PHYCS 6730 Lab Exercise: Orthogonal Polynomials

Exercise 1 Legendre Polynomials

  1. Use Maple to examine the first four Legendre polynomials. You get them by loading the package with(orthopoly). The Legendre polynomial of order n is P(n,x). Copy them to your answer file.
  2. Which of these is an even function and which is odd?
  3. Check the orthogonality of P(2,x) and P(4,x) by integrating explicitly over the interval [-1,1]. Copy your Maple output to your answer file.
  4. Check the normalization of the polynomial by integrating P(n,x)^2 over the interval [-1,1]. Verify in a couple of cases that the result is 2/(2*n+1). Copy your Maple output to your answer file.
  5. Find the zeros of P(8,x).

Exercise 2 Legendre Polynomials as a complete basis

Expand the polynomial 
   f(x) = 6*x^5 - 3*x^3 + x
in terms of Legendre polynomials. That is, write it as a sum of terms of the form 
   a_0 P(0,x) + a_1 P(1,x) + a_2 P(2,x) + ...
Find the coefficients a_0, a_1, ... .
Hint: Here are two ways to do this. (1) Start from the highest degree Legendre polynomial that could occur and do synthetic division, writing the remainder as a polynomial of lesser degree. Do the same with the remainder, etc. (2) Use the orthogonality formula and grind out the integrals. The first method is more fun.

Exercise 3 Lagrange interpolation and Gauss-Legendre Quadrature

Suppose we used the three zeros of the Legendre polynomial as the interpolation points for a function f(x). Write the Lagrange interpolation formula showing the explicit form of the Lagrange functions L(n,x). The zeros of P(3,x) are given by x1 = -sqrt(3/5), x2 = 0, x3 = sqrt(3/5). Remember, the formula looks like 
   f(x) = L(1,x) f(x_1) + L(2,x) f(x_2) + L(3,x) f(x_3).
Maple can help you here. What is the degree of the interpolation polynomial?

If we use the interpolation formula to approximate f(x), then 

  int(f(x),x=-1..1) = c_1 f(x_1) + c_2 f(x_2) + c_3 f(x_3)
where 
  c_i = int(L(i,x),x=-1..1).
Evaluate the coefficients c_i. These are the coefficients required for three-point Gauss-Legendre quadrature. You used them in the last class exercise, so compare your result with the numbers there.

Exercise 4 Chebyshev Polynomials

Examine the first few Chebyshev Polynomials, known to Maple as T(n,x). Check the orthogonality of T(2,x) and T(4,x) over the interval [-1,1] with respect to the measure dx/sqrt(1 - x^2).