
SGEGV(l) ) SGEGV(l)
NAME
SGEGV  routine is deprecated and has been replaced by routine SGGEV
SYNOPSIS
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, VL, LDVL, VR,
LDVR, WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ), BETA( * ), VL( LDVL,
* ), VR( LDVR, * ), WORK( * )
PURPOSE
This routine is deprecated and has been replaced by routine SGGEV. SGEGV computes for a
pair of nbyn real nonsymmetric matrices A and B, the generalized eigenvalues (alphar +/
alphai*i, beta), and optionally, the left and/or right generalized eigenvectors (VL and
VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly speaking, a scalar w or
a ratio alpha/beta = w, such that A  w*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. A good beginning reference is the book, "Matrix Computations", by G. Golub &
C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized eigenvalue w for a pair
of matrices (A,B) is a vector r such that (A  w B) r = 0 . A left generalized eigen
vector is a vector l such that l**H * (A  w B) = 0, where l**H is the
conjugatetranspose of l.
Note: this routine performs "full balancing" on A and B  see "Further Details", below.
ARGUMENTS
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.
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 first of the pair of matrices whose generalized eigenvalues and
(optionally) generalized eigenvectors are to be computed. On exit, the contents
will have been destroyed. (For a description of the contents of A on exit, see
"Further Details", below.)
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices whose generalized eigenvalues and
(optionally) generalized eigenvectors are to be computed. On exit, the contents
will have been destroyed. (For a description of the contents of B on exit, see
"Further Details", below.)
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 generalized eigenvectors. (See "Purpose", above.) Real
eigenvectors take one column, complex take two columns, the first for the real
part and the second for the imaginary part. Complex eigenvectors correspond to an
eigenvalue with positive imaginary part. Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1, *except* that
for eigenvalues with alpha=beta=0, a zero vector will be returned as the corre
sponding eigenvector. 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 generalized eigenvectors. (See "Purpose", above.) Real
eigenvectors take one column, complex take two columns, the first for the real
part and the second for the imaginary part. Complex eigenvectors correspond to an
eigenvalue with positive imaginary part. Each eigenvector will be scaled so the
largest component will have abs(real part) + abs(imag. part) = 1, *except* that
for eigenvalues with alpha=beta=0, a zero vector will be returned as the corre
sponding eigenvector. Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.
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(1,8*N). For good performance,
LWORK must generally be larger. To compute the optimal value of LWORK, call
ILAENV to get blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute: NB 
MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LWORK is: 2*N +
MAX( 6*N, N*(NB+1) ).
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.
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:
errors that usually indicate LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed iteration) =N+7: error return
from STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)
FURTHER DETAILS
Balancing

This driver calls SGGBAL to both permute and scale rows and columns of A and B. The per
mutations PL and PR are chosen so that PL*A*PR and PL*B*R will be upper triangular except
for the diagonal blocks A(i:j,i:j) and B(i:j,i:j), with i and j as close together as pos
sible. The diagonal scaling matrices DL and DR are chosen so that the pair
DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the elements that
start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been computed, SGGBAK
transforms the eigenvectors back to what they would have been (in perfect arithmetic) if
they had not been balanced.
Contents of A and B on Exit
      
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both), then on exit the
arrays A and B will contain the real Schur form[*] of the "balanced" versions of A and B.
If no eigenvectors are computed, then only the diagonal blocks will be correct.
[*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
LAPACK version 3.0 15 June 2000 SGEGV(l) 
