bv_cvxbook_extra_exercises

Bv_cvxbook_extra_exercises

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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 following properties: • 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.

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