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

SGGESX(l)					)					SGGESX(l)

NAME
       SGGESX  -  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 SGGESX( 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 )

	   CHARACTER	  JOBVSL, JOBVSR, SENSE, SORT

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

	   LOGICAL	  BWORK( * )

	   INTEGER	  IWORK( * )

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

	   LOGICAL	  SELCTG

	   EXTERNAL	  SELCTG

PURPOSE
       SGGESX 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 SELCTG).

       SELCTG  (input) LOGICAL FUNCTION of three REAL 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-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) REAL 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) REAL 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 SELCTG is true.  (Complex conjugate pairs for which SELCTG is  true
	       for either eigenvalue count as 2.)

       ALPHAR  (output) REAL array, dimension (N)
	       ALPHAI	(output) REAL array, dimension (N) BETA    (output) REAL 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 conju-
	       gate 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) REAL 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) REAL 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) REAL 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) REAL 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) REAL 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 SHGEQZ
	       =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 STGSEN.

	       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 				SGGESX(l)


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