Question

1. Consider the polynonial Pl (z) of degree 4 interpolating the function f(x) sin(x) on the interval n/4,4 at the equidistant points r--r/4, xi =-r/8, x2 = 0, 3 π/8, and x4 = π/4. Estimate the maximum of the interpolation absolute error for x E [-r/4, π/4 , ie, give an upper bound for this absolute error maxsin(x) P(x) s? Remark: you are not asked to give the interpolation polynomial P(r).

Answer 1

For a polynomial of degree 4, the upper bound for the error is given by

(41

Where,

db n+1= max

Now,

$\max_{-{\pi\over 4}\leq x\leq {\pi\over 4}} (x+\pi/4)(x+\pi/8)(x-0)(x-\pi/8)(x-\pi/4)\approx 0.0339$ found by using graph of

$x (x+\pi/4)(x+\pi/8)(x-\pi/8)(x-\pi/4)$

Hence

Ед <-(0.0339) ︽ะ 0.0002825 5!

Thus,

max |sin(x) - P4)0.0002825