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dggevx.f(3)				      LAPACK				      dggevx.f(3)

NAME
       dggevx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
	   VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK,
	   LWORK, IWORK, BWORK, INFO)
	    DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors
	   for GE matrices

Function/Subroutine Documentation
   subroutine dggevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN,
       double precision, dimension( lda, * )A, integerLDA, double precision, dimension( ldb, *
       )B, integerLDB, double precision, dimension( * )ALPHAR, double precision, dimension( *
       )ALPHAI, double precision, dimension( * )BETA, double precision, dimension( ldvl, * )VL,
       integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, integerILO,
       integerIHI, double precision, dimension( * )LSCALE, double precision, dimension( *
       )RSCALE, double precisionABNRM, double precisionBBNRM, double precision, dimension( *
       )RCONDE, double precision, dimension( * )RCONDV, double precision, dimension( * )WORK,
       integerLWORK, integer, dimension( * )IWORK, logical, dimension( * )BWORK, integerINFO)
	DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for
       GE matrices

       Purpose:

	    DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
	    the generalized eigenvalues, and optionally, the left and/or right
	    generalized eigenvectors.

	    Optionally also, it computes a balancing transformation to improve
	    the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
	    LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
	    the eigenvalues (RCONDE), and reciprocal condition numbers for the
	    right eigenvectors (RCONDV).

	    A generalized eigenvalue for a pair of matrices (A,B) is a scalar
	    lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
	    singular. It is usually represented as the pair (alpha,beta), as
	    there is a reasonable interpretation for beta=0, and even for both
	    being zero.

	    The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
	    of (A,B) satisfies

			     A * v(j) = lambda(j) * B * v(j) .

	    The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
	    of (A,B) satisfies

			     u(j)**H * A  = lambda(j) * u(j)**H * B.

	    where u(j)**H is the conjugate-transpose of u(j).

       Parameters:
	   BALANC

		     BALANC is CHARACTER*1
		     Specifies the balance option to be performed.
		     = 'N':  do not diagonally scale or permute;
		     = 'P':  permute only;
		     = 'S':  scale only;
		     = 'B':  both permute and scale.
		     Computed reciprocal condition numbers will be for the
		     matrices after permuting and/or balancing. Permuting does
		     not change condition numbers (in exact arithmetic), but
		     balancing does.

	   JOBVL

		     JOBVL is CHARACTER*1
		     = 'N':  do not compute the left generalized eigenvectors;
		     = 'V':  compute the left generalized eigenvectors.

	   JOBVR

		     JOBVR is CHARACTER*1
		     = 'N':  do not compute the right generalized eigenvectors;
		     = 'V':  compute the right generalized eigenvectors.

	   SENSE

		     SENSE is CHARACTER*1
		     Determines which reciprocal condition numbers are computed.
		     = 'N': none are computed;
		     = 'E': computed for eigenvalues only;
		     = 'V': computed for eigenvectors only;
		     = 'B': computed for eigenvalues and eigenvectors.

	   N

		     N is INTEGER
		     The order of the matrices A, B, VL, and VR.  N >= 0.

	   A

		     A is DOUBLE PRECISION array, dimension (LDA, N)
		     On entry, the matrix A in the pair (A,B).
		     On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
		     or both, then A contains the first part of the real Schur
		     form of the "balanced" versions of the input A and B.

	   LDA

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

	   B

		     B is DOUBLE PRECISION array, dimension (LDB, N)
		     On entry, the matrix B in the pair (A,B).
		     On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
		     or both, then B contains the second part of the real Schur
		     form of the "balanced" versions of the input A and B.

	   LDB

		     LDB is INTEGER
		     The leading dimension of B.  LDB >= max(1,N).

	   ALPHAR

		     ALPHAR is DOUBLE PRECISION array, dimension (N)

	   ALPHAI

		     ALPHAI is DOUBLE PRECISION array, dimension (N)

	   BETA

		     BETA is DOUBLE PRECISION array, dimension (N)
		     On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
		     be the generalized eigenvalues.  If ALPHAI(j) is zero, then
		     the j-th eigenvalue is real; if positive, then the j-th and
		     (j+1)-st eigenvalues are a complex conjugate pair, with
		     ALPHAI(j+1) negative.

		     Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
		     may easily over- or underflow, and BETA(j) may even be zero.
		     Thus, the user should avoid naively computing the ratio
		     ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
		     than and usually comparable with norm(A) in magnitude, and
		     BETA always less than and usually comparable with norm(B).

	   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 the j-th eigenvalue is real, then
		     u(j) = VL(:,j), the j-th column of VL. If the j-th and
		     (j+1)-th 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).
		     Each eigenvector will be scaled so the largest component have
		     abs(real part) + abs(imag. part) = 1.
		     Not referenced if JOBVL = 'N'.

	   LDVL

		     LDVL is INTEGER
		     The leading dimension of the matrix VL. LDVL >= 1, and
		     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 the j-th eigenvalue is real, then
		     v(j) = VR(:,j), the j-th column of VR. If the j-th and
		     (j+1)-th 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).
		     Each eigenvector will be scaled so the largest component have
		     abs(real part) + abs(imag. part) = 1.
		     Not referenced if JOBVR = 'N'.

	   LDVR

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

	   ILO

		     ILO is INTEGER

	   IHI

		     IHI is INTEGER
		     ILO and IHI are integer values such that on exit
		     A(i,j) = 0 and B(i,j) = 0 if i > j and
		     j = 1,...,ILO-1 or i = IHI+1,...,N.
		     If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

	   LSCALE

		     LSCALE is DOUBLE PRECISION array, dimension (N)
		     Details of the permutations and scaling factors applied
		     to the left side of A and B.  If PL(j) is the index of the
		     row interchanged with row j, and DL(j) is the scaling
		     factor applied to row j, then
		       LSCALE(j) = PL(j)  for j = 1,...,ILO-1
				 = DL(j)  for j = ILO,...,IHI
				 = PL(j)  for j = IHI+1,...,N.
		     The order in which the interchanges are made is N to IHI+1,
		     then 1 to ILO-1.

	   RSCALE

		     RSCALE is DOUBLE PRECISION array, dimension (N)
		     Details of the permutations and scaling factors applied
		     to the right side of A and B.  If PR(j) is the index of the
		     column interchanged with column j, and DR(j) is the scaling
		     factor applied to column j, then
		       RSCALE(j) = PR(j)  for j = 1,...,ILO-1
				 = DR(j)  for j = ILO,...,IHI
				 = PR(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 A.

	   BBNRM

		     BBNRM is DOUBLE PRECISION
		     The one-norm of the balanced matrix B.

	   RCONDE

		     RCONDE is DOUBLE PRECISION array, dimension (N)
		     If SENSE = 'E' or 'B', the reciprocal condition numbers of
		     the eigenvalues, stored in consecutive elements of the array.
		     For a complex conjugate pair of eigenvalues two consecutive
		     elements of RCONDE are set to the same value. Thus RCONDE(j),
		     RCONDV(j), and the j-th columns of VL and VR all correspond
		     to the j-th eigenpair.
		     If SENSE = 'N or 'V', RCONDE is not referenced.

	   RCONDV

		     RCONDV is DOUBLE PRECISION array, dimension (N)
		     If SENSE = 'V' or 'B', the estimated reciprocal condition
		     numbers of the eigenvectors, stored in consecutive elements
		     of the array. For a complex eigenvector two consecutive
		     elements of RCONDV are set to the same value. If the
		     eigenvalues cannot be reordered to compute RCONDV(j),
		     RCONDV(j) is set to 0; this can only occur when the true
		     value would be very small anyway.
		     If SENSE = 'N' or 'E', RCONDV is not referenced.

	   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. LWORK >= max(1,2*N).
		     If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
		     LWORK >= max(1,6*N).
		     If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
		     If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.

		     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 (N+6)
		     If SENSE = 'E', IWORK is not referenced.

	   BWORK

		     BWORK is LOGICAL array, dimension (N)
		     If SENSE = 'N', BWORK is not referenced.

	   INFO

		     INFO is INTEGER
		     = 0:  successful exit
		     < 0:  if INFO = -i, the i-th argument had an illegal value.
		     = 1,...,N:
			   The QZ iteration failed.  No eigenvectors have been
			   calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
			   should be correct for j=INFO+1,...,N.
		     > N:  =N+1: other than QZ iteration failed in DHGEQZ.
			   =N+2: error return from DTGEVC.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   April 2012

       Further Details:

	     Balancing a matrix pair (A,B) includes, first, permuting rows and
	     columns to isolate eigenvalues, second, applying diagonal similarity
	     transformation to the rows and columns to make the rows and columns
	     as close in norm as possible. 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 4.11.1.2 of LAPACK Users' Guide.

	     An approximate error bound on the chordal distance between the i-th
	     computed generalized eigenvalue w and the corresponding exact
	     eigenvalue lambda is

		  chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)

	     An approximate error bound for the angle between the i-th computed
	     eigenvector VL(i) or VR(i) is given by

		  EPS * norm(ABNRM, BBNRM) / DIF(i).

	     For further explanation of the reciprocal condition numbers RCONDE
	     and RCONDV, see section 4.11 of LAPACK User's Guide.

       Definition at line 389 of file dggevx.f.

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

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