subroutine schdd(r,ldr,p,x,z,ldz,nz,y,rho,c,s,info)
integer ldr,p,ldz,nz,info
real r(ldr,1),x(1),z(ldz,1),y(1),s(1)
real rho(1),c(1)
c
c schdd downdates an augmented cholesky decomposition or the
c triangular factor of an augmented qr decomposition.
c specifically, given an upper triangular matrix r of order p, a
c row vector x, a column vector z, and a scalar y, schdd
c determineds a orthogonal matrix u and a scalar zeta such that
c
c (r z ) (rr zz)
c u * ( ) = ( ) ,
c (0 zeta) ( x y)
c
c where rr is upper triangular. if r and z have been obtained
c from the factorization of a least squares problem, then
c rr and zz are the factors corresponding to the problem
c with the observation (x,y) removed. in this case, if rho
c is the norm of the residual vector, then the norm of
c the residual vector of the downdated problem is
c sqrt(rho**2 - zeta**2). schdd will simultaneously downdate
c several triplets (z,y,rho) along with r.
c for a less terse description of what schdd does and how
c it may be applied, see the linpack guide.
c
c the matrix u is determined as the product u(1)*...*u(p)
c where u(i) is a rotation in the (p+1,i)-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 the user is warned that a given downdating problem may
c be impossible to accomplish or may produce
c inaccurate results. for example, this can happen
c if x is near a vector whose removal will reduce the
c rank of r. beware.
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 downdated. the part of r
c below the diagonal is not referenced.
c
c ldr integer.
c ldr is the leading dimension fo 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 vector that is to
c be removed from r. x is not altered by schdd.
c
c z real(ldz,nz), where ldz .ge. p.
c z is an array of nz p-vectors which
c are to be downdated along 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 downdated
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 the downdating
c of the vectors z. y is not altered by schdd.
c
c rho real(nz).
c rho contains the norms of the residual
c vectors that are to be downdated.
c
c on return
c
c r
c z contain the downdated quantities.
c rho
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 info integer.
c info is set as follows.
c
c info = 0 if the entire downdating
c was successful.
c
c info =-1 if r could not be downdated.
c in this case, all quantities
c are left unaltered.
c
c info = 1 if some rho could not be
c downdated. the offending rhos are
c set to -1.
c
c linpack. this version dated 08/14/78 .
c g.w. stewart, university of maryland, argonne national lab.
c
c schdd uses the following functions and subprograms.
c
c fortran abs
c blas sdot, snrm2
c
integer i,ii,j
real a,alpha,azeta,norm,snrm2
real sdot,t,zeta,b,xx
c
c solve the system trans(r)*a = x, placing the result
c in the array s.
c
info = 0
s(1) = x(1)/r(1,1)
if (p .lt. 2) go to 20
do 10 j = 2, p
s(j) = x(j) - sdot(j-1,r(1,j),1,s,1)
s(j) = s(j)/r(j,j)
10 continue
20 continue
norm = snrm2(p,s,1)
if (norm .lt. 1.0e0) go to 30
info = -1
go to 120
30 continue
alpha = sqrt(1.0e0-norm**2)
c
c determine the transformations.
c
do 40 ii = 1, p
i = p - ii + 1
scale = alpha + abs(s(i))
a = alpha/scale
b = s(i)/scale
norm = sqrt(a**2+b**2+0.0e0**2)
c(i) = a/norm
s(i) = b/norm
alpha = scale*norm
40 continue
c
c apply the transformations to r.
c
do 60 j = 1, p
xx = 0.0e0
do 50 ii = 1, j
i = j - ii + 1
t = c(i)*xx + s(i)*r(i,j)
r(i,j) = c(i)*r(i,j) - s(i)*xx
xx = t
50 continue
60 continue
c
c if required, downdate z and rho.
c
if (nz .lt. 1) go to 110
do 100 j = 1, nz
zeta = y(j)
do 70 i = 1, p
z(i,j) = (z(i,j) - s(i)*zeta)/c(i)
zeta = c(i)*zeta - s(i)*z(i,j)
70 continue
azeta = abs(zeta)
if (azeta .le. rho(j)) go to 80
info = 1
rho(j) = -1.0e0
go to 90
80 continue
rho(j) = rho(j)*sqrt(1.0e0-(azeta/rho(j))**2)
90 continue
100 continue
110 continue
120 continue
return
end