCONDV is not referenced.
- 25: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 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
LWORK=-1, 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,
nb×N, where
nb is the optimal
block size.
Constraints:
- if SENSE='N',
- if BALANC='N' or 'P' and JOBVL='N' and JOBVR='N', LWORK≥max1,2×N;
- otherwise LWORK≥max1,6×N;
- if SENSE='E', LWORK≥max1,10×N;
- if SENSE='B' or SENSE='V', LWORK≥max10×N,2×N×N+4+16.
- 27: IWORK(*) – INTEGER arrayWorkspace
-
Note: the dimension of the array
IWORK
must be at least
N+6.
If
SENSE='E',
IWORK is not referenced.
- 28: BWORK(*) – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
max1,N.
If
SENSE='N',
BWORK is not referenced.
- 29: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO=1 to N
The QZ iteration failed. No eigenvectors have been calculated, but ALPHARj, ALPHAIj, and BETAj should be correct for j=INFO+1,…,N.
- INFO=N+1
Unexpected error returned from
F08XEF (DHGEQZ).
- INFO=N+2
Error returned from
F08YKF (DTGEVC).
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrices
A+E and
B+F, where
and
ε is the
machine precision.
An approximate error bound on the chordal distance between the
ith computed generalized eigenvalue
w and the corresponding exact eigenvalue
λ
is
An approximate error bound for the angle between the
ith computed eigenvector
VLi
or
VRi
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
QZ 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
αj and
βj. It should be noted that if
αj and
βj are
both small for any
j, it may be that no reliance can be placed on
any of the computed eigenvalues
λi=αi/βi. 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 Further Comments
The total number of floating point operations is proportional to n3.
The complex analogue of this routine is
F08WPF (ZGGEVX).
9 Example
This example finds all the eigenvalues and right eigenvectors of the matrix pair
A,B,
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 64 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)