DGEEVX (3lapack)

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

SYNOPSIS

  SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL,
                     LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV,
                     WORK, LWORK, IWORK, INFO )

      CHARACTER      BALANC, JOBVL, JOBVR, SENSE

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

      DOUBLE         PRECISION ABNRM

      INTEGER        IWORK( * )

      DOUBLE         PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE(
                     * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ),
                     WR( * )

PURPOSE

  DGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues
  and, optionally, the left and/or right eigenvectors.

  Optionally also, it computes a balancing transformation to improve the
  conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and
  ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and
  reciprocal condition numbers for the right
  eigenvectors (RCONDV).

  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.

  Balancing a matrix means permuting the rows and columns to make it more
  nearly upper triangular, and applying a diagonal similarity transformation
  D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns
  closer in norm and the condition numbers of its eigenvalues and
  eigenvectors smaller.  The computed reciprocal condition numbers correspond
  to the balanced matrix.  Permuting rows and columns will not change the
  condition numbers (in exact arithmetic) but diagonal scaling will.  For
  further explanation of balancing, see section 4.10.2 of the LAPACK Users'
  Guide.

ARGUMENTS

  BALANC  (input) CHARACTER*1
          Indicates how the input matrix should be diagonally scaled and/or
          permuted to improve the conditioning of its eigenvalues.  = 'N': Do
          not diagonally scale or permute;
          = 'P': Perform permutations to make the matrix more nearly upper
          triangular. Do not diagonally scale; = 'S': Diagonally scale the
          matrix, i.e. replace A by D*A*D**(-1), where D is a diagonal matrix
          chosen to make the rows and columns of A more equal in norm. Do not
          permute; = 'B': Both diagonally scale and permute A.

          Computed reciprocal condition numbers will be for the matrix after
          balancing and/or permuting. Permuting does not change condition
          numbers (in exact arithmetic), but balancing does.

  JOBVL   (input) CHARACTER*1
          = 'N': left eigenvectors of A are not computed;
          = 'V': left eigenvectors of A are computed.  If SENSE = 'E' or 'B',
          JOBVL must = 'V'.

  JOBVR   (input) CHARACTER*1
          = 'N': right eigenvectors of A are not computed;
          = 'V': right eigenvectors of A are computed.  If SENSE = 'E' or
          'B', JOBVR must = 'V'.

  SENSE   (input) CHARACTER*1
          Determines which reciprocal condition numbers are computed.  = 'N':
          None are computed;
          = 'E': Computed for eigenvalues only;
          = 'V': Computed for right eigenvectors only;
          = 'B': Computed for eigenvalues and right eigenvectors.

          If SENSE = 'E' or 'B', both left and right eigenvectors must also
          be computed (JOBVL = 'V' and JOBVR = 'V').

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

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the N-by-N matrix A.  On exit, A has been overwritten.
          If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur form of
          the balanced version of the input matrix A.

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

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

  VL      (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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, and if JOBVR =
          'V', LDVR >= N.

          ILO,IHI (output) INTEGER ILO and IHI are integer values determined
          when A was balanced.  The balanced A(i,j) = 0 if I > J and J =
          1,...,ILO-1 or I = IHI+1,...,N.

  SCALE   (output) DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied when
          balancing A.  If P(j) is the index of the row and column
          interchanged with row and column j, and D(j) is the scaling factor
          applied to row and column j, then SCALE(J) = P(J),    for J =
          1,...,ILO-1 = D(J),    for J = ILO,...,IHI = P(J)     for J =
          IHI+1,...,N.  The order in which the interchanges are made is N to
          IHI+1, then 1 to ILO-1.

  ABNRM   (output) DOUBLE PRECISION
          The one-norm of the balanced matrix (the maximum of the sum of
          absolute values of elements of any column).

  RCONDE  (output) DOUBLE PRECISION array, dimension (N)
          RCONDE(j) is the reciprocal condition number of the j-th
          eigenvalue.

  RCONDV  (output) DOUBLE PRECISION array, dimension (N)
          RCONDV(j) is the reciprocal condition number of the j-th right
          eigenvector.

  WORK    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.   If SENSE = 'N' or 'E', LWORK >=
          max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.  If
          SENSE = 'V' or 'B', LWORK >= N*(N+6).  For good performance, LWORK
          must generally be larger.

  IWORK   (workspace) INTEGER array, dimension (2*N-2)
          If SENSE = 'N' or 'E', not referenced.

  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 or condition numbers have been
          computed; elements 1:ILO-1 and i+1:N of WR and WI contain
          eigenvalues which have converged.