Chapter 10, Problem 25P

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

$f\left(x\right)=p_1{x}^{n-1}+p_2{x}^{n-2}+\cdot \cdot \cdot +p_{n-1}x+p_n\left(\text{P}\text{10}\text{.25}\right)$

where the p's are constant coefficients. A straight forward 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 $f\left(x\right)=p_1{x}^4+p_2{x}^3+p_3{x}^2+p_{4}x+p_5$ that passes through the following five points: $\left(200,0.746\right),\left(250,0.675\right),\left(300,0.616\right),\left(400,0.525\right),\text{ and }\left(500,0.457\right).$ Each of these pairs can be substituted into Eq. $\left(\text{P10}\text{.25}\right)$ 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.