F08WAF (DGGEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08WAF (DGGEV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

F08WAF (DGGEV) 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.

2  Specification

SUBROUTINE F08WAF ( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
INTEGER  N, LDA, LDB, LDVL, LDVR, LWORK, INFO
REAL (KIND=nag_wp)  A(LDA,*), B(LDB,*), ALPHAR(N), ALPHAI(N), BETA(N), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK))
CHARACTER(1)  JOBVL, JOBVR
The routine may be called by its LAPACK name dggev.

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
A vj = λj B vj .
The left eigenvector uj corresponding to the eigenvalue λj of A,B satisfies
ujH A = λj ujH B ,
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:
  1. A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
  2. 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 .
  3. 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
    λj=αj/βj,  j=1,2,,n.
    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.
  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 (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:     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'.
2:     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'.
3:     N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint: N0.
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 matrix A in the pair A,B.
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 F08WAF (DGGEV) is called.
Constraint: LDAmax1,N.
6:     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.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F08WAF (DGGEV) is called.
Constraint: LDBmax1,N.
8:     ALPHAR(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element ALPHARj contains the real part of αj.
9:     ALPHAI(N) – REAL (KIND=nag_wp) arrayOutput
On exit: the element ALPHAIj contains the imaginary part of αj.
10:   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.
11:   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.
12:   LDVL – INTEGERInput
On entry: the first dimension of the array VL as declared in the (sub)program from which F08WAF (DGGEV) is called.
Constraints:
  • if JOBVL='V', LDVLmax1,N;
  • otherwise LDVL1.
13:   VR(LDVR,*) – REAL (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 the corresponding 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.
14:   LDVR – INTEGERInput
On entry: the first dimension of the array VR as declared in the (sub)program from which F08WAF (DGGEV) is called.
Constraints:
  • if JOBVR='V', LDVRmax1,N;
  • otherwise LDVR1.
15:   WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
16:   LWORK – INTEGERInput
On entry: the dimension of the array WORK as declared in the (sub)program from which F08WAF (DGGEV) 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.
Constraint: LWORKmax1,8×N.
17:   INFO – INTEGEROutput

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
INFO=N+2

7  Accuracy

The computed eigenvalues and eigenvectors are exact for a nearby matrices A+E and B+F, where
E,F F = Oε A,B F ,
and ε is the machine precision. See Section 4.11 of
Anderson et al. (1999) for further details.
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 F08WNF (ZGGEV).

9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix pair A,B, where
A = 3.9 12.5 -34.5 -0.5 4.3 21.5 -47.5 7.5 4.3 21.5 -43.5 3.5 4.4 26.0 -46.0 6.0   and   B= 1.0 2.0 -3.0 1.0 1.0 3.0 -5.0 4.0 1.0 3.0 -4.0 3.0 1.0 3.0 -4.0 4.0 .
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 (f08wafe.f90)

9.2  Program Data

Program Data (f08wafe.d)

9.3  Program Results

Program Results (f08wafe.r)


F08WAF (DGGEV) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011