bb_qz_dggev: Generalized Eigenvalues Decomposition for Real Paired...

In QZ: Generalized Eigenvalues and QZ Decomposition

Generalized Eigenvalues Decomposition for Real Paired Matrices

Description

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

Usage

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

Arguments

A	a 'real' matrix, dim = c(N, N).
B	a 'real' matrix, dim = c(N, N).
vl	if compute left 'real' eigen vector. (U)
vr	if compute right 'real' eigen vector. (V)
LWORK	optional, dimension of array WORK for workspace. (>= 8N)

Details

See 'dggev.f' for all details.

DGGEV computes for a pair of N-by-N real 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 eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

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

The left eigenvector u(j) corresponding to the eigenvalue 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:

'ALPHAR'	original returns from 'dggev.f'.
'ALPHAI'	original returns from 'dggev.f'.
'BETA'	original returns from 'dggev.f'.
'VL'	original returns from 'dggev.f'.
'VR'	original returns from 'dggev.f'.
'WORK'	optimal LWORK (for dggev.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.
'U'	the left eigen vectors.
'V'	the right eigen vectors.

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.

If the j-th eigenvalue is real, then U[, j] = VL[, j], the j-th column of VL. If the j-th and (j+1)-th eigenvalues form a complex conjugate pair, then U[, j] = VL[, j] + i * VL[, j+1] and U[, j+1] = VL[, j] - i * VL[, j+1]. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1.

Similarly, for the right eigenvectors of V and VR.

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/dggev.f

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

See Also

qz.dgges

Examples

library(QZ, quiet = TRUE)

### https://www.nag.com/lapack-ex/node117.html 
A <- exAB2$A
B <- exAB2$B
ret <- qz.dggev(A, B)

# Verify
(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))

QZ documentation built on Sept. 8, 2023, 5:43 p.m.