Answered: Question 2 Let f(x) = cos() and P(x) be…

Question 2 Let f(x) = cos() and P(x) be the interpolation polynomial of degree at most 3 which agrees with f(x) at To = -1, x1 = 0, x2 = 1, x3 = 2. (a) Compute P(0.5) using Neville's method.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.3: Zeros Of Polynomials

Problem 4E

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Question

Part B

all computations, you could give your answers to five decimal places if needed

Question 2
Let f(x) = cos(T) and P(x) be the interpolation polynomial of degree at most 3 which agrees with f(x) at
To = -1, x1 = 0, x2 = 1, x3 = 2.
(a)
Compute P(0.5) using Neville's method.
(b)
Express P(x) using Newton's divided difference formula. Use this result to compute P(0.5).
(c)
Give a bound for the absolute error |f(0.5) – P(0.5)|.
Page 3

Expert Solution

Step 1

Given:

$f (x) = \cos (\frac{πx}{2})$

x_o=-1

x₁=0

x₂=1

x₃=2

Step 2

$f (x_{o}) = f (- 1) = \cos (\frac{- π}{2}) = 0$

$f (0) = 1, f (1) = 0, f (2) = - 1$

Newton's divided difference table,

x	f(x)	first order	second-order	third order
-1	0	$\frac{1 - 0}{0 + 1} = 1$	$\frac{- 1 - 1}{1 + 1} = - 1$	$\frac{0 + 1}{3} = 0.333$
0	1	$\frac{0 - 1}{1} = - 1$	0
1	0	$\frac{- 1 - 0}{2 - 1} = - 1$
2	-1

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