CXML
        	    
LAPACK version 3.0
dgeevx(3)

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

SYNTAX
  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( * )

DESCRIPTION
  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.

	  If LWORK = -1, then a workspace query is assumed; the routine only
	  calculates the optimal size of the WORK array, returns this value
	  as the first entry of the WORK array, and no error message related
	  to LWORK is issued by XERBLA.

  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.