subroutine bsplvd ( t, k, x, left, a, dbiatx, nderiv ) c from * a practical guide to splines * by c. de Boor (7 may 92) calls bsplvb calculates value and deriv.s of all b-splines which do not vanish at x c c****** i n p u t ****** c t the knot array, of length left+k (at least) c k the order of the b-splines to be evaluated c x the point at which these values are sought c left an integer indicating the left endpoint of the interval of c interest. the k b-splines whose support contains the interval c (t(left), t(left+1)) c are to be considered. c a s s u m p t i o n - - - it is assumed that c t(left) .lt. t(left+1) c division by zero will result otherwise (in b s p l v b ). c also, the output is as advertised only if c t(left) .le. x .le. t(left+1) . c nderiv an integer indicating that values of b-splines and their c derivatives up to but not including the nderiv-th are asked c for. ( nderiv is replaced internally by the integer m h i g h c in (1,k) closest to it.) c c****** w o r k a r e a ****** c a an array of order (k,k), to contain b-coeff.s of the derivat- c ives of a certain order of the k b-splines of interest. c c****** o u t p u t ****** c dbiatx an array of order (k,nderiv). its entry (i,m) contains c value of (m-1)st derivative of (left-k+i)-th b-spline of c order k for knot sequence t , i=1,...,k, m=1,...,nderiv. c c****** m e t h o d ****** c values at x of all the relevant b-splines of order k,k-1,..., c k+1-nderiv are generated via bsplvb and stored temporarily in c dbiatx . then, the b-coeffs of the required derivatives of the b- c splines of interest are generated by differencing, each from the pre- c ceding one of lower order, and combined with the values of b-splines c of corresponding order in dbiatx to produce the desired values . c integer k,left,nderiv, i,ideriv,il,j,jlow,jp1mid,kp1,kp1mm * ,ldummy,m,mhigh real a(k,k),dbiatx(k,nderiv),t(1),x, factor,fkp1mm,sum mhigh = max0(min0(nderiv,k),1) c mhigh is usually equal to nderiv. kp1 = k+1 call bsplvb(t,kp1-mhigh,1,x,left,dbiatx) if (mhigh .eq. 1) go to 99 c the first column of dbiatx always contains the b-spline values c for the current order. these are stored in column k+1-current c order before bsplvb is called to put values for the next c higher order on top of it. ideriv = mhigh do 15 m=2,mhigh jp1mid = 1 do 11 j=ideriv,k dbiatx(j,ideriv) = dbiatx(jp1mid,1) 11 jp1mid = jp1mid + 1 ideriv = ideriv - 1 call bsplvb(t,kp1-ideriv,2,x,left,dbiatx) 15 continue c c at this point, b(left-k+i, k+1-j)(x) is in dbiatx(i,j) for c i=j,...,k and j=1,...,mhigh ('=' nderiv). in particular, the c first column of dbiatx is already in final form. to obtain cor- c responding derivatives of b-splines in subsequent columns, gene- c rate their b-repr. by differencing, then evaluate at x. c jlow = 1 do 20 i=1,k do 19 j=jlow,k 19 a(j,i) = 0. jlow = i 20 a(i,i) = 1. c at this point, a(.,j) contains the b-coeffs for the j-th of the c k b-splines of interest here. c do 40 m=2,mhigh kp1mm = kp1 - m fkp1mm = float(kp1mm) il = left i = k c c for j=1,...,k, construct b-coeffs of (m-1)st derivative of c b-splines from those for preceding derivative by differencing c and store again in a(.,j) . the fact that a(i,j) = 0 for c i .lt. j is used. do 25 ldummy=1,kp1mm factor = fkp1mm/(t(il+kp1mm) - t(il)) c the assumption that t(left).lt.t(left+1) makes denominator c in factor nonzero. do 24 j=1,i 24 a(i,j) = (a(i,j) - a(i-1,j))*factor il = il - 1 25 i = i - 1 c c for i=1,...,k, combine b-coeffs a(.,i) with b-spline values c stored in dbiatx(.,m) to get value of (m-1)st derivative of c i-th b-spline (of interest here) at x , and store in c dbiatx(i,m). storage of this value over the value of a b-spline c of order m there is safe since the remaining b-spline derivat- c ives of the same order do not use this value due to the fact c that a(j,i) = 0 for j .lt. i . do 40 i=1,k sum = 0. jlow = max0(i,m) do 35 j=jlow,k 35 sum = a(j,i)*dbiatx(j,m) + sum 40 dbiatx(i,m) = sum 99 return end