Polynomial interpolation consists of determining the unique (n 1)th-order polynomial

Chapter 11, Problem 11.14

(choose chapter or problem)

Polynomial interpolation consists of determining the unique (n 1)th-order polynomial that fits n data points. Such polynomials have the general form, f (x) = p1xn1 + p2xn2 ++ pn1x + pn (P11.14) where the ps are constant coefficients. A straightforward way for computing the coefficients is to generate n linear algebraic equations that we can solve simultaneously for the coefficients. Suppose that we want to determine the coefficients of the fourth-order polynomial f (x) = p1x4 + p2x3 + p3x2 + p4x + p5 that passes through the following five points: (200, 0.746), (250, 0.675), (300, 0.616), (400, 0.525), and (500, 0.457). Each of these pairs can be substituted into Eq. (P11.14) to yield a system of five equations with five unknowns (the ps). Use this approach to solve for the coefficients. In addition, determine and interpret the condition number.