NAG FL Interface
f08wbf (dggevx)

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1 Purpose

f08wbf computes for a pair of n×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

Fortran Interface
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, Intent (In) :: n, lda, ldb, ldvl, ldvr, lwork
Integer, Intent (Inout) :: iwork(*)
Integer, Intent (Out) :: ilo, ihi, info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), vl(ldvl,*), vr(ldvr,*), rconde(*), rcondv(*)
Real (Kind=nag_wp), Intent (Out) :: alphar(n), alphai(n), beta(n), lscale(n), rscale(n), abnrm, bbnrm, work(max(1,lwork))
Logical, Intent (Inout) :: bwork(*)
Character (1), Intent (In) :: balanc, jobvl, jobvr, sense
C Header Interface
#include <nag.h>
void  f08wbf_ (const char *balanc, const char *jobvl, const char *jobvr, const char *sense, const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, double alphar[], double alphai[], double beta[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, Integer *ilo, Integer *ihi, double lscale[], double rscale[], double *abnrm, double *bbnrm, double rconde[], double rcondv[], double work[], const Integer *lwork, Integer iwork[], logical bwork[], Integer *info, const Charlen length_balanc, const Charlen length_jobvl, const Charlen length_jobvr, const Charlen length_sense)
The routine may be called by the names f08wbf, nagf_lapackeig_dggevx or 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
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. 1.A is reduced to upper Hessenberg form and at the same time B is reduced to upper triangular form.
  2. 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. 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. 4.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.

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (2012) Matrix Computations (4th 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 Arguments

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 Integer Input
On entry: n, the order of the matrices A and B.
Constraint: n0.
6: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,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 Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08wbf is called.
Constraint: ldamax(1,n).
8: b(ldb,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least max(1,n).
On entry: the matrix B in the pair (A,B).
On exit: b has been overwritten.
9: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08wbf is called.
Constraint: ldbmax(1,n).
10: alphar(n) Real (Kind=nag_wp) array Output
On exit: the element alphar(j) contains the real part of αj.
11: alphai(n) Real (Kind=nag_wp) array Output
On exit: the element alphai(j) contains the imaginary part of αj.
12: beta(n) Real (Kind=nag_wp) array Output
On exit: (alphar(j)+alphai(j)×i)/beta(j), for j=1,2,,n, will be the generalized eigenvalues.
If alphai(j) is zero, then the jth eigenvalue is real; if positive, then the jth and (j+1)st eigenvalues are a complex conjugate pair, with alphai(j+1) negative.
Note:  the quotients alphar(j)/beta(j) and alphai(j)/beta(j) may easily overflow or underflow, and beta(j) may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, max|αj| will always be less than and usually comparable with A2 in magnitude, and max|βj| will always be less than and usually comparable with B2.
13: vl(ldvl,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vl must be at least max(1,n) if jobvl='V', and at least 1 otherwise.
On exit: if jobvl='V', the left generalized eigenvectors uj are stored one after another in the columns of vl, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have |real part|+|imag. part|=1.
If jobvl='N', vl is not referenced.
14: ldvl Integer Input
On entry: the first dimension of the array vl as declared in the (sub)program from which f08wbf is called.
Constraints:
  • if jobvl='V', ldvl max(1,n) ;
  • otherwise ldvl1.
15: vr(ldvr,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vr must be at least max(1,n) if jobvr='V', and at least 1 otherwise.
On exit: if jobvr='V', the right generalized eigenvectors vj are stored one after another in the columns of vr, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have |real part|+|imag. part|=1.
If jobvr='N', vr is not referenced.
16: ldvr Integer Input
On entry: the first dimension of the array vr as declared in the (sub)program from which f08wbf is called.
Constraints:
  • if jobvr='V', ldvr max(1,n) ;
  • otherwise ldvr1.
17: ilo Integer Output
18: ihi Integer Output
On exit: ilo and ihi are integer values such that a(i,j)=0 and b(i,j)=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) array Output
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:
  • lscale(j) = 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) array Output
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:
  • rscale(j)=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) array Output
Note: the dimension of the array rconde must be at least max(1,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 rconde(j), rcondv(j), 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) array Output
Note: the dimension of the array rcondv must be at least max(1,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(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
26: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08wbf 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', lworkmax(1,2×n);
    • otherwise lworkmax(1,6×n);
  • if sense='E', lworkmax(1,10×n);
  • if sense='B' or sense='V', lworkmax(10×n,2×n×(n+4)+16).
27: iwork(*) Integer array Workspace
Note: the dimension of the array iwork must be at least n+6.
If sense='E', iwork is not referenced.
28: bwork(*) Logical array Workspace
Note: the dimension of the array bwork must be at least max(1,n).
If sense='N', bwork is not referenced.
29: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

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,,n
The QZ iteration failed. No eigenvectors have been calculated but alphar(j), alphai(j) and beta(j) should be correct from element value.
info=n+1
The QZ iteration failed with an unexpected error, please contact NAG.
info=n+2
A failure occurred in f08ykf while computing generalized eigenvectors.

7 Accuracy

The computed eigenvalues and eigenvectors are exact for nearby matrices (A+E) and (B+F), where
(E,F) F = O(ε) (A,B) F ,  
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
ε × abnrm,bbnrm2 / rconde(i) .  
An approximate error bound for the angle between the ith computed eigenvector uj or vj is given by
ε × abnrm,bbnrm2 / rcondv(i) .  
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 Parallelism and Performance

f08wbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08wbf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is proportional to n3.
The complex analogue of this routine is f08wpf.

10 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 ) ,  
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.

10.1 Program Text

Program Text (f08wbfe.f90)

10.2 Program Data

Program Data (f08wbfe.d)

10.3 Program Results

Program Results (f08wbfe.r)