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CGEEVX(l)			       )			     CGEEVX(l)

NAME
       CGEEVX  -  compute for an N-by-N complex nonsymmetric matrix A, the ei‐
       genvalues and, optionally, the left and/or right eigenvectors

SYNOPSIS
       SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL,
			  VR,  LDVR,  ILO,  IHI, SCALE, ABNRM, RCONDE, RCONDV,
			  WORK, LWORK, RWORK, INFO )

	   CHARACTER	  BALANC, JOBVL, JOBVR, SENSE

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

	   REAL		  ABNRM

	   REAL		  RCONDE( * ), RCONDV( * ), RWORK( * ), SCALE( * )

	   COMPLEX	  A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), W(	 *  ),
			  WORK( * )

PURPOSE
       CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigen‐
       values and, optionally, the left and/or right eigenvectors.  Optionally
       also,  it computes a balancing transformation to improve the condition‐
       ing of the eigenvalues and eigenvectors (ILO, IHI, SCALE,  and  ABNRM),
       reciprocal condition numbers for the eigenvalues (RCONDE), and recipro‐
       cal 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 transforma‐
       tion  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,  ie. 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) COMPLEX array, dimension (LDA,N)
	       On entry, the N-by-N matrix A.  On exit, A has  been  overwrit‐
	       ten.   If JOBVL = 'V' or JOBVR = 'V', A contains the Schur form
	       of the balanced version of the matrix A.

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

       W       (output) COMPLEX array, dimension (N)
	       W contains the computed eigenvalues.

       VL      (output) COMPLEX 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 eigen‐
	       values.	If JOBVL = 'N', VL is not referenced.  u(j) = VL(:,j),
	       the j-th column of VL.

       LDVL    (input) INTEGER
	       The  leading  dimension of the array VL.	 LDVL >= 1; if JOBVL =
	       'V', LDVL >= N.

       VR      (output) COMPLEX 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.   v(j)  =
	       VR(:,j), the j-th column of VR.

       LDVR    (input) INTEGER
	       The  leading  dimension of the array VR.	 LDVR >= 1; if JOBVR =
	       'V', LDVR >= N.

	       ILO,IHI (output) INTEGER ILO and IHI are integer values	deter‐
	       mined  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 inter‐
	       changed 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) REAL
	       The one-norm of the balanced matrix (the maximum of the sum  of
	       absolute values of elements of any column).

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

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

       WORK    (workspace/output) COMPLEX 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 SENSE = 'V' or  'B',  LWORK  >=  N*N+2*N.
	       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.

       RWORK   (workspace) REAL array, dimension (2*N)

       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 W  contain  eigenvalues
	       which have converged.

LAPACK version 3.0		 15 June 2000			     CGEEVX(l)
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