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In approximation problems with splines it is convenient to parametrize a spline as a linear combination of basis functions, called B-splines. The precise deﬁnition of B-splines is not important for
our purposes; it is suﬃcient to know that every cubic spline can be written as a linear combination
of M + 3 cubic B-splines gk (t), i.e., in the form
f (t) = x1 g1 (t) + · · · + xM +3 gM +3 (t) = xT g (t),
and that there exist eﬃcient algorithms for computing g (t) = (g1 (t), . . . , gM +3 (t)). The next ﬁgure
shows the 13 B-splines for the breakpoints 0, 1, . . . , 10.
44 1 0.8 g k ( t) 0.6 0.4 0.2 0
0 2 4 6 8 10 t
In this exercise we study the problem of ﬁtting a cubic spline to a set of data points, subject to the
constraint that the spline is a convex function. Speciﬁcally, the breakpoints α0 , . . . , αM are ﬁxed,
and we are given N data points (tk , yk ) with tk ∈ [α0 , αM ]. We are asked to ﬁnd the convex cubic
spline f (t) that minimizes the least-square...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
- Fall '13
- The Aeneid