| double precision function bvalue(t,bcoef,n,k,x,jderiv) |
| |
| 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. |
| c |
| C calls interv() (from ../../../appl/interv.c ) |
| 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 s(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 <= kmax (=20) is imposed |
| c arbitrarily by the dimension statement for aj, dm, dm below, |
| c but is n o w h e r e c h e c k e d for. |
| c however in R, this is only called from bvalus() with k=4 anyway! |
| 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 > 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 |
| c be the desired number (d^j)f(x). (see x.(17)-(19) of text). |
| c |
| C Arguments |
| integer n,k, jderiv |
| DOUBLE precision t(*),bcoef(n),x |
| c dimension t(n+k) |
| c current fortran standard makes it impossible to specify the length of |
| c t precisely without the introduction of otherwise superfluous |
| c additional arguments. |
| |
| C Local Variables |
| integer kmax |
| parameter(kmax = 20) |
| |
| DOUBLE precision aj(kmax),dm(kmax),dp(kmax),fkmj |
| |
| integer i,ilo,imk,j,jc,jcmin,jcmax,jj,km1,kmj,mflag,nmi, jdrvp1 |
| c |
| integer interv |
| external interv |
| |
| c initialize |
| data i/1/ |
| |
| bvalue = 0.d0 |
| if (jderiv .ge. k) go to 99 |
| c |
| c *** find i s.t. 1 <= i < n+k and t(i) < t(i+1) and |
| c t(i) <= x < t(i+1) . if no such i can be found, x lies |
| c outside the support of the spline f and bvalue = 0. |
| c {this case is handled in the calling R code} |
| c (the asymmetry in this choice of i makes f rightcontinuous) |
| if( (x.ne.t(n+1)) .or. (t(n+1).ne.t(n+k)) ) then |
| i = interv ( t, n+k, x, 0, 0, i, mflag) |
| if (mflag .ne. 0) then |
| call rwarn('bvalue() mflag != 0: should never happen!') |
| go to 99 |
| endif |
| else |
| i = n |
| endif |
| |
| c *** if k = 1 (and jderiv = 0), bvalue = bcoef(i). |
| km1 = k - 1 |
| if (km1 .le. 0) then |
| bvalue = bcoef(i) |
| go to 99 |
| endif |
| 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 dm(j) = x - t(i+1-j), |
| c dp(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. |
| jcmin = 1 |
| imk = i - k |
| if (imk .ge. 0) then |
| do 9 j=1,km1 |
| dm(j) = x - t(i+1-j) |
| 9 continue |
| else |
| jcmin = 1 - imk |
| do 5 j=1,i |
| dm(j) = x - t(i+1-j) |
| 5 continue |
| do 6 j=i,km1 |
| aj(k-j) = 0.d0 |
| dm(j) = dm(i) |
| 6 continue |
| endif |
| c |
| jcmax = k |
| nmi = n - i |
| if (nmi .ge. 0) then |
| do 19 j=1,km1 |
| C the following if() happens; e.g. in pp <- predict(cars.spl, xx) |
| c - if( (i+j) .gt. lent) write(6,9911) i+j,lent |
| c - 9911 format(' i+j, lent ',2(i6,1x)) |
| dp(j) = t(i+j) - x |
| 19 continue |
| else |
| jcmax = k + nmi |
| do 15 j=1,jcmax |
| dp(j) = t(i+j) - x |
| 15 continue |
| do 16 j=jcmax,km1 |
| aj(j+1) = 0.d0 |
| dp(j) = dp(jcmax) |
| 16 continue |
| endif |
| |
| c |
| do 21 jc=jcmin,jcmax |
| aj(jc) = bcoef(imk + jc) |
| 21 continue |
| c |
| c *** difference the coefficients jderiv times. |
| if (jderiv .ge. 1) then |
| do 23 j=1,jderiv |
| kmj = k-j |
| fkmj = dble(kmj) |
| ilo = kmj |
| do 24 jj=1,kmj |
| aj(jj) = ((aj(jj+1) - aj(jj))/(dm(ilo) + dp(jj)))*fkmj |
| ilo = ilo - 1 |
| 24 continue |
| 23 continue |
| endif |
| |
| 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). |
| |
| if (jderiv .ne. km1) then |
| jdrvp1 = jderiv + 1 |
| do 33 j=jdrvp1,km1 |
| kmj = k-j |
| ilo = kmj |
| do 34 jj=1,kmj |
| aj(jj) = (aj(jj+1)*dm(ilo) + aj(jj)*dp(jj)) / |
| * (dm(ilo)+dp(jj)) |
| ilo = ilo - 1 |
| 34 continue |
| 33 continue |
| endif |
| |
| bvalue = aj(1) |
| c |
| 99 return |
| end |
