function on this interval and then plot the three polynomial interpolants you constructed on the same graph (using hold on). Your graph should include a legend identifying the gamma function and the three different interpolants. Be sure to use different line types for each of these. Use 1000 equally spaced points in the plot.

Second, compute the interpolant on the interval [.5, 3.5] using 10 equally spaced points in this interval. Plot the Gamma function and the interpolating polynomial on the same graph as above. Examine the result and judiciously add a few more interpolation points with x_j in the interval [.1, 4] to improve the quality of the interpolation approximation. When you have a decent fit, plot the improved interpolating polynomial on the same graph as the gamma function and first interpolating polynomial. This plot should be over the in- terval [.1, 4] using 1000 equally spaced points as before. Again a legend and distinct line types are required.

Problem 2 Let {x_i = ih}n

with h = 1/n be a uniform partition

of the interval [0

i=0

, 1]. Consider a function f

œ C²([0, 1]), and its piecewise

linear polynomial interpolation p(x). You are going to prove that the L² error estimate for piecewise linear interpolation if

Î ≠ Î

= Û⁄  1[

( ) ≠

(  )]2 Æ

.  ÕÕ.

For that, follow the steps below:

1. Show that the Taylor expansion of f (x) for x close to zero can be written as

x

f (x) = f (0) + f (0)x + 0 f (s)(x ≠ s)ds.

Hint: You should use the integration by parts formula

⁄  u(x)v^Õ(x)dx = u(x)v(x) ≠ ⁄  u^Õ(x)v(x)dx.

2. Then, show that since p(x) is the linear interpolation of f œ C([0, h]),

p(x) = f (0) + f ^Õ(–)x, – œ [0, h].

3. Use the results from parts 1 and 2 to show that for x œ [0, h],

Ô       A⁄  h -

-2 B1/2

|f (x) ≠ p(x)| Æ

3hx

0   -f ^ÕÕ(s)-  ds .

Hint: notice that

⁄ x Õ

Hint: use the following result (without proof)

h h 1/2

0  |f (x)| dx Æ h 0  f (x)dx .

4. Now prove that

Îf ≠ pÎL²([0,1]) Æ h2 .f ÕÕ.L2([0,1]) .

Problem 3 This problem is pledged!

The Hermite interpolant h_n P_2n+1 of f C¹([a, b]) at the points x_j : j = 0, 1, 2,. .., n is the interpolating polynomail such that h_n(x_j) = f (x_j) and h^Õn(x_j) = f ^Õ(x_j).

It can be written in the form

n

h_n(x) = A_j(x)f (x_k)+ B_j(x)f ^Õ(x_j)

j=0

where the functions A_j and B_j generalize the Lagrange basis functions:

A_j(x) = 11 ≠ 2ljÕ (x_j)(x ≠ x_j)2 l²j(x) (4)

B_j(x) = (x ≠ x_j)l²(x) (5)

where

l_j(x) = Ÿ

 x ≠ x_k . (6)

x_j ≠ x_k

k=0,k  j

Calculate explicitly (by hand) the cubic hermite polynomials A₀, A₁, B₀ and

B₁ for x₀ = 0 and x₁ = h, for h> 0.  For  that, assume that A₀ is of the form

A₀(x) = a + bx + cx² + dx³.

Then, calculate the coefficients a, b, c and d by superimposing the constraints A₀(0) = 1, A₀(h) = 0, A^Õ(0) = A^Õ(h) = 0.  Do  the  same  thing  for  A₁, B₀ and B₁ to get the final result. Verify that the results correspond to the formulas 4 and 5.

We want to build a piecewise hermite interpolation H_N (x) of degree 3 for the function f (x) = 1/(1 + x²) in x = [ 4, 4]. For that, we divide [ 4, 4] into N intervals of size 8/N . In each of those intervals, we approximate f using cubic Hermite polynomials. The result is the piecewise cubic function H_N (x).