Construct a sequence of interpolating values yn to f (1 + 10), where f (x) = (1 + x2)1

Chapter 3, Problem 11

(choose chapter or problem)

Construct a sequence of interpolating values yn to f (1 + 10), where f (x) = (1 + x2)1 for 5 x 5, as follows: For each n = 1, 2, ... , 10, let h = 10/n and yn = Pn(1+ 10), where Pn(x) is the interpolating polynomial for f (x) at the nodes x (n) 0 , x (n) 1 , ... , x(n) n and x (n) j = 5 + jh, for each j = 0, 1, 2, ... , n. Does the sequence {yn} appear to converge to f (1 + 10)? Inverse Interpolation Suppose f C1[a, b], f (x) = 0 on [a, b] and f has one zero p in [a, b]. Let x0, ... , xn, be n + 1 distinct numbers in [a, b] with f (xk ) = yk , for each k = 0, 1, ... , n. To approximate p construct the interpolating polynomial of degree n on the nodes y0, ... , yn for f 1. Since yk = f (xk ) and 0 = f (p), it follows that f 1(yk ) = xk and p = f 1(0). Using iterated interpolation to approximate f 1(0) is called iterated inverse interpolation.