SGEGV - routine is deprecated and has been replaced by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This routine is deprecated and has been replaced by rou
tine SGGEV. SGEGV computes for a pair of n-by-n real non
symmetric 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 inter
pretation for beta=0, and even for both being zero. A
good beginning reference is the book, "Matrix Computa
tions", by G. Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a gener
alized eigenvalue w for a pair of matrices (A,B) is a
vector r such that (A - w B) r = 0 . A left general
ized eigenvector is a vector l such that l**H * (A - w B)
= 0, where l**H is the
conjugate-transpose 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 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.
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) general
ized 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) general
ized 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 j-th eigenvalue is real; if posi
tive, then the j-th 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 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) 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 compo
nent will have abs(real part) + abs(imag. part) =
1, *except* that for eigenvalues with
the corresponding 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 eigenvec
tors. (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 corresponding 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 gen
erally 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 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: errors that usually indi
cate LAPACK problems:
=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 permutations 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 possible. 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 eigen
vectors 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 Computa
tions",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
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