7.1.bSuppose we evaluatefn(x) at thexito obtain the datafn(x0), . . . , fn(xn).We can thenwritefn(x) in its Lagrange form,fn(x) =Ln(x) =fn(x0)ℓ0(x) +. . .+fn(xn)ℓn(x)Since theℓ0(x), . . . , ℓn(x) are an orthogonal basis forPn, they also can be used to compute,fn(x), the unique least squares approximation tof(x). As with the Legendre polynomials,using the generalized Fourier series, yieldsfn(x) =σ0ℓ0(x) +σ1ℓ1(x) +· · ·+σnℓn(x) whereσi=(f, ℓi)(ℓi, ℓi)Show that these last two forms offn(x) give the same polynomial by showing thatσi=(f, ℓi)(ℓi, ℓi)=fn(xi)Hint:Consider the relationship betweenf(x) andfn(x). 2
Problem 7.2Considerf(x) =exon the interval−1≤x≤1. Suppose we want to approximatef(x) witha polynomial. Generate the following polynomials:(a)F1(x) andF3(x):the first and third order Taylor series approximations off(x) expandedaboutx= 0.(b)N1(x): the linear near-minimax approximation tof(x) on the interval.(c)C1(x) andC2(x) – the linear and quadratic polynomials that result from Chebysheveconomization applied toF3(x), the third order Taylor series approximation off(x)expanded aboutx= 0.(d)p1(x) andp2(x) – the linear and quadratic polynomials that result from Legendreeconomization applied toF3(x
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