Numerical Analysis Burden 9th Edition. Show the next equaliy. Show Strictly all Algebraic Steps.

A, = [X0,X1,43] = f[x1, x2] - f[X0,X1) 13-10 126 CHAPTER 3. Interpolation and Polynomial Approximation As might be expected from the evaluation of a, and at the required constants are 0x = f(x0, x1,x2,...,x), for each k = 0,1,...,n. So P(x) can be rewritten in a form called Newton's Divided- Difference: P.(x) = f[xl+ [260, X1, ... ,**)(x xo) ...(x Xx-1). (3.10) The value of f[xo,X1,...,xk] is independent of the order of the numbers X0,X1,...,. as shown in Exercise 21. The generation of the divided differences is outlined in Table 3.9. Two fourth and one fifth difference can also be determined from these data. Table 3.9 First divided differences Second divided differences Third divided differences I f(x) 20 f[x] - [[x] f1XX) = X-XO X 14 f(x,x21 - x0, x] X-XO f[X1,2)= f123] - [X] [X1,12,131 - fx...] I-X