NAG Library Routine Document
F08NAF (DGEEV)
1 Purpose
F08NAF (DGEEV) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an n by n real nonsymmetric matrix A.
2 Specification
SUBROUTINE F08NAF ( |
JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO) |
INTEGER |
N, LDA, LDVL, LDVR, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBVL, JOBVR |
|
The routine may be called by its
LAPACK
name dgeev.
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.
The matrix A is first reduced to upper Hessenberg form by means of orthogonal similarity transformations, and the QR algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, T, with 1 by 1 and 2 by 2 blocks on the main diagonal. The eigenvalues are computed from T, the 2 by 2 blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of T are computed and backtransformed to the eigenvectors 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/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: 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.
Constraint:
JOBVL='N' or 'V'.
- 2: 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.
Constraint:
JOBVR='N' or 'V'.
- 3: N – INTEGERInput
On entry: n, the order of the matrix A.
Constraint:
N≥0.
- 4: 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 n by n matrix A.
On exit:
A has been overwritten.
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08NAF (DGEEV) is called.
Constraint:
LDA≥max1,N.
- 6: WR(*) – REAL (KIND=nag_wp) arrayOutput
- 7: WI(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the arrays
WR and
WI
must be at least
max1,N.
On exit:
WR and
WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
- 8: 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 their corresponding eigenvalues. If the
jth eigenvalue is real, then
uj=VL:j, the
jth column of
VL. If the
jth and
j+1st eigenvalues form a complex conjugate pair, then
uj=VL:j+i×VL:j+1 and
uj+1=VL:j-i×VL:j+1.
If
JOBVL='N',
VL is not referenced.
- 9: LDVL – INTEGERInput
On entry: the first dimension of the array
VL as declared in the (sub)program from which F08NAF (DGEEV) is called.
Constraints:
- if JOBVL='V', LDVL≥ max1,N ;
- otherwise LDVL≥1.
- 10: 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 corresponding eigenvalues. If the
jth eigenvalue is real, then
vj=VR:j, the
jth column of
VR. If the
jth and
j+1st eigenvalues form a complex conjugate pair, then
vj=VR:j+i×VR:j+1 and
vj+1=VR:j-i×VR:j+1.
If
JOBVR='N',
VR is not referenced.
- 11: LDVR – INTEGERInput
On entry: the first dimension of the array
VR as declared in the (sub)program from which F08NAF (DGEEV) is called.
Constraints:
- if JOBVR='V', LDVR≥ max1,N ;
- otherwise LDVR≥1.
- 12: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 13: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08NAF (DGEEV) 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, say,
4×N+nb×N , where
nb is the optimal
block size of
F08NEF (DGEHRD).
Constraints:
- if JOBVL='V' or JOBVR='V', LWORK≥4×N;
- otherwise LWORK≥max1,3×N.
- 14: 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 have been computed; elements
i+1:N of
WRWI
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 complex analogue of this routine is
F08NNF (ZGEEV).
9 Example
This example finds all the eigenvalues and right eigenvectors of the matrix
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 (f08nafe.f90)
9.2 Program Data
Program Data (f08nafe.d)
9.3 Program Results
Program Results (f08nafe.r)