qz.dgges {QZ}

QZ Decomposition for Real Paired Matrices

Description

This function call 'dgges' in Fortran to decompose 'real' matrices (A,B).

Usage

  qz.dgges(A, B, vsl = TRUE, vsr = TRUE, LWORK = NULL)

Arguments

A

a 'real' matrix, dim = c(N, N).

B

a 'real' matrix, dim = c(N, N).

vsl

if compute left 'real' Schur vectors. (Q)

vsr

if compute right 'real' Schur vectors. (Z)

LWORK

optional, dimension of array WORK for workspace. (>= 8N+16)

Details

See 'dgges.f' for all details.

DGGES computes for a pair of N-by-N real non-symmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T.The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver DGGEV instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be "standardized" by making the corresponding elements of T have the form:

[ a 0 ]
[ 0 b ]

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

Value

Return a list contains next:

'S'

A's generalized Schur form.

'T'

B's generalized Schur form.

'ALPHAR'

original returns from 'dgges.f'.

'ALPHAI'

original returns from 'dgges.f'.

'BETA'

original returns from 'dgges.f'.

'VSL'

original returns from 'dgges.f'.

'VSR'

original returns from 'dgges.f'.

'WORK'

optimal LWORK (for dgges.f only)

'INFO'

= 0: successful. < 0: if INFO = -i, the i-th argument had an illegal value. =1,...,N: QZ iteration failed. =N+1: other than QZ iteration failed in DHGEQZ. =N+2: reordering problem. =N+3: reordering failed.

Extra returns in the list:

'ALPHA'

ALPHAR + ALPHAI * i.

'Q'

the left Schur vectors.

'Z'

the right Schur vectors.

The ALPHA[j]/BETA[j] are generalized eigenvalues.

If ALPHAI[j] is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI[j+1] negative.

Author(s)

Wei-Chen Chen [e-mail]

References

Anderson, E., et al. (1999) LAPACK User's Guide, 3rd edition, SIAM, Philadelphia.

https://www.netlib.org/lapack/double/dgges.f

https://en.wikipedia.org/wiki/Schur_decomposition

See Also

qz.dggev

Examples

library(QZ, quiet = TRUE)

### https://www.nag.com/lapack-ex/node119.html 
A <- exAB2$A
B <- exAB2$B
ret <- qz.dgges(A, B)

# Verify 1
A.new <- ret$Q %*% ret$S %*% t(ret$Z)
B.new <- ret$Q %*% ret$T %*% t(ret$Z)
round(A - A.new)
round(B - B.new)

# verify 2
round(ret$Q %*% t(ret$Q))
round(ret$Z %*% t(ret$Z))