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Unformatted text preview: 1, 2, and 3 internal knot points, evenly spaced in [0, 1]. (For example, for 3 internal
knot points we have a0 = 0, a1 = 0.25, a2 = 0.50, a3 = 0.75, a4 = 1.) Give the least-squares
ﬁtting cost for each one. Plot the data and the piecewise-linear ﬁts found. Express each function
in the form
f (x) = max (αi x + βi ).
i=1...,K (In this form the function is easily incorporated into an optimization problem.)
5.8 Least-squares ﬁtting with convex splines. A cubic spline (or fourth-order spline ) with breakpoints
α0 , α1 , . . . , αM (that satisfy α0 < α1 < · · · < αM ) is a piecewise-polynomial function with the
• the function is a cubic polynomial on each interval [αi , αi+1 ] • the function values, and the ﬁrst and second derivatives are continuous on the interval (α0 , αM ).
The ﬁgure shows an example of a cubic spline f (t) with M = 10 segments and breakpoints α0 = 0,
α1 = 1, . . . , α10 = 10.
10 f ( t) 5 0 −5 −10
0 2 4 6 8 1...
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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