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sgeev

```
NAME
SGEEV - compute for an N-by-N real nonsymmetric matrix A,
the eigenvalues and, optionally, the left and/or right
eigenvectors

SYNOPSIS
SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL,
VR, LDVR, WORK, LWORK, INFO )

CHARACTER     JOBVL, JOBVR

INTEGER       INFO, LDA, LDVL, LDVR, LWORK, N

REAL          A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ),
WI( * ), WORK( * ), WR( * )

PURPOSE
SGEEV computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvec-
tors.

The left eigenvectors of A are the same as the right eigen-
vectors of A**T.  If u(j) and v(j) are the left and right
eigenvectors, respectively, corresponding to the eigenvalue
lambda(j), then (u(j)**T)*A = lambda(j)*(u(j)**T) and A*v(j)
= lambda(j) * v(j).

The computed eigenvectors are normalized to have Euclidean
norm equal to 1 and largest component real.

ARGUMENTS
JOBVL   (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed.

JOBVR   (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed.

N       (input) INTEGER
The order of the matrix A. N >= 0.

A       (input/output) REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.  On exit, A has been
overwritten.

LDA     (input) INTEGER
The leading dimension of the array A.  LDA >=
max(1,N).

WR      (output) REAL array, dimension (N)

WI      (output) REAL array, dimension (N) WR and WI
contain the real and imaginary parts, respectively,
of the computed eigenvalues.  Complex conjugate
pairs of eigenvalues appear consecutively with the
eigenvalue having the positive imaginary part first.

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 JOBVL =
'N', VL is not referenced.  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)-st 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).

LDVL    (input) INTEGER
The leading dimension of the array VL.  LDVL >= 1;
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 JOBVR =
'N', VR is not referenced.  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)-st eigenvalues form a complex
conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1)
and
v(j+1) = VR(:,j) = i*VR(:,j+1).

LDVR    (input) INTEGER
The leading dimension of the array VR.  LDVR >= 1;
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(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK
>= 4*N.  For good performance, LWORK must generally
be larger.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value.
> 0:  if INFO = i, the QR algorithm failed to com-
pute all the eigenvalues, and no eigenvectors have

been computed; elements i+1:N of WR and WI contain
eigenvalues which have converged.
```