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

DGGESX(l)					)					DGGESX(l)

NAME
       DGGESX  -  compute  for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized
       eigenvalues, the real Schur form (S,T), and,

SYNOPSIS
       SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, DELCTG, SENSE, N, A, LDA, B, LDB,  SDIM,  ALPHAR,
			  ALPHAI,  BETA,  VSL,	LDVSL,	VSR,  LDVSR, RCONDE, RCONDV, WORK, LWORK,
			  IWORK, LIWORK, BWORK, INFO )

	   CHARACTER	  JOBVSL, JOBVSR, SENSE, SORT

	   INTEGER	  INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, SDIM

	   LOGICAL	  BWORK( * )

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),  BETA(	*
			  ),  RCONDE( 2 ), RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( *
			  )

	   LOGICAL	  DELCTG

	   EXTERNAL	  DELCTG

PURPOSE
       DGGESX computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized ei-
       genvalues,  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 lead-
       ing  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.

ARGUMENTS
       JOBVSL  (input) CHARACTER*1
	       = 'N':  do not compute the left Schur vectors;
	       = 'V':  compute the left Schur vectors.

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

       SORT    (input) CHARACTER*1
	       Specifies whether or not to order the eigenvalues on the diagonal of the  general-
	       ized Schur form.  = 'N':  Eigenvalues are not ordered;
	       = 'S':  Eigenvalues are ordered (see DELZTG).

       DELZTG  (input) LOGICAL FUNCTION of three DOUBLE PRECISION arguments
	       DELZTG must be declared EXTERNAL in the calling subroutine.  If SORT = 'N', DELZTG
	       is not referenced.  If SORT = 'S', DELZTG 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 DELZTG(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
	       DELZTG(ALPHAR(j),ALPHAI(j),BETA(j))  =  .TRUE.  after ordering, since ordering may
	       change the value of complex eigenvalues (especially if the eigenvalue is  ill-con-
	       ditioned), in this case INFO is set to N+3.

       SENSE   (input) CHARACTER
	       Determines which reciprocal condition numbers are computed.  = 'N' : None are com-
	       puted;
	       = '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       (input) INTEGER
	       The order of the matrices A, B, VSL, and VSR.  N >= 0.

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

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

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

       ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
	       ALPHAI  (output) DOUBLE PRECISION array, dimension  (N)	BETA	 (output)  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 com-
	       puting 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 compa-
	       rable with norm(B).

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

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

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

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

       RCONDE  (output) 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  (output) 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    (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  dimension  of  the array WORK.  LWORK >= 8*(N+1)+16.  If SENSE = 'E', 'V', or
	       'B', LWORK >= MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).

       IWORK   (workspace) INTEGER array, dimension (LIWORK)
	       Not referenced if SENSE = 'N'.

       LIWORK  (input) INTEGER
	       The dimension of the array WORK.  LIWORK >= N+6.

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

       INFO    (output) 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
	       DELZTG=.TRUE.  This could also be caused due to scaling.  =N+3: reordering  failed
	       in DTGSEN.

	       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.

LAPACK version 3.0			   15 June 2000 				DGGESX(l)


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