Question

1. Suppose that we would like to approximate Sof(x)dx by QU) = 0 P2(x)dx, (1) where P2(x) is the polynomial of degree at most two which interpolates f at 0, 1/2, and 1. (a) Write P2(x) in Lagrange form and prove that Q[F] o [s0 f(0) + 4f 45 (2) +scn)] (2) (b) Consider now a general interval [a, b] and the integral só f(x)dx. Do the change of variables x = a + (b − a)t to transform the integral to one in [0, 1] and use (2) to obtain the more general quadrature (Simpson's Rule): Qºln=66 * [s(a) + 45 () +50) (3) (c) Explain why the quadrature is exact, i.e. QS[f] = [[f] when f is a polynomial of degree 2 or less and prove that it is actually exact for polynomials of degree 3 or less. (d) If we use an interpolating polynomial of much higher degree than 2 and equidis- tributed nodes, does the accuracy of the quadrature improve for a general contin- uous integrand? Explain.

Answer 1

al can P2 (x) is lagrange. form be written as P20) (x-x.) (x - 12) f(x0) + (x-360) (0C-X2) f(x) (xo-x)(X-X2) (20.-26) (x,-362) + (x-xo) (x-x,) f (x2) (x2-Xo) (7. - X) sc, = x₂ = 1 P2 (22) = (x-1/2) (x-1) flo) + (x-6) (x-1) f (2) + Co-Y2) Co- (Y2-0 (1/2) 2020, リュ + (oc-o) (x - / f (1) (-o) (1-₂) (xx²-3/22+4) flot (x2-x) (12) + (x²+4) f (1) 1/ J.P 2 (x) dx = ('[2(x²-322 + 4) fro) - 4(x** *[/]+2(32-42) fco)] = [26%*- 99**%.] flv) - 4(- x)a) f(x)+2(03-2) 2(/-3/4+1/2) f (o)-4 (1/3-1/2) + 2(2-14) f) I flo) + 3 f (2) + Įf (1) & Polasche % [fco + 4 f(1/2) + f(1)] b) Transtorm So Pe (se) doe to integral P2 (2) dx we make substitution na a4c6-ast >= a t=o x=6 => t=1

So P2 ( x ) abc = Sla[a + (6-0) [] Coral dt og 2₂ = b. se cora, za 20, = a +6 > 2 a’() = 6-a [fcal + 4f (&#6) + f(b)] 2 Let f be polynomial of degre it (is) f(x) = co +52 + (z 22 ) So foxo dx = so Cco + Cisc + case) cse (Cox+ (x² + ( ₂x½ 133.0 Coli (10 6 + C, 67 (10 6 + C, 1/2 + aby - coa - 4 Coa- c. q² - (₂ 93 teht - Co (6-4) + 6 (6²-42) + C₂ 163-2²) so feed doc = (6-a) [lot Ci (6+9) + Ce (6²+ ab + a²) 2 3 2 3 2 Coi By qurdzatie termula s fro) dx = (6-0) (Cot Categ²+66 +0,6+ (26²7 + 9(6.64044C) ( + 6

Date 6 (6-a) [olo + c [a + 6 + 2(a+b)] + Ca[a2+6+ catw?]] (b-a2 [6 Co + 3 ca+676 + (2a2+262+206)Ce I So f(x) doc (6-a) [ Co + Cyp. (670) + Czy (62+ab)] un Value of so how dx is some by direct integration and by quadraturze method. So durdsature is eat exact when it is polynomial of degree 2 less. We can prove it for f when I s polynomial of degree 3 by d> It we use polynomial of degree hyher than 2, the accuracy will not improve same way.