DGGEVX - compute for a pair of N-by-N real nonsymmetric
matrices (A,B)
SYNOPSIS
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA,
B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL,
VR, LDVR, ILO, IHI, LSCALE, RSCALE,
ABNRM, BBNRM, RCONDE, RCONDV, WORK,
LWORK, 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 A( LDA, * ), ALPHAI( * ),
ALPHAR( * ), B( LDB, * ), BETA( * ),
LSCALE( * ), RCONDE( * ), RCONDV( * ),
RSCALE( * ), VL( LDVL, * ), VR( LDVR, *
), WORK( * )
PURPOSE
DGGEVX computes for a pair of N-by-N real nonsymmetric
matrices (A,B) the generalized eigenvalues, and option
ally, the left and/or right generalized eigenvectors.
Optionally also, it 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
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 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) DOUBLE PRECISION 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 real 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) DOUBLE PRECISION array, dimension
(LDB, N)
On entry, the matrix B in the pair (A,B). On
JOBVR='V' or both, then B contains the second part
of the real Schur form of the "balanced" versions
of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension
(N) BETA (output) DOUBLE PRECISION array,
dimension (N) On exit, (ALPHAR(j) +
ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen
eralized eigenvalues. 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 conju
gate pair, with ALPHAI(j+1) negative.
Note: the quotients ALPHAR(j)/BETA(j) and
ALPHAI(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, ALPHAR and ALPHAI 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) DOUBLE PRECISION array, dimension
(LDVL,N)
If JOBVL = 'V', the left eigenvectors u(j) are
stored one after another in the columns of VL, in
the same order as their eigenvalues. 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 will be
scaled so the largest component 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) DOUBLE PRECISION array, dimension
(LDVR,N)
If JOBVR = 'V', the right eigenvectors v(j) are
stored one after another in the columns of VR, in
the same order as their eigenvalues. If the j-th
eigenvalue is real, then v(j) = VR(:,j), the j-th
column of VR. If the j-th and (j+1)-th eigenvalues
form a complex conjugate pair, then v(j) =
VR(:,j)+i*VR(:,j+1) and v(j+1) =
scaled so the largest component 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) 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,...,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) 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 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) DOUBLE PRECISION
The one-norm of the balanced matrix A.
BBNRM (output) DOUBLE PRECISION
The one-norm of the balanced matrix B.
RCONDE (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition
numbers of the selected eigenvalues, stored in
consecutive elements of the array. For a complex
conjugate pair of eigenvalues two consecutive ele
ments of RCONDE are set to the same value. Thus
RCONDE(j), RCONDV(j), and the j-th columns of VL
and VR all correspond to the same eigenpair (but
not in general the j-th eigenpair, unless all
eigenpairs are selected). If SENSE = 'V', RCONDE
is not referenced.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal
stored in consecutive elements of the array. For a
complex eigenvector two consecutive elements of
RCONDV are set to the same value. If the eigenval
ues 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.
WORK (workspace/output) DOUBLE PRECISION 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(1,6*N). If SENSE = 'E', LWORK >= 12*N. If
SENSE = 'V' or 'B', LWORK >= 2*N*N+12*N+16.
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.
IWORK (workspace) INTEGER array, dimension (N+6)
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 ille
gal value.
= 1,...,N: The QZ iteration failed. No eigenvec
tors have been calculated, but ALPHAR(j),
ALPHAI(j), and BETA(j) should be correct for
j=INFO+1,...,N. > N: =N+1: other than QZ itera
tion failed in DHGEQZ.
=N+2: error return from DTGEVC.
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
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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