Legendre and Chebyshev Polynomials

1. In Problem [13], you are asked to construct the Quadratic Least Squares Approximation in terms of the Legendre polynomials. It can be done similarly to the solution with the “traditional” polynomials considered in class. Here, the approximation is:

g(x) = c_0P_0(x) + c_1P_1(x) + c_2P_2(x),

where P_0, P_1, and P_2 are the first three Legendre polynomials. So, the coefficients c_0, c_1, c_2 are determined using three integrals of which the first one is:

c_0 = (1/2) INT [-1,1] { f(x) P_0(x) } dx.

2. In Problem [14], you work with Chebyshev polynomials. These polynomials could be used in absolutely the same manner as the Legendre polynomials in Problem [13].

3. In Problems [C18], [C19], you are asked to visualize first 6 (from 0 to 5) Legendre and Chebyshev polynomials and compare them with polynomials x_k. From the graphs, it should be clearly seen that x_k is a poor choice for numerical work since these functions are much alike, so when you wish to analyze a given function in its components [Sigma( c_i x_i)], it is difficult to determine coefficients c_i precisely.

November 13, 2014:

Handling Piecewise Cubic Interpolation

1. In Problem [6], you are asked to construct a cubic spline for the set of 5 points. One of the convenient ways to do that is to directly use the class formulas (CS1)-(CS3). Here are a few more specific suggestions about how to solve this problem.

For the given points, (CS2) yields (as a0 = a4 = 0):

(2/3)a1 + (1/6)a2 = (y3 - y2) - (y2 - y1)
(1/6)a1 + (2/3)a2 + (1/6)a3 = (y4 - y3) - (y3 - y2)
(1/6)a2 + (2/3)a3 = (y5 - y4) - (y4 - y3)

Substituting the values of yi and simplify, we obtain three equations with 3 unknowns a1, a2, a3. The system is solved by Gaussian elimination.

Then the coefficients b1, .., b4 and c1,..., c4 are determined with (CS3):

b1 = y1 - a0/6 = y1; b2 = y2 - a1/6; etc.
c1 = y2 - a1/6; c2 = y3 - a2/6; etc.

With all the coefficients determined, the required cubic spline interpolation (for all 4 intervals, -2 to -1, -2 to 0, etc.) is constructed from (CS1).

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Last modified: Fri, Dec 12, 2014