real function bvalue ( t, bcoef, n, k, x, jderiv )
c from * a practical guide to splines * by c. de boor
calls interv
c
calculates value at x of jderiv-th derivative of spline from b-repr.
c the spline is taken to be continuous from the right, EXCEPT at the
c rightmost knot, where it is taken to be continuous from the left.
c
c****** i n p u t ******
c t, bcoef, n, k......forms the b-representation of the spline f to
c be evaluated. specifically,
c t.....knot sequence, of length n+k, assumed nondecreasing.
c bcoef.....b-coefficient sequence, of length n .
c n.....length of bcoef and dimension of spline(k,t),
c a s s u m e d positive .
c k.....order of the spline .
c
c w a r n i n g . . . the restriction k .le. kmax (=20) is imposed
c arbitrarily by the dimension statement for aj, dl, dr below,
c but is n o w h e r e c h e c k e d for.
c
c x.....the point at which to evaluate .
c jderiv.....integer giving the order of the derivative to be evaluated
c a s s u m e d to be zero or positive.
c
c****** o u t p u t ******
c bvalue.....the value of the (jderiv)-th derivative of f at x .
c
c****** m e t h o d ******
c The nontrivial knot interval (t(i),t(i+1)) containing x is lo-
c cated with the aid of interv . The k b-coeffs of f relevant for
c this interval are then obtained from bcoef (or taken to be zero if
c not explicitly available) and are then differenced jderiv times to
c obtain the b-coeffs of (d**jderiv)f relevant for that interval.
c Precisely, with j = jderiv, we have from x.(12) of the text that
c
c (d**j)f = sum ( bcoef(.,j)*b(.,k-j,t) )
c
c where
c / bcoef(.), , j .eq. 0
c /
c bcoef(.,j) = / bcoef(.,j-1) - bcoef(.-1,j-1)
c / ----------------------------- , j .gt. 0
c / (t(.+k-j) - t(.))/(k-j)
c
c Then, we use repeatedly the fact that
c
c sum ( a(.)*b(.,m,t)(x) ) = sum ( a(.,x)*b(.,m-1,t)(x) )
c with
c (x - t(.))*a(.) + (t(.+m-1) - x)*a(.-1)
c a(.,x) = ---------------------------------------
c (x - t(.)) + (t(.+m-1) - x)
c
c to write (d**j)f(x) eventually as a linear combination of b-splines
c of order 1 , and the coefficient for b(i,1,t)(x) must then be the
c desired number (d**j)f(x). (see x.(17)-(19) of text).
c
integer jderiv,k,n, i,ilo,imk,j,jc,jcmin,jcmax,jj,kmax,kmj,km1
* ,mflag,nmi,jdrvp1
parameter (kmax = 20)
real bcoef(n),t(n+k),x, aj(kmax),dl(kmax),dr(kmax),fkmj
bvalue = 0.
if (jderiv .ge. k) go to 99
c
c *** Find i s.t. 1 .le. i .lt. n+k and t(i) .lt. t(i+1) and
c t(i) .le. x .lt. t(i+1) . If no such i can be found, x lies
c outside the support of the spline f , hence bvalue = 0.
c (The asymmetry in this choice of i makes f rightcontinuous, except
c at t(n+k) where it is leftcontinuous.)
call interv ( t, n+k, x, i, mflag )
if (mflag .ne. 0) go to 99
c *** if k = 1 (and jderiv = 0), bvalue = bcoef(i).
km1 = k - 1
if (km1 .gt. 0) go to 1
bvalue = bcoef(i)
go to 99
c
c *** store the k b-spline coefficients relevant for the knot interval
c (t(i),t(i+1)) in aj(1),...,aj(k) and compute dl(j) = x - t(i+1-j),
c dr(j) = t(i+j) - x, j=1,...,k-1 . set any of the aj not obtainable
c from input to zero. set any t.s not obtainable equal to t(1) or
c to t(n+k) appropriately.
1 jcmin = 1
imk = i - k
if (imk .ge. 0) go to 8
jcmin = 1 - imk
do 5 j=1,i
5 dl(j) = x - t(i+1-j)
do 6 j=i,km1
aj(k-j) = 0.
6 dl(j) = dl(i)
go to 10
8 do 9 j=1,km1
9 dl(j) = x - t(i+1-j)
c
10 jcmax = k
nmi = n - i
if (nmi .ge. 0) go to 18
jcmax = k + nmi
do 15 j=1,jcmax
15 dr(j) = t(i+j) - x
do 16 j=jcmax,km1
aj(j+1) = 0.
16 dr(j) = dr(jcmax)
go to 20
18 do 19 j=1,km1
19 dr(j) = t(i+j) - x
c
20 do 21 jc=jcmin,jcmax
21 aj(jc) = bcoef(imk + jc)
c
c *** difference the coefficients jderiv times.
if (jderiv .eq. 0) go to 30
do 23 j=1,jderiv
kmj = k-j
fkmj = float(kmj)
ilo = kmj
do 23 jj=1,kmj
aj(jj) = ((aj(jj+1) - aj(jj))/(dl(ilo) + dr(jj)))*fkmj
23 ilo = ilo - 1
c
c *** compute value at x in (t(i),t(i+1)) of jderiv-th derivative,
c given its relevant b-spline coeffs in aj(1),...,aj(k-jderiv).
30 if (jderiv .eq. km1) go to 39
jdrvp1 = jderiv + 1
do 33 j=jdrvp1,km1
kmj = k-j
ilo = kmj
do 33 jj=1,kmj
aj(jj) = (aj(jj+1)*dl(ilo) + aj(jj)*dr(jj))/(dl(ilo)+dr(jj))
33 ilo = ilo - 1
39 bvalue = aj(1)
c
99 return
end