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c parameter(nn=20,kk=4)
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c real ta(nn+kk),bcoef(nn)
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c k=kk
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c n=nn
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c do i=1,n+k
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c ta(i)=float(i)
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c end do
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c do i=1,n
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c bcoef(i)=float(i)
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c end do
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c do i=3,n+1
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c write(*,*) ta(i)+0.5, bvalue (ta, bcoef, n, k, ta(i)+0.5, 0 )
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c end do
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c stop
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c end
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real function bvalue ( t, bcoef, n, k, x, jderiv )
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c from * a practical guide to splines * by c. de boor
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calls interv
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c
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calculates value at x of jderiv-th derivative of spline from b-repr.
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c the spline is taken to be continuous from the right, EXCEPT at the
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c rightmost knot, where it is taken to be continuous from the left.
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c
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c****** i n p u t ******
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c t, bcoef, n, k......forms the b-representation of the spline f to
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c be evaluated. specifically,
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c t.....knot sequence, of length n+k, assumed nondecreasing.
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c bcoef.....b-coefficient sequence, of length n .
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c n.....length of bcoef and dimension of spline(k,t),
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c a s s u m e d positive .
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c k.....order of the spline .
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c
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c w a r n i n g . . . the restriction k .le. kmax (=20) is imposed
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c arbitrarily by the dimension statement for aj, dl, dr below,
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c but is n o w h e r e c h e c k e d for.
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c
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c x.....the point at which to evaluate .
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c jderiv.....integer giving the order of the derivative to be evaluated
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c a s s u m e d to be zero or positive.
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c
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c****** o u t p u t ******
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c bvalue.....the value of the (jderiv)-th derivative of f at x .
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c
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c****** m e t h o d ******
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c The nontrivial knot interval (t(i),t(i+1)) containing x is lo-
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c cated with the aid of interv . The k b-coeffs of f relevant for
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c this interval are then obtained from bcoef (or taken to be zero if
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c not explicitly available) and are then differenced jderiv times to
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c obtain the b-coeffs of (d**jderiv)f relevant for that interval.
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c Precisely, with j = jderiv, we have from x.(12) of the text that
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c
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c (d**j)f = sum ( bcoef(.,j)*b(.,k-j,t) )
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c
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c where
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c / bcoef(.), , j .eq. 0
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c /
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c bcoef(.,j) = / bcoef(.,j-1) - bcoef(.-1,j-1)
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c / ----------------------------- , j .gt. 0
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c / (t(.+k-j) - t(.))/(k-j)
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c
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c Then, we use repeatedly the fact that
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c
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c sum ( a(.)*b(.,m,t)(x) ) = sum ( a(.,x)*b(.,m-1,t)(x) )
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c with
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c (x - t(.))*a(.) + (t(.+m-1) - x)*a(.-1)
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c a(.,x) = ---------------------------------------
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c (x - t(.)) + (t(.+m-1) - x)
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c
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c to write (d**j)f(x) eventually as a linear combination of b-splines
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c of order 1 , and the coefficient for b(i,1,t)(x) must then be the
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c desired number (d**j)f(x). (see x.(17)-(19) of text).
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c
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parameter (kmax = 20)
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integer jderiv,k,n,i,ilo,imk,j,jc,jcmin,jcmax,jj,
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c * kmax,
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* kmj,km1
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* ,mflag,nmi,jdrvp1
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C real bcoef(n),t(1),x, aj(20),dl(20),dr(20),fkmj
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real bcoef(n),x, aj(kmax),dl(kmax),dr(kmax),fkmj
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dimension t(n+k)
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c former fortran standard made it impossible to specify the length of t
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c precisely without the introduction of otherwise superfluous addition-
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c al arguments.
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bvalue = 0.
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if (jderiv .ge. k) go to 99
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c
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c *** Find i s.t. 1 .le. i .lt. n+k and t(i) .lt. t(i+1) and
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c t(i) .le. x .lt. t(i+1) . If no such i can be found, x lies
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c outside the support of the spline f , hence bvalue = 0.
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c (The asymmetry in this choice of i makes f rightcontinuous, except
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c at t(n+k) where it is leftcontinuous.)
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call interv ( t, n+k, x, i, mflag )
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if (mflag .ne. 0) go to 99
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c *** if k = 1 (and jderiv = 0), bvalue = bcoef(i).
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km1 = k - 1
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if (km1 .gt. 0) go to 1
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bvalue = bcoef(i)
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go to 99
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c
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c *** store the k b-spline coefficients relevant for the knot interval
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c (t(i),t(i+1)) in aj(1),...,aj(k) and compute dl(j) = x - t(i+1-j),
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c dr(j) = t(i+j) - x, j=1,...,k-1 . set any of the aj not obtainable
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c from input to zero. set any t.s not obtainable equal to t(1) or
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c to t(n+k) appropriately.
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1 jcmin = 1
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imk = i - k
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if (imk .ge. 0) go to 8
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jcmin = 1 - imk
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do 5 j=1,i
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5 dl(j) = x - t(i+1-j)
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do 6 j=i,km1
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aj(k-j) = 0.
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6 dl(j) = dl(i)
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go to 10
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8 do 9 j=1,km1
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9 dl(j) = x - t(i+1-j)
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c
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10 jcmax = k
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nmi = n - i
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if (nmi .ge. 0) go to 18
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jcmax = k + nmi
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do 15 j=1,jcmax
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15 dr(j) = t(i+j) - x
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do 16 j=jcmax,km1
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aj(j+1) = 0.
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16 dr(j) = dr(jcmax)
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go to 20
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18 do 19 j=1,km1
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19 dr(j) = t(i+j) - x
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c
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20 do 21 jc=jcmin,jcmax
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21 aj(jc) = bcoef(imk + jc)
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c
|
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c *** difference the coefficients jderiv times.
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if (jderiv .eq. 0) go to 30
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do 23 j=1,jderiv
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|
kmj = k-j
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|
fkmj = float(kmj)
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|
ilo = kmj
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|
do 23 jj=1,kmj
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|
aj(jj) = ((aj(jj+1) - aj(jj))/(dl(ilo) + dr(jj)))*fkmj
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|
23 ilo = ilo - 1
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|
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).
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|
|
30 if (jderiv .eq. km1) go to 39
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|
jdrvp1 = jderiv + 1
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|
|
do 33 j=jdrvp1,km1
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|
kmj = k-j
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|
|
ilo = kmj
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|
do 33 jj=1,kmj
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|
|
aj(jj) = (aj(jj+1)*dl(ilo) + aj(jj)*dr(jj))/(dl(ilo)+dr(jj))
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|
|
33 ilo = ilo - 1
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39 bvalue = aj(1)
|
|
|
c
|
|
|
99 return
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|
|
end
|
|
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|
