Exercises 21.4 Homework
1.
(Theoretical) Write down explicitly the Newton polynomial that interpolates the points \((-1, u)\text{,}\) \((0, v)\text{,}\) \((1, w)\text{.}\) Its coefficients \(c_1, c_2, c_3\) will depend on \(u, v, w\text{.}\)
2.
(Theoretical) Integrate the polynomial found in Exercise 21.4.1 over the interval \([-1, 1]\text{.}\) (It is easier to integrate its general form \(c_1 + c_2(x+1) + c_3(x+1)x\) first and then plug in the formulas for coefficients.) Explain how this is related to Simpson's integration rule.
3.
Use either Lagrange method or Newton method to interpolate the points \((k, \cos k)\text{,}\) where \(k=1, \dots, 15\text{.}\) Then do the same with the points \((k, \cos 2k)\text{,}\) where \(k=1, \dots, 15\text{.}\) Plot both side by side using subplot. Do you observe a difference in the quality of interpolating curves?