subroutine bsplpp ( t, bcoef, n, k, scrtch, break, coef, l )
c from * a practical guide to splines * by c. de Boor (7 may 92)
calls bsplvb
c
converts the b-representation t, bcoef, n, k of some spline into its
c pp-representation break, coef, l, k .
c
c****** i n p u t ******
c t.....knot sequence, of length n+k
c bcoef.....b-spline coefficient sequence, of length n
c n.....length of bcoef and dimension of spline space spline(k,t)
c k.....order of the spline
c
c w a r n i n g . . . the restriction k .le. kmax (= 20) is impo-
c sed by the arbitrary dimension statement for biatx below, but
c is n o w h e r e c h e c k e d for.
c
c****** w o r k a r e a ******
c scrtch......of size (k,k) , needed to contain bcoeffs of a piece of
c the spline and its k-1 derivatives
c
c****** o u t p u t ******
c break.....breakpoint sequence, of length l+1, contains (in increas-
c ing order) the distinct points in the sequence t(k),...,t(n+1)
c coef.....array of size (k,l), with coef(i,j) = (i-1)st derivative of
c spline at break(j) from the right
c l.....number of polynomial pieces which make up the spline in the in-
c terval (t(k), t(n+1))
c
c****** m e t h o d ******
c for each breakpoint interval, the k relevant b-coeffs of the
c spline are found and then differenced repeatedly to get the b-coeffs
c of all the derivatives of the spline on that interval. the spline and
c its first k-1 derivatives are then evaluated at the left end point
c of that interval, using bsplvb repeatedly to obtain the values of
c all b-splines of the appropriate order at that point.
c
integer k,l,n, i,j,jp1,kmax,kmj,left,lsofar
parameter (kmax = 20)
real bcoef(n),break(l+1),coef(k,l),t(n+k), scrtch(k,k)
* ,biatx(kmax),diff,factor,sum
c
lsofar = 0
break(1) = t(k)
do 50 left=k,n
c find the next nontrivial knot interval.
if (t(left+1) .eq. t(left)) go to 50
lsofar = lsofar + 1
break(lsofar+1) = t(left+1)
if (k .gt. 1) go to 9
coef(1,lsofar) = bcoef(left)
go to 50
c store the k b-spline coeff.s relevant to current knot interval
c in scrtch(.,1) .
9 do 10 i=1,k
10 scrtch(i,1) = bcoef(left-k+i)
c
c for j=1,...,k-1, compute the k-j b-spline coeff.s relevant to
c current knot interval for the j-th derivative by differencing
c those for the (j-1)st derivative, and store in scrtch(.,j+1) .
do 20 jp1=2,k
j = jp1 - 1
kmj = k - j
do 20 i=1,kmj
diff = t(left+i) - t(left+i - kmj)
if (diff .gt. 0.) scrtch(i,jp1) =
* (scrtch(i+1,j)-scrtch(i,j))/diff
20 continue
c
c for j = 0, ..., k-1, find the values at t(left) of the j+1
c b-splines of order j+1 whose support contains the current
c knot interval from those of order j (in biatx ), then comb-
c ine with the b-spline coeff.s (in scrtch(.,k-j) ) found earlier
c to compute the (k-j-1)st derivative at t(left) of the given
c spline.
c note. if the repeated calls to bsplvb are thought to gene-
c rate too much overhead, then replace the first call by
c biatx(1) = 1.
c and the subsequent call by the statement
c j = jp1 - 1
c followed by a direct copy of the lines
c deltar(j) = t(left+j) - x
c ......
c biatx(j+1) = saved
c from bsplvb . deltal(kmax) and deltar(kmax) would have to
c appear in a dimension statement, of course.
c
call bsplvb ( t, 1, 1, t(left), left, biatx )
coef(k,lsofar) = scrtch(1,k)
do 30 jp1=2,k
call bsplvb ( t, jp1, 2, t(left), left, biatx )
kmj = k+1 - jp1
sum = 0.
do 28 i=1,jp1
28 sum = biatx(i)*scrtch(i,kmj) + sum
30 coef(kmj,lsofar) = sum
50 continue
l = lsofar
if (k .eq. 1) return
factor = 1.
do 60 i=2,k
factor = factor*float(k+1-i)
do 60 j=1,lsofar
60 coef(i,j) = coef(i,j)*factor
return
end