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dgeevx.f(3)				      LAPACK				      dgeevx.f(3)

NAME
       dgeevx.f -

SYNOPSIS
   Functions/Subroutines
       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)
	    DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
	   for GE matrices

Function/Subroutine Documentation
   subroutine dgeevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN,
       double precision, dimension( lda, * )A, integerLDA, double precision, dimension( * )WR,
       double precision, dimension( * )WI, double precision, dimension( ldvl, * )VL, integerLDVL,
       double precision, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, double
       precision, dimension( * )SCALE, double precisionABNRM, double precision, dimension( *
       )RCONDE, double precision, dimension( * )RCONDV, double precision, dimension( * )WORK,
       integerLWORK, integer, dimension( * )IWORK, integerINFO)
	DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       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.

       Parameters:
	   BALANC

		     BALANC is 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

		     JOBVL is 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

		     JOBVR is 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

		     SENSE is 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

		     N is INTEGER
		     The order of the matrix A. N >= 0.

	   A

		     A is 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

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

	   WR

		     WR is DOUBLE PRECISION array, dimension (N)

	   WI

		     WI is 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

		     VL is 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

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

	   VR

		     VR is 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

		     LDVR is INTEGER
		     The leading dimension of the array VR.  LDVR >= 1, and if
		     JOBVR = 'V', LDVR >= N.

	   ILO

		     ILO is INTEGER

	   IHI

		     IHI is 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

		     SCALE is 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

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

	   RCONDE

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

	   RCONDV

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

	   WORK

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

	   LWORK

		     LWORK is 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

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

	   INFO

		     INFO is 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.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   September 2012

       Definition at line 302 of file dgeevx.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dgeevx.f(3)
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