SUBROUTINE SCHUD(R,LDR,P,X,Z,LDZ,NZ,Y,RHO,C,S)
C***BEGIN PROLOGUE  SCHUD
C***DATE WRITTEN   780814   (YYMMDD)
C***REVISION DATE  820801   (YYMMDD)
C***CATEGORY NO.  D7B
C***KEYWORDS  CHOLESKY DECOMPOSITION,LINEAR ALGEBRA,LINPACK,MATRIX,
C             UPDATE
C***AUTHOR  STEWART, G. W., (U. OF MARYLAND)
C***PURPOSE  Updates an augmented Cholesky decomposition of the
C            triangular part of an augmented QR decomposition.
C***DESCRIPTION
C
C     SCHUD updates an augmented Cholesky decomposition of the
C     triangular part of an augmented QR decomposition.  Specifically,
C     given an upper triangular matrix R of order P, a row vector
C     X, a column vector Z, and a scalar Y, SCHUD determines a
C     unitary matrix U and a scalar ZETA such that
C
C
C                              (R  Z)     (RR   ZZ )
C                         U  * (    )  =  (        ) ,
C                              (X  Y)     ( 0  ZETA)
C
C     where RR is upper triangular.  If R and Z have been
C     obtained from the factorization of a least squares
C     problem, then RR and ZZ are the factors corresponding to
C     the problem with the observation (X,Y) appended.  In this
C     case, if RHO is the norm of the residual vector, then the
C     norm of the residual vector of the updated problem is
C     SQRT(RHO**2 + ZETA**2).  SCHUD will simultaneously update
C     several triplets (Z,Y,RHO).
C     For a less terse description of what SCHUD does and how
C     it may be applied, see the LINPACK guide.
C
C     The matrix U is determined as the product U(P)*...*U(1),
C     where U(I) is a rotation in the (I,P+1) plane of the
C     form
C
C                       (     C(I)      S(I) )
C                       (                    ) .
C                       (    -S(I)      C(I) )
C
C     The rotations are chosen so that C(I) is real.
C
C     On Entry
C
C         R      REAL(LDR,P), where LDR .GE. P.
C                R contains the upper triangular matrix
C                that is to be updated.  The part of R
C                below the diagonal is not referenced.
C
C         LDR    INTEGER.
C                LDR is the leading dimension of the array R.
C
C         P      INTEGER.
C                P is the order of the matrix R.
C
C         X      REAL(P).
C                X contains the row to be added to R.  X is
C                not altered by SCHUD.
C
C         Z      REAL(LDZ,NZ), where LDZ .GE. P.
C                Z is an array containing NZ P-vectors to
C                be updated with R.
C
C         LDZ    INTEGER.
C                LDZ is the leading dimension of the array Z.
C
C         NZ     INTEGER.
C                NZ is the number of vectors to be updated.
C                NZ may be zero, in which case Z, Y, and RHO
C                are not referenced.
C
C         Y      REAL(NZ).
C                Y contains the scalars for updating the vectors
C                Z.  Y is not altered by SCHUD.
C
C         RHO    REAL(NZ).
C                RHO contains the norms of the residual
C                vectors that are to be updated.  If RHO(J)
C                is negative, it is left unaltered.
C
C     On Return
C
C         RC
C         RHO    contain the updated quantities.
C         Z
C
C         C      REAL(P).
C                C contains the cosines of the transforming
C                rotations.
C
C         S      REAL(P).
C                S contains the sines of the transforming
C                rotations.
C
C     LINPACK.  This version dated 08/14/78 .
C     G. W. Stewart, University of Maryland, Argonne National Lab.
C
C     SCHUD uses the following functions and subroutines.
C
C     Extended BLAS SROTG
C     Fortran SQRT
C***REFERENCES  DONGARRA J.J., BUNCH J.R., MOLER C.B., STEWART G.W.,
C                 *LINPACK USERS  GUIDE*, SIAM, 1979.
C***ROUTINES CALLED  SROTG
C***END PROLOGUE  SCHUD