Transcription of Unit 5: Cubic Splines - LTH
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Unit 5: Cubic Splines - LTH
unit 5: Cubic SplinesLetK={x0, .. , xm}be a set of given knots witha=x0< x1< < xm=bDefinition. [ ]A functions C2[a, b]is called acubic splineon[a, b],ifsis a Cubic polynomialsiin each interval[xi, xi+1].It is called acubic interpolating splineifs(xi) =yifor given F uhrer: : Cubic SplinesInterpolating Cubic Splines need two additional conditions to be uniquelydefinedDefinition. [ ]An Cubic interpolatory spilnesis called anatural splineifs (x0) =s (xm) = 0C. F uhrer: : Cubic Splines - ConstructionWe construct an interpolating in a different but equivalent way than in thetextbook:Ansatz formthe piecewise polynomialssi(x) =ai(x xi)3+bi(x xi)2+ci(x xi) +diBy fixing the4mfree coefficientsai, bi, ci, di, i= 0 :m 1the entire splineis F uhrer: : Cubic Splines -ConstructionWe need4mconditions to fix the coefficients(1)si(xi) =yi,fori= 0 :m 1,(2)sm 1=ym,1condition(3)si(xi+1)
Unit 5: Cubic Splines Let K = {x 0,...,x m} be a set of given knots with a = x 0 < x 1 < ··· < x m = b Definition. [11.2] A function s ∈ C2[a,b] is called a cubic spline on [a,b], if s is a cubic polynomial s i in each interval [x i,x i+1]. It is called a cubic interpolating spline if s(x
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