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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