CCHDD(3F)CCHDD(3F)NAMECCHDD - CCHDD 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, CCHDD determines a unitary 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).
CCHDD will simultaneously downdate several triplets (Z,Y,RHO) along with
R. For a less terse description of what CCHDD 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) -CONJG(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
SUBROUTINE CCHDD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)
DESCRIPTION
On Entry
R COMPLEX(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 COMPLEX(P).
X contains the row vector that is to
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CCHDD(3F)CCHDD(3F)
be removed from R. X is not altered by CCHDD.
Z COMPLEX(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 COMPLEX(NZ).
Y contains the scalars for the downdating
of the vectors Z. Y is not altered bY CCHDD.
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
C REAL(P).
C contains the cosines of the transforming
rotations.
S COMPLEX(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.
CCHDD uses the following functions and subprograms. Fortran
AIMAG,CABS,CONJG BLAS CDOTC, SCNRM2
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