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# sgegv

NAME
SGEGV - a pair of N-by-N real nonsymmetric matrices A, B

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
For a pair of N-by-N real nonsymmetric matrices A, B:

compute the generalized eigenvalues (alphar +/- alphai*i,
beta)
compute the left and/or right generalized eigenvectors
(VL and VR)

The second action is optional -- see the description of
JOBVL and JOBVR below.

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 interpre-
tation 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 general-
ized 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
H
l  such that  (A - w B) l = 0 .

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 eigen-
vectors;
= 'V':  compute the right generalized eigenvectors.

N       (input) INTEGER
The number of rows and columns in the matrices A, B,
VL, and VR.  N >= 0.

A       (input/workspace) 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 con-
tents 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/workspace) 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 con-
tents 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 positive, then the j-th and (j+1)-st eigenvalues
are a complex conjugate pair, with ALPHAI(j+1) nega-
tive.

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.  How-
ever, 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 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 JOBVL = '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 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) ).

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 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 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 entries close to one
(except for the entries that start out zero.)

After the eigenvalues and eigenvectors of the balanced
matrices have been computed, SGGBAK transforms the eigenvec-
tors back to what they would have been (in perfect arith-
metic) 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.