Problems-linear algebra and numerical analysis

Question

Problems:

Linear Algebra and Numerical Analysis

Questions on a Sequence of Polynomials.

Let T_k be the sequence of polynomials defined by

              T₀(x) = 1, T₁(x) = x, T_k+1(x) = 2xT_k(x) - T_k-1(x) k>1

1) Show that Tk is a polynomial of degree k. Calculate the coefficient of x^k of T_k .

2) Show by induction that T_k(cosθ) = cos(kθ)for all real θ.

3) Deduce that if x∈[-1,1], |Tk(x)|< 1.

4) Show that for all whole natural numbers n, we have

                           (x-x₀)(x-x₁)...(x-x_n) = 2^-nT_n+1(x)

where xi = cos[(1+2i)∏/2(n+1)], i = 0,1,....n.Give a numerical approximation of these numbers for n = 4, to the precision of your calculator.

5) Let us consider the function f defined as ƒ(x) = e^x.Evaluate the Lagrange interpolation polynomial P of f at the points that we calculated in the previous question.

6) Estimate the error for ƒ(x) - P(x) for x∈[-1,1].