DCHDD(3F)

NAME
     DCHDD   - DCHDD 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, DCHDD 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 DSQRT(RHO**2 - ZETA**2).
     DCHDD will simultaneously downdate several triplets (Z,Y,RHO) along with
     R.  For a less terse description of what DCHDD 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 double precision.

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

DESCRIPTION
     On Entry

     R DOUBLE PRECISION(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.
     X DOUBLE PRECISION(P).
        X contains the row vector that is to
        be removed from R.  X is not altered by DCHDD.

     Z DOUBLE PRECISION(LDZ,N)Z), 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 DOUBLE PRECISION(NZ).
        Y contains the scalars for the downdating
        of the vectors Z.  Y is not altered by DCHDD.

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

     R


     Z contain the downdated quantities.

     RHO


     C DOUBLE PRECISION(P).
        C contains the cosines of the transforming
        rotations.

     S DOUBLE PRECISION(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 RHO's are
        set to -1.  LINPACK.  This version dated 08/14/78 .  Stewart, G. W.,
     University of Maryland, Argonne National Lab.
     DCHDD uses the following functions and subprograms. Fortran DABS BLAS
     DDOT, DNRM2