Polynomial interpolation consists of determining the unique (n – 1)th-order polynomial that fits n data points. Such polynomials have the general form,

where the p’s 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 p’s). Use this approach to solve for the coefficients. In addition, determine and interpret the condition number.

f(x) = Px"-1 + Pxn- + ... + Pn-1X + Pn