CGGEVX - compute for a pair of N-by-N complex nonsymmetric
matrices (A,B) the generalized eigenvalues, and option
ally, the left and/or right generalized eigenvectors
SYNOPSIS
SUBROUTINE CGGEVX( 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
REAL ABNRM, BBNRM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL LSCALE( * ), RCONDE( * ), RCONDV( * ),
RSCALE( * ), RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, *
), WORK( * )
PURPOSE
CGGEVX computes for a pair of N-by-N complex nonsymmetric
matrices (A,B) the generalized eigenvalues, and option
ally, the left and/or right generalized eigenvectors.
Optionally, it also computes a balancing transformation to
improve the conditioning of the eigenvalues and eigenvec
tors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), recip
rocal condition numbers for the eigenvalues (RCONDE), and
reciprocal condition numbers for the right eigenvectors
(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 conjugate-transpose of u(j).
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 recipro
cal condition numbers will be for the matrices
after permuting and/or balancing. Permuting does
not change condition numbers (in exact arith
metic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigen
vectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized
eigenvectors;
= 'V': compute the right generalized eigenvec
tors.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are
computed. = 'N': none are computed;
= '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 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" ver
sions of the input A and B.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) COMPLEX 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" ver
sions of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
BETA (output) COMPLEX 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 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 eigenvector 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 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 eigenvector 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,...,ILO-1 or i = IHI+1,...,N.
If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE (output) REAL 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,...,ILO-1 = 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 ILO-1.
RSCALE (output) REAL array, dimension (N)
Details of the permutations and scaling factors
the index of the column interchanged with column
j, and DR(j) is the scaling factor applied to col
umn j, then RSCALE(j) = PR(j) for j = 1,...,ILO-1
= 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 ILO-1.
ABNRM (output) REAL
The one-norm of the balanced matrix A.
BBNRM (output) REAL
The one-norm of the balanced matrix B.
RCONDE (output) REAL array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition
numbers of the selected eigenvalues, stored in
consecutive elements of the array. If SENSE =
'V', RCONDE is not referenced.
RCONDV (output) REAL array, dimension (N)
If JOB = 'V' or 'B', the estimated reciprocal con
dition numbers of the selected eigenvectors,
stored in consecutive elements of the array. If
the eigenvalues cannot be reordered to compute
RCONDV(j), RCONDV(j) is set to 0; this can only
occur when the true value would be very small any
way. If SENSE = 'E', RCONDV is not referenced.
Not referenced if JOB = 'E'.
WORK (workspace/output) COMPLEX 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) REAL 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.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille
gal value.
= 1,...,N: The QZ iteration failed. No eigenvec
tors 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
CHGEQZ.
=N+2: error return from CTGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting
rows and columns to isolate eigenvalues, 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 explanation of balanc
ing, see section 4.11.1.2 of LAPACK Users' Guide.
An approximate error bound on the chordal distance between
the i-th computed generalized eigenvalue w and the corre
sponding exact eigenvalue lambda is
chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) /
RCONDE(I)
An approximate error bound for the angle between the i-th
computed eigenvector VL(i) or VR(i) is given by
EPS * norm(ABNRM, BBNRM) / DIF(i).
For further explanation of the reciprocal condition num
bers RCONDE and RCONDV, see section 4.11 of LAPACK User's
Guide.
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