qz.dtrsen {QZ}

Reordered QZ Decomposition for a Real Matrix

Description

This function call 'dtrsend' in Fortran to reorder 'double' matrices (T,Q).

Usage

  qz.dtrsen(T, Q, select, job = c("B", "V", "E", "N"),
            want.Q = TRUE, LWORK = NULL, LIWORK = NULL)

Arguments

T

a 'double' generalized Schur form, dim = c(N, N).

Q

a 'double' Schur vectors, dim = c(N, N).

select

specifies the eigenvalues in the selected cluster.

job

Specifies whether condition numbers are required for the cluster of eigenvalues (S) or the invariant subspace (SEP).

want.Q

if update Q.

LWORK

optional, dimension of array WORK for workspace. (>= N(N+1)/2)

LIWORK

optional, dimension of array IWORK for workspace. (>= N(N+1)/4)

Details

See 'dtrsen.f' for all details.

DTRSEN reorders the real Schur factorization of a real matrix A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix T, and the leading columns of Q form an orthonormal basis of the corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of the cluster of eigenvalues and/or the invariant subspace.

T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign.

Value

Return a list contains next:

'T'

T's reorded generalized Schur form.

'WR'

original returns from 'dtrsen.f'.

'WI'

original returns from 'dtrsen.f'.

'M'

original returns from 'dtrsen.f'.

'S'

original returns from 'dtrsen.f'.

'SEP'

original returns from 'dtrsen.f'.

'WORK'

optimal LWORK (for dtrsen.f only)

'IWORK'

optimal LIWORK (for dtrsen.f only)

'INFO'

= 0: successful. < 0: if INFO = -i, the i-th argument had an illegal value. =1: reordering of T failed.

Extra returns in the list:

'W'

WR + WI * i.

'Q'

the reorded Schur vectors.

Warning(s)

There is no format checking for T and Q which are usually returned by qz.dgees.

There is also no checking for select which is usually according to the returns of qz.dgeev.

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/dtrsen.f

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

See Also

qz.zgees, qz.dgees, qz.ztrsen.

Examples

library(QZ, quiet = TRUE)

### https://www.nag.com/numeric/fl/nagdoc_fl22/xhtml/f08/f08qgf.xml
T <- exA4$T
Q <- exA4$Q
select <- c(TRUE, FALSE, FALSE, TRUE)
ret <- qz.dtrsen(T, Q, select)

# Verify 1
A <- Q %*% T %*% solve(Q)
A.new <- ret$Q %*% ret$T %*% solve(ret$Q)
round(A - A.new)

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