Question

We want to produce an evenly spaced table of values for the function f(x) sin(x) for x E [0,Tt/2] such that, with cubic interpolation, we can give the values of the function at any point in the interval with an error less than 5 10-12. That means finding a number n such that with h = π/2n and Xk-kh, k-0, , n the cubic interpolation polynomial with the interpolation points XK-1,XK, X+1 XK+2 for x has an error less than the number given above for x E [xk,x+1 Hint: One can show by translation that xe[xk,xk+1 3

Answer 1

Given that we want to produce the spaced table values for the function f (x) = sin x elementof [0, pi/2] The values of function at any point in the interval with an error less than 5 times 10^- 12 h = pi/2 n x_k = k h, k = 0, . . . ., n firstly we do this for general function 'f' by using Taylor series. let P (x) be the linear interpolating polynomial for a continues twice differential function 'f' in the interval [x_-k - 1, x_0] Than p (x) is simply the straight line passing through (x_k, f (x_k), (x_k - 1, f (x_k - 1)) P (x) = f (x_k - 1) + f (x_k) - f (x-k - 1)/x_k - x_k - 1 (x - x_k - 1) \r\n in the limiting case, that is when, x_k rightarrow x-k - 1, 'P' is just P (x)