(a)

(b)

Thank you!

Consider computing the fitting coefficients (a, b, c) for the following formula. f (x) = a + bx + Derive a matrix system [A] {:} = {r} for a dataset (l'i, f (xi)) for i = 1~5 using the following two methods. (a) A regular method in page 5 of Interpolation" handout (b) An efficient method in page 8 of "Interpolation handout 1 5 2. 3 4. 2.8 3.6 4.5 y 2.2. 5.5 Interpolating polynomials fn(x) = do + 2 x + ... + anxa from n+1 data fi(x) = ao + a1x f2(x) = ao + a1x + azx f3(x) f(x) + f(x) = Inx J. :) 2 2 f(x) - Inx- 2 f) - In True value 1 True value True value Cubic estimate Linear estimates Quadratic estimate Linear estimate 0 0 0 0 0 5 5 Note: Higher-order polynomials have better accuracy, but ... For numerical implementation, the higher-order polynomials are complicated . For some cases, the higher-order polynomials are not good. Spline interpolation 2. Quadratic splines A function between two neighboring points is quadratic ft(x) = ax2 + bx + Xi-13n equations f(x) -> n+1 equations Interval 1. Interval 2 Interval 3 fi(xi) = given II fi(xi) = fi+1(xi) - >n-1 equations Xo i = 0 21 i = 1 X2 i = 2 13 i = 3 fi'(xi) = fi+1(xi) ->n-1 equations Ex: 9 unknowns f"(x) = 0 a1 = 0 fi(xi) i = 0,1,2,3 fi(xi) = fi+1(xi) i = 1,2 f'(xi) = fi+1(xi) i = 1,2 f"(x) = 0 Consider computing the fitting coefficients (a, b, c) for the following formula. f (x) = a + bx + Derive a matrix system [A] {:} = {r} for a dataset (l'i, f (xi)) for i = 1~5 using the following two methods. (a) A regular method in page 5 of Interpolation" handout (b) An efficient method in page 8 of "Interpolation handout 1 5 2. 3 4. 2.8 3.6 4.5 y 2.2. 5.5 Interpolating polynomials fn(x) = do + 2 x + ... + anxa from n+1 data fi(x) = ao + a1x f2(x) = ao + a1x + azx f3(x) f(x) + f(x) = Inx J. :) 2 2 f(x) - Inx- 2 f) - In True value 1 True value True value Cubic estimate Linear estimates Quadratic estimate Linear estimate 0 0 0 0 0 5 5 Note: Higher-order polynomials have better accuracy, but ... For numerical implementation, the higher-order polynomials are complicated . For some cases, the higher-order polynomials are not good. Spline interpolation 2. Quadratic splines A function between two neighboring points is quadratic ft(x) = ax2 + bx + Xi-13n equations f(x) -> n+1 equations Interval 1. Interval 2 Interval 3 fi(xi) = given II fi(xi) = fi+1(xi) - >n-1 equations Xo i = 0 21 i = 1 X2 i = 2 13 i = 3 fi'(xi) = fi+1(xi) ->n-1 equations Ex: 9 unknowns f"(x) = 0 a1 = 0 fi(xi) i = 0,1,2,3 fi(xi) = fi+1(xi) i = 1,2 f'(xi) = fi+1(xi) i = 1,2 f"(x) = 0