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dggesx.f(3)				      LAPACK				      dggesx.f(3)

NAME
       dggesx.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dggesx (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM, ALPHAR,
	   ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
	   BWORK, INFO)
	    DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
	   vectors for GE matrices

Function/Subroutine Documentation
   subroutine dggesx (characterJOBVSL, characterJOBVSR, characterSORT, logical, externalSELCTG,
       characterSENSE, integerN, double precision, dimension( lda, * )A, integerLDA, double
       precision, dimension( ldb, * )B, integerLDB, integerSDIM, double precision, dimension( *
       )ALPHAR, double precision, dimension( * )ALPHAI, double precision, dimension( * )BETA,
       double precision, dimension( ldvsl, * )VSL, integerLDVSL, double precision, dimension(
       ldvsr, * )VSR, integerLDVSR, double precision, dimension( 2 )RCONDE, double precision,
       dimension( 2 )RCONDV, double precision, dimension( * )WORK, integerLWORK, integer,
       dimension( * )IWORK, integerLIWORK, logical, dimension( * )BWORK, integerINFO)
	DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

	    DGGESX computes for a pair of N-by-N real nonsymmetric matrices
	    (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
	    optionally, the left and/or right matrices of Schur vectors (VSL and
	    VSR).  This gives the generalized Schur factorization

		 (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

	    Optionally, it also orders the eigenvalues so that a selected cluster
	    of eigenvalues appears in the leading diagonal blocks of the upper
	    quasi-triangular matrix S and the upper triangular matrix T; computes
	    a reciprocal condition number for the average of the selected
	    eigenvalues (RCONDE); and computes a reciprocal condition number for
	    the right and left deflating subspaces corresponding to the selected
	    eigenvalues (RCONDV). The leading columns of VSL and VSR then form
	    an orthonormal basis for the corresponding left and right eigenspaces
	    (deflating subspaces).

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

	    A pair of matrices (S,T) is in generalized real Schur form if T is
	    upper triangular with non-negative diagonal and S is block upper
	    triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
	    to real generalized eigenvalues, while 2-by-2 blocks of S will be
	    "standardized" by making the corresponding elements of T have the
	    form:
		    [  a  0  ]
		    [  0  b  ]

	    and the pair of corresponding 2-by-2 blocks in S and T will have a
	    complex conjugate pair of generalized eigenvalues.

       Parameters:
	   JOBVSL

		     JOBVSL is CHARACTER*1
		     = 'N':  do not compute the left Schur vectors;
		     = 'V':  compute the left Schur vectors.

	   JOBVSR

		     JOBVSR is CHARACTER*1
		     = 'N':  do not compute the right Schur vectors;
		     = 'V':  compute the right Schur vectors.

	   SORT

		     SORT is CHARACTER*1
		     Specifies whether or not to order the eigenvalues on the
		     diagonal of the generalized Schur form.
		     = 'N':  Eigenvalues are not ordered;
		     = 'S':  Eigenvalues are ordered (see SELCTG).

	   SELCTG

		     SELCTG is procedure) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
		     SELCTG must be declared EXTERNAL in the calling subroutine.
		     If SORT = 'N', SELCTG is not referenced.
		     If SORT = 'S', SELCTG is used to select eigenvalues to sort
		     to the top left of the Schur form.
		     An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
		     SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
		     one of a complex conjugate pair of eigenvalues is selected,
		     then both complex eigenvalues are selected.
		     Note that a selected complex eigenvalue may no longer satisfy
		     SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) = .TRUE. after ordering,
		     since ordering may change the value of complex eigenvalues
		     (especially if the eigenvalue is ill-conditioned), in this
		     case INFO is set to N+3.

	   SENSE

		     SENSE is CHARACTER*1
		     Determines which reciprocal condition numbers are computed.
		     = 'N' : None are computed;
		     = 'E' : Computed for average of selected eigenvalues only;
		     = 'V' : Computed for selected deflating subspaces only;
		     = 'B' : Computed for both.
		     If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

	   N

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

	   A

		     A is DOUBLE PRECISION array, dimension (LDA, N)
		     On entry, the first of the pair of matrices.
		     On exit, A has been overwritten by its generalized Schur
		     form S.

	   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 second of the pair of matrices.
		     On exit, B has been overwritten by its generalized Schur
		     form T.

	   LDB

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

	   SDIM

		     SDIM is INTEGER
		     If SORT = 'N', SDIM = 0.
		     If SORT = 'S', SDIM = number of eigenvalues (after sorting)
		     for which SELCTG is true.	(Complex conjugate pairs for which
		     SELCTG is true for either eigenvalue count as 2.)

	   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.  ALPHAR(j) + ALPHAI(j)*i
		     and BETA(j),j=1,...,N  are the diagonals of the complex Schur
		     form (S,T) that would result if the 2-by-2 diagonal blocks of
		     the real Schur form of (A,B) were further reduced to
		     triangular form using 2-by-2 complex unitary transformations.
		     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.
		     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).

	   VSL

		     VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
		     If JOBVSL = 'V', VSL will contain the left Schur vectors.
		     Not referenced if JOBVSL = 'N'.

	   LDVSL

		     LDVSL is INTEGER
		     The leading dimension of the matrix VSL. LDVSL >=1, and
		     if JOBVSL = 'V', LDVSL >= N.

	   VSR

		     VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
		     If JOBVSR = 'V', VSR will contain the right Schur vectors.
		     Not referenced if JOBVSR = 'N'.

	   LDVSR

		     LDVSR is INTEGER
		     The leading dimension of the matrix VSR. LDVSR >= 1, and
		     if JOBVSR = 'V', LDVSR >= N.

	   RCONDE

		     RCONDE is DOUBLE PRECISION array, dimension ( 2 )
		     If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
		     reciprocal condition numbers for the average of the selected
		     eigenvalues.
		     Not referenced if SENSE = 'N' or 'V'.

	   RCONDV

		     RCONDV is DOUBLE PRECISION array, dimension ( 2 )
		     If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
		     reciprocal condition numbers for the selected deflating
		     subspaces.
		     Not referenced if SENSE = 'N' or 'E'.

	   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 N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
		     LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
		     LWORK >= max( 8*N, 6*N+16 ).
		     Note that 2*SDIM*(N-SDIM) <= N*N/2.
		     Note also that an error is only returned if
		     LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
		     this may not be large enough.

		     If LWORK = -1, then a workspace query is assumed; the routine
		     only calculates the bound on the optimal size of the WORK
		     array and the minimum size of the IWORK array, returns these
		     values as the first entries of the WORK and IWORK arrays, and
		     no error message related to LWORK or LIWORK is issued by
		     XERBLA.

	   IWORK

		     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
		     On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

	   LIWORK

		     LIWORK is INTEGER
		     The dimension of the array IWORK.
		     If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
		     LIWORK >= N+6.

		     If LIWORK = -1, then a workspace query is assumed; the
		     routine only calculates the bound on the optimal size of the
		     WORK array and the minimum size of the IWORK array, returns
		     these values as the first entries of the WORK and IWORK
		     arrays, and no error message related to LWORK or LIWORK is
		     issued by XERBLA.

	   BWORK

		     BWORK is LOGICAL array, dimension (N)
		     Not referenced if SORT = 'N'.

	   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.  (A,B) are not in Schur
			   form, 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: after reordering, roundoff changed values of
				 some complex eigenvalues so that leading
				 eigenvalues in the Generalized Schur form no
				 longer satisfy SELCTG=.TRUE.  This could also
				 be caused due to scaling.
			   =N+3: reordering failed in DTGSEN.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Further Details:

	     An approximate (asymptotic) bound on the average absolute error of
	     the selected eigenvalues is

		  EPS * norm((A, B)) / RCONDE( 1 ).

	     An approximate (asymptotic) bound on the maximum angular error in
	     the computed deflating subspaces is

		  EPS * norm((A, B)) / RCONDV( 2 ).

	     See LAPACK User's Guide, section 4.11 for more information.

       Definition at line 363 of file dggesx.f.

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

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