SGEEV (3lapack)

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 eigenvectors.

  The right eigenvector v(j) of A satisfies
                   A * v(j) = lambda(j) * v(j)
  where lambda(j) is its eigenvalue.
  The left eigenvector u(j) of A satisfies
                u(j)**H * A = lambda(j) * u(j)**H
  where u(j)**H denotes the conjugate transpose of u(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 compute all the
          eigenvalues, and no eigenvectors have been computed; elements i+1:N
          of WR and WI contain eigenvalues which have converged.