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RedHat 9 (Linux i386) - man page for zgeevx (redhat section l)

ZGEEVX(l)					)					ZGEEVX(l)

NAME
       ZGEEVX - compute for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, option-
       ally, the left and/or right eigenvectors

SYNOPSIS
       SUBROUTINE ZGEEVX( 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

	   DOUBLE	  PRECISION ABNRM

	   DOUBLE	  PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ), SCALE( * )

	   COMPLEX*16	  A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ), WORK( * )

PURPOSE
       ZGEEVX  computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, option-
       ally, the left and/or right eigenvectors.  Optionally also, it computes a balancing trans-
       formation  to  improve  the  conditioning  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  tri-
       angular,  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 cor-
       respond 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 bal-
       ancing, 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  arith-
	       metic), 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 com-
	       puted;
	       = '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*16 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 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*16 array, dimension (N)
	       W contains the computed eigenvalues.

       VL      (output) COMPLEX*16 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.  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*16 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 determined when A was bal-
	       anced.  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) COMPLEX*16 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 gener-
	       ally 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) DOUBLE PRECISION 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 				ZGEEVX(l)


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