| 1 | module modbvalue |
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| 2 | contains |
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| 3 | real function bvalue ( t, bcoef, n, k, x, jderiv ) |
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| 4 | & bind(C,name='bvalue') |
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| 5 | USE ISO_C_BINDING |
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| 6 | !DEC$ ATTRIBUTES DLLEXPORT::bvalue |
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| 7 | c from * a practical guide to splines * by c. de boor |
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| 8 | calls interv |
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| 9 | c |
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| 10 | calculates value at x of jderiv-th derivative of spline from b-repr. |
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| 11 | c the spline is taken to be continuous from the right, EXCEPT at the |
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| 12 | c rightmost knot, where it is taken to be continuous from the left. |
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| 13 | c |
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| 14 | c****** i n p u t ****** |
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| 15 | c t, bcoef, n, k......forms the b-representation of the spline f to |
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| 16 | c be evaluated. specifically, |
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| 17 | c t.....knot sequence, of length n+k, assumed nondecreasing. |
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| 18 | c bcoef.....b-coefficient sequence, of length n . |
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| 19 | c n.....length of bcoef and dimension of spline(k,t), |
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| 20 | c a s s u m e d positive . |
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| 21 | c k.....order of the spline . |
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| 22 | c |
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| 23 | c w a r n i n g . . . the restriction k .le. kmax (=20) is imposed |
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| 24 | c arbitrarily by the dimension statement for aj, dl, dr below, |
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| 25 | c but is n o w h e r e c h e c k e d for. |
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| 26 | c |
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| 27 | c x.....the point at which to evaluate . |
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| 28 | c jderiv.....integer giving the order of the derivative to be evaluated |
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| 29 | c a s s u m e d to be zero or positive. |
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| 30 | c |
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| 31 | c****** o u t p u t ****** |
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| 32 | c bvalue.....the value of the (jderiv)-th derivative of f at x . |
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| 33 | c |
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| 34 | c****** m e t h o d ****** |
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| 35 | c The nontrivial knot interval (t(i),t(i+1)) containing x is lo- |
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| 36 | c cated with the aid of interv . The k b-coeffs of f relevant for |
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| 37 | c this interval are then obtained from bcoef (or taken to be zero if |
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| 38 | c not explicitly available) and are then differenced jderiv times to |
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| 39 | c obtain the b-coeffs of (d**jderiv)f relevant for that interval. |
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| 40 | c Precisely, with j = jderiv, we have from x.(12) of the text that |
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| 41 | c |
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| 42 | c (d**j)f = sum ( bcoef(.,j)*b(.,k-j,t) ) |
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| 43 | c |
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| 44 | c where |
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| 45 | c / bcoef(.), , j .eq. 0 |
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| 46 | c / |
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| 47 | c bcoef(.,j) = / bcoef(.,j-1) - bcoef(.-1,j-1) |
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| 48 | c / ----------------------------- , j .gt. 0 |
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| 49 | c / (t(.+k-j) - t(.))/(k-j) |
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| 50 | c |
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| 51 | c Then, we use repeatedly the fact that |
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| 52 | c |
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| 53 | c sum ( a(.)*b(.,m,t)(x) ) = sum ( a(.,x)*b(.,m-1,t)(x) ) |
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| 54 | c with |
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| 55 | c (x - t(.))*a(.) + (t(.+m-1) - x)*a(.-1) |
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| 56 | c a(.,x) = --------------------------------------- |
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| 57 | c (x - t(.)) + (t(.+m-1) - x) |
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| 58 | c |
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| 59 | c to write (d**j)f(x) eventually as a linear combination of b-splines |
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| 60 | c of order 1 , and the coefficient for b(i,1,t)(x) must then be the |
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| 61 | c desired number (d**j)f(x). (see x.(17)-(19) of text). |
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| 62 | c |
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| 63 | integer jderiv,k,n, i,ilo,imk,j,jc,jcmin,jcmax,jj,kmax,kmj,km1 |
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| 64 | * ,mflag,nmi,jdrvp1 |
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| 65 | parameter (kmax = 20) |
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| 66 | real bcoef(n),t(n+k),x, aj(kmax),dl(kmax),dr(kmax),fkmj |
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| 67 | bvalue = 0. |
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| 68 | if (jderiv .ge. k) go to 99 |
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| 69 | c |
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| 70 | c *** Find i s.t. 1 .le. i .lt. n+k and t(i) .lt. t(i+1) and |
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| 71 | c t(i) .le. x .lt. t(i+1) . If no such i can be found, x lies |
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| 72 | c outside the support of the spline f , hence bvalue = 0. |
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| 73 | c (The asymmetry in this choice of i makes f rightcontinuous, except |
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| 74 | c at t(n+k) where it is leftcontinuous.) |
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| 75 | call interv ( t, n+k, x, i, mflag ) |
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| 76 | if (mflag .ne. 0) go to 99 |
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| 77 | c *** if k = 1 (and jderiv = 0), bvalue = bcoef(i). |
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| 78 | km1 = k - 1 |
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| 79 | if (km1 .gt. 0) go to 1 |
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| 80 | bvalue = bcoef(i) |
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| 81 | go to 99 |
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| 82 | c |
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| 83 | c *** store the k b-spline coefficients relevant for the knot interval |
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| 84 | c (t(i),t(i+1)) in aj(1),...,aj(k) and compute dl(j) = x - t(i+1-j), |
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| 85 | c dr(j) = t(i+j) - x, j=1,...,k-1 . set any of the aj not obtainable |
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| 86 | c from input to zero. set any t.s not obtainable equal to t(1) or |
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| 87 | c to t(n+k) appropriately. |
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| 88 | 1 jcmin = 1 |
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| 89 | imk = i - k |
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| 90 | if (imk .ge. 0) go to 8 |
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| 91 | jcmin = 1 - imk |
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| 92 | do 5 j=1,i |
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| 93 | 5 dl(j) = x - t(i+1-j) |
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| 94 | do 6 j=i,km1 |
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| 95 | aj(k-j) = 0. |
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| 96 | 6 dl(j) = dl(i) |
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| 97 | go to 10 |
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| 98 | 8 do 9 j=1,km1 |
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| 99 | 9 dl(j) = x - t(i+1-j) |
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| 100 | c |
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| 101 | 10 jcmax = k |
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| 102 | nmi = n - i |
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| 103 | if (nmi .ge. 0) go to 18 |
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| 104 | jcmax = k + nmi |
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| 105 | do 15 j=1,jcmax |
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| 106 | 15 dr(j) = t(i+j) - x |
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| 107 | do 16 j=jcmax,km1 |
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| 108 | aj(j+1) = 0. |
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| 109 | 16 dr(j) = dr(jcmax) |
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| 110 | go to 20 |
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| 111 | 18 do 19 j=1,km1 |
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| 112 | 19 dr(j) = t(i+j) - x |
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| 113 | c |
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| 114 | 20 do 21 jc=jcmin,jcmax |
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| 115 | 21 aj(jc) = bcoef(imk + jc) |
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| 116 | c |
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| 117 | c *** difference the coefficients jderiv times. |
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| 118 | if (jderiv .eq. 0) go to 30 |
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| 119 | do 23 j=1,jderiv |
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| 120 | kmj = k-j |
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| 121 | fkmj = float(kmj) |
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| 122 | ilo = kmj |
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| 123 | do 23 jj=1,kmj |
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| 124 | aj(jj) = ((aj(jj+1) - aj(jj))/(dl(ilo) + dr(jj)))*fkmj |
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| 125 | 23 ilo = ilo - 1 |
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| 126 | c |
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| 127 | c *** compute value at x in (t(i),t(i+1)) of jderiv-th derivative, |
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| 128 | c given its relevant b-spline coeffs in aj(1),...,aj(k-jderiv). |
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| 129 | 30 if (jderiv .eq. km1) go to 39 |
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| 130 | jdrvp1 = jderiv + 1 |
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| 131 | do 33 j=jdrvp1,km1 |
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| 132 | kmj = k-j |
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| 133 | ilo = kmj |
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| 134 | do 33 jj=1,kmj |
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| 135 | aj(jj) = (aj(jj+1)*dl(ilo) + aj(jj)*dr(jj))/(dl(ilo)+dr(jj)) |
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| 136 | 33 ilo = ilo - 1 |
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| 137 | 39 bvalue = aj(1) |
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| 138 | c |
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| 139 | 99 return |
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| 140 | end |
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| 141 | end |
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