
ZGGEVX(l) ) ZGGEVX(l)
NAME
ZGGEVX  compute for a pair of NbyN complex nonsymmetric matrices (A,B) the generalized
eigenvalues, and optionally, the left and/or right generalized eigenvectors
SYNOPSIS
SUBROUTINE ZGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL,
VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK,
LWORK, RWORK, IWORK, BWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM, BBNRM
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ), RSCALE( * ), RWORK( *
)
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL( LDVL, * ), VR(
LDVR, * ), WORK( * )
PURPOSE
ZGGEVX computes for a pair of NbyN complex nonsymmetric matrices (A,B) the generalized
eigenvalues, and optionally, the left and/or right generalized eigenvectors. Optionally,
it also computes a balancing transformation to improve the conditioning of the eigenvalues
and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition num
bers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigen
vectors (RCONDV).
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 conjugatetranspose of u(j).
ARGUMENTS
BALANC (input) CHARACTER*1
Specifies the balance option to be performed:
= 'N': do not diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed reciprocal condition numbers will be for
the matrices after permuting and/or balancing. Permuting does not change condition
numbers (in exact arithmetic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are computed. = 'N': none are com
puted;
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If
JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex
Schur form of the "balanced" versions of the input A and B.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX*16 array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If
JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex
Schur form of the "balanced" versions of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHA (output) COMPLEX*16 array, dimension (N)
BETA (output) COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j),
j=1,...,N, will be the generalized eigenvalues.
Note: the quotient ALPHA(j)/BETA(j) ) may easily over or underflow, and BETA(j)
may even be zero. Thus, the user should avoid naively computing the ratio
ALPHA/BETA. However, ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and usually comparable with
norm(B).
VL (output) COMPLEX*16 array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after
another in the columns of VL, in the same order as their eigenvalues. Each eigen
vector will be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1. Not referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.
VR (output) COMPLEX*16 array, dimension (LDVR,N)
If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after
another in the columns of VR, in the same order as their eigenvalues. Each eigen
vector will be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1. Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
ILO,IHI (output) INTEGER ILO and IHI are integer values such that on exit A(i,j) =
0 and B(i,j) = 0 if i > j and j = 1,...,ILO1 or i = IHI+1,...,N. If BALANC = 'N'
or 'S', ILO = 1 and IHI = N.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to the left side of A and
B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the
scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO1 =
DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the
interchanges are made is N to IHI+1, then 1 to ILO1.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to the right side of A and
B. If PR(j) is the index of the column interchanged with column j, and DR(j) is
the scaling factor applied to column j, then RSCALE(j) = PR(j) for j =
1,...,ILO1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in
which the interchanges are made is N to IHI+1, then 1 to ILO1.
ABNRM (output) DOUBLE PRECISION
The onenorm of the balanced matrix A.
BBNRM (output) DOUBLE PRECISION
The onenorm of the balanced matrix B.
RCONDE (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of the selected eigenval
ues, stored in consecutive elements of the array. If SENSE = 'V', RCONDE is not
referenced.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected
eigenvectors, stored in consecutive elements of the array. If the eigenvalues can
not be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur
when the true value would be very small anyway. If SENSE = 'E', RCONDV is not
referenced. Not referenced if JOB = 'E'.
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,2*N). If SENSE = 'N' or 'E',
LWORK >= 2*N. If SENSE = 'V' or 'B', LWORK >= 2*N*N+2*N.
If LWORK = 1, then 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 by XERBLA.
RWORK (workspace) DOUBLE PRECISION array, dimension (6*N)
Real workspace.
IWORK (workspace) INTEGER array, dimension (N+2)
If SENSE = 'E', IWORK is not referenced.
BWORK (workspace) LOGICAL array, dimension (N)
If SENSE = 'N', BWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value.
= 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but
ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than
QZ iteration failed in ZHGEQZ.
=N+2: error return from ZTGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate ei
genvalues, second, applying diagonal similarity transformation to the rows and columns to
make the rows and columns as close in norm as possible. The computed reciprocal condition
numbers correspond to the balanced matrix. Permuting rows and columns will not change the
condition numbers (in exact arithmetic) but diagonal scaling will. For further explana
tion of balancing, see section 4.11.1.2 of LAPACK Users' Guide.
An approximate error bound on the chordal distance between the ith computed generalized
eigenvalue w and the corresponding exact eigenvalue lambda is
chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
An approximate error bound for the angle between the ith computed eigenvector VL(i) or
VR(i) is given by
EPS * norm(ABNRM, BBNRM) / DIF(i).
For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section
4.11 of LAPACK User's Guide.
LAPACK version 3.0 15 June 2000 ZGGEVX(l) 
