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calls |
interv
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calculates value at | | |
x of jderiv-th derivative of spline from b-repr.
the spline is taken to be continuous from the right, EXCEPT at the
rightmost knot, where it is taken to be continuous from the left.
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****** |
i n p u t ******
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t, bcoef, n, k......forms the b-representation of the spline | | |
f to
be evaluated. specifically,
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t.....knot sequence, of length | | |
n+k, assumed nondecreasing.
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bcoef.....b-coefficient sequence, of length | | |
n .
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n.....length of | | |
bcoef and dimension of spline(k,t),
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a s s u m e d | | |
positive .
k.....order of the spline .
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w a r n i n g . . . | | |
the restriction k .le. kmax (=20) is imposed
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arbitrarily by the dimension statement for | | |
aj, dl, dr below,
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but is |
n o w h e r e c h e c k e d for.
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x.....the point at which to evaluate .
jderiv.....integer giving the order of the derivative to be evaluated
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a s s u m e d | | |
to be zero or positive.
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****** |
o u t p u t ******
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bvalue.....the value of the (jderiv)-th derivative of | | |
f at x .
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****** |
m e t h o d ******
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The nontrivial knot interval | | |
(t(i),t(i+1)) containing x is lo-
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cated with the aid of | | |
interv . The k b-coeffs of f relevant for
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this interval are then obtained from | | |
bcoef (or taken to be zero if
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not explicitly available) and are then differenced | | |
jderiv times to
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obtain the b-coeffs of | | |
(d**jderiv)f relevant for that interval.
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Precisely, with | | |
j = jderiv, we have from x.(12) of the text that
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(d**j)f = sum ( bcoef(.,j)*b(.,k-j,t) )
where
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bcoef(.,j) | | |
= / bcoef(.,j-1) - bcoef(.-1,j-1)
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/ ----------------------------- , | | |
j .gt. 0
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/ |
(t(.+k-j) - t(.))/(k-j)
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Then, we use repeatedly the fact that
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sum ( a(.)*b(.,m,t)(x) ) | | |
= sum ( a(.,x)*b(.,m-1,t)(x) )
with
(x - t(.))*a(.) + (t(.+m-1) - x)*a(.-1)
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a(.,x) |
= ---------------------------------------
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(x - t(.)) | | |
+ (t(.+m-1) - x)
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to write | | |
(d**j)f(x) eventually as a linear combination of b-splines
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of order | | |
1 , and the coefficient for b(i,1,t)(x) must then be the
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desired number | | |
(d**j)f(x). (see x.(17)-(19) of text).
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