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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.