aa_qz_zggev: Generalized Eigenvalues Decomposition for Complex Paired...

In snoweye/QZ: Generalized Eigenvalues and QZ Decomposition

Generalized Eigenvalues Decomposition for Complex Paired Matrices

Description

This function call 'zggev' in Fortran to decompose 'complex' matrices (A,B).

Usage

  qz.zggev(A, B, vl = TRUE, vr = TRUE, LWORK = NULL)

Arguments

A	a 'complex' matrix, dim = c(N, N).
B	a 'complex' matrix, dim = c(N, N).
vl	if compute left 'complex' eigen vectors. (U)
vr	if compute right 'complex' eigen vectors. (V)
LWORK	optional, dimension of array WORK for workspace. (>= 2N)

Details

See 'zggev.f' for all details.

ZGGEV computes for a pair of N-by-N complex non-symmetric matrices (A,B), the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

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

The right generalized eigenvector v(j) corresponding to the generalized eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the generalized eigenvalues lambda(j) of (A,B) satisfies

u(j)**H * A = lambda(j) * u(j)**H * B

where u(j)**H is the conjugate-transpose of u(j).

Value

Return a list contains next:

'ALPHA'	original returns from 'zggev.f'.
'BETA'	original returns from 'zggev.f'.
'VL'	original returns from 'zggev.f'.
'VR'	original returns from 'zggev.f'.
'WORK'	optimal LWORK (for zggev.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 ZHGEQZ. =N+2: reordering problem. =N+3: reordering failed.

Extra returns in the list:

'U'	the left eigen vectors.
'V'	the right eigen vectors.

Note that 'VL' and 'VR' are scaled so the largest component has abs(real part) + abs(imag. part) = 1.

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/complex16/zggev.f

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

See Also

qz.zgges

Examples

library(QZ, quiet = TRUE)

### https://www.nag.com/lapack-ex/node122.html
A <- exAB1$A
B <- exAB1$B
ret <- qz.zggev(A, B)

# Verify 1
(lambda <- ret$ALPHA / ret$BETA)    # Unstable
diff.R <- matrix(ret$BETA, 4, 4, byrow = TRUE) * A %*% ret$V -
          matrix(ret$ALPHA, 4, 4, byrow = TRUE) * B %*% ret$V
diff.L <- matrix(ret$BETA, 4, 4) * H(ret$U) %*% A -
          matrix(ret$ALPHA, 4, 4) * H(ret$U) %*% B
round(diff.R)
round(diff.L)

# Verify 2
round(ret$U %*% solve(ret$U))
round(ret$V %*% solve(ret$V))

snoweye/QZ documentation built on Sept. 12, 2023, 4:59 a.m.