CGGEVX(l) ) CGGEVX(l)NAME
CGGEVX - compute for a pair of N-by-N complex nonsymmetric matrices
(A,B) the generalized eigenvalues, and optionally, 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 optionally, the left and/or
right generalized eigenvectors. Optionally, it also computes a balanc‐
ing transformation to improve the conditioning of the eigenvalues and
eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal
condition numbers for the eigenvalues (RCONDE), and reciprocal condi‐
tion 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 singu‐
lar. 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).
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 balanc‐
ing. 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 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"
versions of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHA (output) COMPLEX array, dimension (N)
BETA (output) COMPLEX array, dimension (N) On exit,
ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
ues.
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 inter‐
changed 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 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,...,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 condition 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 anyway. 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th 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
CHGEQZ.
=N+2: error return from CTGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
umns 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 balancing, 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 corresponding exact eigenval‐
ue 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 numbers RCONDE and
RCONDV, see section 4.11 of LAPACK User's Guide.
LAPACK version 3.0 15 June 2000 CGGEVX(l)
