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