## Description

Problem 1. Find the polynomial of least degree that interpolates the data

(1) x 3 7

y 5 −1

(2) x 7 1 2

y 146 2 1

(3) x 3 7 1 2

y 10 146 2 1

Problem 2. The polynomial p of degree ≤ n that interpolates a given function f at n + 1

given nodes is uniquely defined. Hence, there is a mapping f → p. Denote this mapping by

L and show that

Lf =

Xn

i=0

f(xi)li(x).

Show also that L is linear; that is L(af + bg) = aLf + bLg for any functions f, g, and

constants a, b.

Problem 3. Write the Lagrange and Newton interpolating polynomials for the data

x 2 0 3

y 11 7 28

Problem 4. For n = 5, 10, 15, program the Newton interpolating polynomial pn for the

function f(x) = 1/(1 + x

2

) on the interval [−5, 5]. Use equally spaced notes. In each case,

compute f(x) − pn(x) for 30 equally spaced points in [−5, 5] in order to see the divergence

of pn from f.

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