
SGGEVX(l) ) SGGEVX(l)
NAME
SGGEVX  compute for a pair of NbyN real nonsymmetric matrices (A,B)
SYNOPSIS
SUBROUTINE SGGEVX( 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
REAL ABNRM, BBNRM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), LSCALE(
* ), RCONDE( * ), RCONDV( * ), RSCALE( * ), VL( LDVL, * ), VR( LDVR, *
), WORK( * )
PURPOSE
SGGEVX computes for a pair of NbyN real nonsymmetric matrices (A,B) the generalized ei
genvalues, and optionally, the left and/or right generalized eigenvectors.
Optionally also, it computes a balancing transformation to improve the conditioning of the
eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal con
dition 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 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) REAL 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) REAL 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 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) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA (output) REAL array, dimension
(N) On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized
eigenvalues. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive,
then the jth and (j+1)st eigenvalues are a complex conjugate 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 com
puting 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) REAL 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 jth eigenvalue is
real, then u(j) = VL(:,j), the jth column of VL. If the jth and (j+1)th eigen
values 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) REAL 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 jth eigenvalue is
real, then v(j) = VR(:,j), the jth column of VR. If the jth and (j+1)th eigen
values form a complex conjugate pair, then v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) =
VR(:,j)i*VR(:,j+1). Each eigenvector will be 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,...,ILO1 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,...,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) 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,...,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) REAL
The onenorm of the balanced matrix A.
BBNRM (output) REAL
The onenorm of the balanced matrix B.
RCONDE (output) REAL array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of the selected eigenval
ues, stored in consecutive elements of the array. For a complex conjugate pair of
eigenvalues two consecutive elements of RCONDE are set to the same value. Thus
RCONDE(j), RCONDV(j), and the jth columns of VL and VR all correspond to the same
eigenpair (but not in general the jth eigenpair, unless all eigenpairs are
selected). If SENSE = 'V', RCONDE is not referenced.
RCONDV (output) REAL array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition numbers of the selected
eigenvectors, stored in consecutive elements of the array. For a complex eigenvec
tor 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) REAL 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 ith argument had an illegal value.
= 1,...,N: The QZ iteration failed. No eigenvectors 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 iteration failed in SHGEQZ.
=N+2: error return from STGEVC.
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 SGGEVX(l) 
