NAG Library Routine Document
F08NPF (ZGEEVX)
1 Purpose
F08NPF (ZGEEVX) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n complex nonsymmetric matrix A.
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 F08NPF ( |
BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, INFO) |
| INTEGER |
N, LDA, LDVL, LDVR, ILO, IHI, LWORK, INFO |
| REAL (KIND=nag_wp) |
SCALE(*), ABNRM, RCONDE(*), RCONDV(*), RWORK(*) |
| COMPLEX (KIND=nag_wp) |
A(LDA,*), W(*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK)) |
| CHARACTER(1) |
BALANC, JOBVL, JOBVR, SENSE |
|
The routine may be called by its
LAPACK
name zgeevx.
3 Description
The right eigenvector
vj of
A satisfies
where
λj is the
jth eigenvalue of
A. The left eigenvector
uj of
A satisfies
where
ujH denotes the conjugate transpose of
uj.
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation
DAD-1, where
D is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of
Anderson et al. (1999).
Following the optional balancing, the matrix A is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper triangular Schur form, T, from which the eigenvalues are computed. Optionally, the eigenvectors of T are also computed and backtransformed to those of A.
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/lugGolub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: BALANC – CHARACTER(1)Input
-
On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
- BALANC='N'
- Do not diagonally scale or permute.
- BALANC='P'
- Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
- BALANC='S'
- Diagonally scale the matrix, i.e., replace A by DAD-1, where D is a diagonal matrix chosen to make the rows and columns of A more equal in norm. Do not permute.
- BALANC='B'
- Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
Constraint:
BALANC='N', 'P', 'S' or 'B'.
- 2: JOBVL – CHARACTER(1)Input
-
On entry: if
JOBVL='N', the left eigenvectors of
A are not computed.
If JOBVL='V', the left eigenvectors of A are computed.
If
SENSE='E' or
'B',
JOBVL must be set to
JOBVL='V'.
Constraint:
JOBVL='N' or 'V'.
- 3: JOBVR – CHARACTER(1)Input
-
On entry: if
JOBVR='N', the right eigenvectors of
A are not computed.
If JOBVR='V', the right eigenvectors of A are computed.
If
SENSE='E' or
'B',
JOBVR must be set to
JOBVR='V'.
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 right eigenvectors only.
- SENSE='B'
- Computed for eigenvalues and right eigenvectors.
If SENSE='E' or 'B', both left and right eigenvectors must also be computed (JOBVL='V' and JOBVR='V').
Constraint:
SENSE='N', 'E', 'V' or 'B'.
- 5: N – INTEGERInput
-
On entry:
n, the order of the matrix A.
Constraint:
N≥0.
- 6: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the n by n matrix A.
On exit:
A has been overwritten. If
JOBVL='V' or
JOBVR='V',
A contains the
Schur
form of the balanced version of the
matrix
A.
- 7: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08NPF (ZGEEVX) is called.
Constraint:
LDA≥max1,N.
- 8: W(*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
W
must be at least
max1,N.
On exit: contains the computed eigenvalues.
- 9: VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VL
must be at least
max1,N.
On exit: if
JOBVL='V', the left eigenvectors
uj are stored one after another in the columns of
VL, in the same order as their corresponding eigenvalues.
If
JOBVL='N',
VL is not referenced.
uj=VL:,j, the
jth column of
VL.
- 10: LDVL – INTEGERInput
-
On entry: the first dimension of the array
VL as declared in the (sub)program from which F08NPF (ZGEEVX) is called.
Constraints:
- if JOBVL='V', LDVL≥max1,N;
- otherwise LDVL≥1.
- 11: VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VR
must be at least
max1,N.
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 corresponding eigenvalues.
If
JOBVR='N',
VR is not referenced.
vj=VR:,j, the
jth column of
VR.
- 12: LDVR – INTEGERInput
-
On entry: the first dimension of the array
VR as declared in the (sub)program from which F08NPF (ZGEEVX) is called.
Constraints:
- if JOBVR='V', LDVR≥max1,N;
- otherwise LDVR≥1.
- 13: ILO – INTEGEROutput
- 14: IHI – INTEGEROutput
-
On exit:
ILO and
IHI are integer values determined when
A was balanced. The balanced
A has
aij=0 if
i>j and
j=1,2,…,ILO-1 or
i=IHI+1,…,N.
- 15: SCALE(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
SCALE
must be at least
max1,N.
On exit: details of the permutations and scaling factors applied when balancing
A.
If
pj is the index of the row and column interchanged with row and column
j, and
dj is the scaling factor applied to row and column
j, then
- SCALEj=pj, for j=1,2,…,ILO-1;
- SCALEj=dj, for j=ILO,…,IHI;
- SCALEj=pj, for j=IHI+1,…,N.
The order in which the interchanges are made is
N to
IHI+1, then
1 to
ILO-1.
- 16: ABNRM – REAL (KIND=nag_wp)Output
-
On exit: the 1-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
- 17: RCONDE(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDE
must be at least
max1,N.
On exit: RCONDEj is the reciprocal condition number of the jth eigenvalue.
- 18: RCONDV(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDV
must be at least
max1,N.
On exit: RCONDVj is the reciprocal condition number of the jth right eigenvector.
- 19: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 20: LWORK – INTEGERInput
-
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08NPF (ZGEEVX) 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
LWORK by, say,
N×nb
, where
nb
is the optimal
block size for
F08NEF (DGEHRD).
Constraints:
- if SENSE='N' or 'E', LWORK≥max1,2×N;
- if SENSE='V' or 'B', LWORK≥max1,N×N+2×N.
- 21: RWORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
RWORK
must be at least
max1,2×N.
- 22: 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>0
If
INFO=i, the
QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements
1:ILO-1 and
i+1:N of
W contain eigenvalues which have converged.
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrix
A+E, where
and
ε is the
machine precision. See Section 4.8 of
Anderson et al. (1999) for further details.
8 Further Comments
Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real and positive.
The total number of floating point operations is proportional to n3.
The real analogue of this routine is
F08NBF (DGEEVX).
9 Example
This example finds all the eigenvalues and right eigenvectors of the matrix
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.
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 (f08npfe.f90)
9.2 Program Data
Program Data (f08npfe.d)
9.3 Program Results
Program Results (f08npfe.r)