SCHDD(3F)							     SCHDD(3F)

NAME    [Toc]    [Back]

     SCHDD   - SCHDD downdates an augmented Cholesky decomposition or the
     triangular	factor of an augmented QR decomposition.  Specifically,	given
     an	upper triangular matrix	R of order P, a	row vector X, a	column vector
     Z,	and a scalar Y,	SCHDD determines an orthogonal matrix U	and a scalar
     ZETA such that

	(R   Z )     (RR  ZZ)
	U * (	   )  =	 (	) ,
	(0 ZETA)     ( X   Y)

     where RR is upper triangular.  If R and Z have been obtained from the
     factorization of a	least squares problem, then RR and ZZ are the factors
     corresponding to the problem with the observation (X,Y) removed.  In this
     case, if RHO is the norm of the residual vector, then the norm of the
     residual vector of	the downdated problem is SQRT(RHO**2 - ZETA**2). SCHDD
     will simultaneously downdate several triplets (Z,Y,RHO) along with	R.
     For a less	terse description of what SCHDD	does and how it	may be
     applied, see the LINPACK guide.

     The matrix	U is determined	as the product U(1)*...*U(P) where U(I)	is a
     rotation in the (P+1,I)-plane of the form

	( C(I)	   -S(I)     )
	(		     ) .
	( S(I)	     C(I)    )

     The rotations are chosen so that C(I) is real.

     The user is warned	that a given downdating	problem	may be impossible to
     accomplish	or may produce inaccurate results.  For	example, this can
     happen if X is near a vector whose	removal	will reduce the	rank of	R.
     Beware.

SYNOPSYS    [Toc]    [Back]

      SUBROUTINE SCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)

DESCRIPTION    [Toc]    [Back]

     On	Entry

     R REAL(LDR,P), where LDR .GE. P.
	R contains the upper triangular	matrix
	that is	to be downdated.  The part of  R
	below the diagonal is not referenced.

     LDR INTEGER.
	LDR is the leading dimension of	the array R.

     P INTEGER.
	P is the order of the matrix R.




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SCHDD(3F)							     SCHDD(3F)



     X REAL(P).
	X contains the row vector that is to
	be removed from	R.  X is not altered by	SCHDD.

     Z REAL(LDZ,NZ), where LDZ .GE. P.
	Z is an	array of NZ P-vectors which
	are to be downdated along with R.

     LDZ INTEGER.
	LDZ is the leading dimension of	the array Z.

     NZ	INTEGER.
	NZ is the number of vectors to be downdated
	NZ may be zero,	in which case Z, Y, and	RHO
	are not	referenced.

     Y REAL(NZ).
	Y contains the scalars for the downdating
	of the vectors Z.  Y is	not altered by SCHDD.

     RHO REAL(NZ).
	RHO contains the norms of the residual
	vectors	that are to be downdated.  On Return

     R


     Z contain the downdated quantities.

     RHO    [Toc]    [Back]


     C REAL(P).
	C contains the cosines of the transforming
	rotations.

     S REAL(P).
	S contains the sines of	the transforming
	rotations.

     INFO INTEGER.
	INFO is	set as follows.
	INFO = 0  if the entire	downdating
	was successful.
	INFO =-1  if R could not be downdated.
	In this	case, all quantities
	are left unaltered.
	INFO = 1  if some RHO could not	be
	downdated.  The	offending RHOs are
	set to -1.  LINPACK.  This version dated 08/14/78 .  G.	W. Stewart,
     University	of Maryland, Argonne National Lab.




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SCHDD(3F)							     SCHDD(3F)



     SCHDD uses	the following functions	and subprograms. Fortran ABS BLAS
     SDOT, SNRM2


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