MATH4141/MATH6651/PHYS5070

FW02

Assignment 5                                                              Due date: Friday, Nov. 29

1.  Use the Legendre polynomials to find the least squares polynomial approximation to exp(x) of degree 2 on [-1, 1] with weight function w(x) = 1.

2.  Find the Taylor series of degree 6 about x = 0 for cosh(x) and use Chebyshev economization to obtain a polynomial approximation of degree 4.  Estimate the error of this approximation on [-1, 1].

3.  Find the first four Hermite polynomials, i.e. the orthogonal polynomials H0, H1, H2 and H3 on

[-¥, ¥] with weight function exp(-x2).  Take H0 = 1.  The usual normalization for the Hermite polynomial Hn(x) is to take the coefficient of xn as 2n which makes

4.  Using the results of problem 3 find the least squares polynomial approximation of order 2 to the function e-x on [-¥, ¥] with weight function w(x) = exp(-x2).  Show (i.e. plot or produce a table of) the relative error of this approximation on [-2, 2].

5. (a)  Find the [2/2] Padé approximant to x cot(x).  Compare the error in this approximation with the Taylor series of degree 4 about x = 0 on [0, 1].  Also compare with the error in the approximation to x cot(x) obtained by writing it as x cos(x)/sin(x) and using the Maclaurin series of degree 2 for both sin(x) and cos(x).

(b)  Find the location of the singularity in this Padé approximant closest to zero.  How does this compare to the location of the first singularity in x cot(x)?

(c) Write out this Padé approximant in continued fraction form.

6.  Use the Fast Fourier Transform to find the Discrete Fourier Transform of the set of data

(xj, yj), j = 0 ..3 where xi = -p + jp/2 and yj = sin(xj).

Integration formula: