Chapter 3.3, Problem 2ES

Use Eq. (3.10) or Algorithm 3.2 to construct interpolating polynomials of degree one, two, and three for the following data. Approximate the specified value using each of the polynomials.

1. a. f(0.43) if f(0) = 1, f(0.25) = 1.64872, f(0.5) = 2.71828, f(0.75) = 4.48169
2. b. f(0) if f(−0.5) = 1.93750, f(−0.25) = 1.33203, f(0.25) = 0.800781, f(0.5) = 0.687500

$P_n(x)=f\left[x_0\right]+{\displaystyle \underset{k=1}{\overset{n}{\sum }}f}\left[x_0,x_1,\dots ,x_k\right]\left(x-x_0\right)\cdots \left(x-x_{k-1}\right)\text{ (3}\text{.10)}$

ALGORITHM 3.2

Newton’s Divided-Difference Formula

To obtain the divided-difference coefficients of the interpolatory polynomial P on the (n + 1) distinct numbers x₀, x₁, … x_n, for the function f:

INPUT numbers x₀, x₁, … x_n; values f(x₀), f(x₁), …, f(x_n) as F_0,0, F_1,0, …, F_n,0.

OUTPUT the numbers F_0,0, F_1,1, …, F_n,n where

$P_n(x)=F_{0,0}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{F}_{ij}}{\displaystyle \underset{j=0}{\overset{i-1}{\prod }}\left(x-x_j\right)}.\left(F_{ij}\text{ is }f\left[x_0,x_1,\dots ,x_i\right].\right)$

Step 1 For i = 1, 2, …, n

For j = 1, 2, …, i

set $F_{i,j}=\frac{F_{i,j-1}-F_{i-1,j-1}}{x_i-x_{i-j}},\left(F_{i,j}=f\left[x_{i-j},\dots ,x_i\right].\right)$

Step 2 OUTPUT (F_0,0, F_1,1, …, F_n,n);

STOP.