Unformatted text preview: (c) The cubic spline through the four points will have three segments, each different cubic polynomials and in general not the same as the interpolating cubic polynomial through the four points. With notaknot end conditions, the first and second and second and third segments will have the same cubic polynomial, so all three segments will have a single cubic polynomial. Because this polynomial interpolates all four points and the interpolating cubic polynomial is unique, this must be the same as the interpolating polynomial. 3] (a) f(0) = 0, f(2.5) = 0.794, f(5) = 0.893, f(7.5) = 0.935, f(10) = 0.958 composite trapezoid rule with 2 segments: T 2 = 10 * (f(0) + 2f(5) + f(10)) / 4 = 6.86 composite trapezoid rule with 4 segments: T 4 = 10 * (f(0) + 2f(2.5) + 2f(5) + 2f(7.5) + f(10)) / 8 = 7.75 (b) We can combine T 2 and T 4 to cancel out the lowestorder error term (Romberg integration): T = (4T 4  T 2 )/3 = 8.05...
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 Spring '11
 Lybas
 Polynomials, Polynomial interpolation, Romberg's method

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