NAG FL Interface
f08xcf (dgges3)

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1 Purpose

f08xcf computes the generalized eigenvalues, the generalized real Schur form (S,T) and, optionally, the left and/or right generalized Schur vectors for a pair of n×n real nonsymmetric matrices (A,B) .

2 Specification

Fortran Interface
Subroutine f08xcf ( jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)
Integer, Intent (In) :: n, lda, ldb, ldvsl, ldvsr, lwork
Integer, Intent (Out) :: sdim, info
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*), vsl(ldvsl,*), vsr(ldvsr,*)
Real (Kind=nag_wp), Intent (Out) :: alphar(n), alphai(n), beta(n), work(max(1,lwork))
Logical, External :: selctg
Logical, Intent (Inout) :: bwork(*)
Character (1), Intent (In) :: jobvsl, jobvsr, sort
C Header Interface
#include <nag.h>
void  f08xcf_ (const char *jobvsl, const char *jobvsr, const char *sort,
logical (NAG_CALL *selctg)(const double *ar, const double *ai, const double *b),
const Integer *n, double a[], const Integer *lda, double b[], const Integer *ldb, Integer *sdim, double alphar[], double alphai[], double beta[], double vsl[], const Integer *ldvsl, double vsr[], const Integer *ldvsr, double work[], const Integer *lwork, logical bwork[], Integer *info, const Charlen length_jobvsl, const Charlen length_jobvsr, const Charlen length_sort)
The routine may be called by the names f08xcf, nagf_lapackeig_dgges3 or its LAPACK name dgges3.

3 Description

The generalized Schur factorization for a pair of real matrices (A,B) is given by
A = QSZT ,   B = QTZT ,  
where Q and Z are orthogonal, T is upper triangular and S is upper quasi-triangular with 1×1 and 2×2 diagonal blocks. The generalized eigenvalues, λ , of (A,B) are computed from the diagonals of S and T and satisfy
Az = λBz ,  
where z is the corresponding generalized eigenvector. λ is actually returned as the pair (α,β) such that
λ = α/β  
since β , or even both α and β can be zero. The columns of Q and Z are the left and right generalized Schur vectors of (A,B) .
Optionally, f08xcf can order the generalized eigenvalues on the diagonals of (S,T) so that selected eigenvalues are at the top left. The leading columns of Q and Z then form an orthonormal basis for the corresponding eigenspaces, the deflating subspaces.
f08xcf computes T to have non-negative diagonal elements, and the 2×2 blocks of S correspond to complex conjugate pairs of generalized eigenvalues. The generalized Schur factorization, before reordering, is computed by the QZ algorithm.

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

5 Arguments

1: jobvsl Character(1) Input
On entry: if jobvsl='N', do not compute the left Schur vectors.
If jobvsl='V', compute the left Schur vectors.
Constraint: jobvsl='N' or 'V'.
2: jobvsr Character(1) Input
On entry: if jobvsr='N', do not compute the right Schur vectors.
If jobvsr='V', compute the right Schur vectors.
Constraint: jobvsr='N' or 'V'.
3: sort Character(1) Input
On entry: specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.
sort='N'
Eigenvalues are not ordered.
sort='S'
Eigenvalues are ordered (see selctg).
Constraint: sort='N' or 'S'.
4: selctg Logical Function, supplied by the user. External Procedure
If sort='S', selctg is used to select generalized eigenvalues to be moved to the top left of the generalized Schur form.
If sort='N', selctg is not referenced by f08xcf, and may be called with the dummy function f08xaz.
The specification of selctg is:
Fortran Interface
Function selctg ( ar, ai, b)
Logical :: selctg
Real (Kind=nag_wp), Intent (In) :: ar, ai, b
C Header Interface
Nag_Boolean  selctg (const double *ar, const double *ai, const double *b)
1: ar Real (Kind=nag_wp) Input
2: ai Real (Kind=nag_wp) Input
3: b Real (Kind=nag_wp) Input
On entry: an eigenvalue (ar(j)+-1×ai(j)) / b(j) is selected if selctg(ar(j),ai(j),b(j))=.TRUE. . If either one of a complex conjugate pair is selected, then both complex generalized eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex generalized eigenvalue may no longer satisfy selctg(ar(j),ai(j),b(j))=.TRUE. after ordering. info=n+2 in this case.
selctg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which f08xcf is called. Arguments denoted as Input must not be changed by this procedure.
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 first of the pair of matrices, A.
On exit: a has been overwritten by its generalized Schur form S.
7: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08xcf 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 second of the pair of matrices, B.
On exit: b has been overwritten by its generalized Schur form T.
9: ldb Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08xcf is called.
Constraint: ldbmax(1,n).
10: sdim Integer Output
On exit: if sort='N', sdim=0.
If sort='S', sdim= number of eigenvalues (after sorting) for which selctg is .TRUE.. (Complex conjugate pairs for which selctg is .TRUE. for either eigenvalue count as 2.)
11: alphar(n) Real (Kind=nag_wp) array Output
On exit: see the description of beta.
12: alphai(n) Real (Kind=nag_wp) array Output
On exit: see the description of beta.
13: 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. alphar(j)+alphai(j)×i, and beta(j), for j=1,2,,n, are the diagonals of the complex Schur form (S,T) that would result if the 2×2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2×2 complex unitary transformations.
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 α/β. However, alphar and alphai will always be less than and usually comparable with A2 in magnitude, and beta will always be less than and usually comparable with B2.
14: vsl(ldvsl,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vsl must be at least max(1,n) if jobvsl='V', and at least 1 otherwise.
On exit: if jobvsl='V', vsl will contain the left Schur vectors, Q.
If jobvsl='N', vsl is not referenced.
15: ldvsl Integer Input
On entry: the first dimension of the array vsl as declared in the (sub)program from which f08xcf is called.
Constraints:
  • if jobvsl='V', ldvsl max(1,n) ;
  • otherwise ldvsl1.
16: vsr(ldvsr,*) Real (Kind=nag_wp) array Output
Note: the second dimension of the array vsr must be at least max(1,n) if jobvsr='V', and at least 1 otherwise.
On exit: if jobvsr='V', vsr will contain the right Schur vectors, Z.
If jobvsr='N', vsr is not referenced.
17: ldvsr Integer Input
On entry: the first dimension of the array vsr as declared in the (sub)program from which f08xcf is called.
Constraints:
  • if jobvsr='V', ldvsr max(1,n) ;
  • otherwise ldvsr1.
18: 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.
19: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08xcf 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 mimimum; add, say nb×(n×6), where nb is the optimal block size.
Constraints:
if lwork−1,
  • if n=0, lwork1;
  • otherwise lwork max(8×n,6×n+16) .
20: bwork(*) Logical array Workspace
Note: the dimension of the array bwork must be at least 1 if sort='N', and at least max(1,n) otherwise.
If sort='N', bwork is not referenced.
21: 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
After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the generalized Schur form no longer satisfy selctg=.TRUE.. This could also be caused by underflow due to scaling.
info=n+3
The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned).

7 Accuracy

The computed generalized Schur factorization satisfies
A+E = QS ZT ,   B+F = QT ZT ,  
where
(E,F) F = O(ε) (A,B) F  
and ε is the machine precision. See Section 4.11 of Anderson et al. (1999) for further details.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08xcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08xcf 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 f08xqf.

10 Example

This example finds the generalized Schur factorization 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 ) ,  
such that the real positive eigenvalues of (A,B) correspond to the top left diagonal elements of the generalized Schur form, (S,T).

10.1 Program Text

Program Text (f08xcfe.f90)

10.2 Program Data

Program Data (f08xcfe.d)

10.3 Program Results

Program Results (f08xcfe.r)