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Unformatted text preview: 0 t 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.

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