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sgeevx


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

 SYNOPSIS
      SUBROUTINE SGEEVX( 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

          REAL           ABNRM

          INTEGER        IWORK( * )

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

 PURPOSE
      SGEEVX 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 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.

      Balancing a matrix means permuting the rows and columns to
      make it more nearly upper triangular, and applying a diago-
      nal 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 eigenvec-
      tors 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 explana-
      tion 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. Permut-
              ing 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 eigenvec-
              tors.

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

      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.  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) 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 will 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,
              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) REAL 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 inter-
              changes are made is N to IHI+1, then 1 to ILO-1.

      ABNRM   (output) REAL
              The one-norm of the balanced matrix (the maximum of
              the sum of absolute values of entries of any
              column).

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

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

      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.   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 com-
              pute 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.