AMSC/CMSC 666 Dr. Wolfe ASSIGNMENT #2 Due March 9, 2009
1. Use the Gram-Schmidt procedure to determine the first three orthogonal polynomials
for [0,1] with weight function w(x) = ln(1/x).
2. Define Sn(x) = 1
n+1 T′
n+1(x), n ≥0,with Tn(x) the Chebychev polynomial of degree
n. The polynomials Sn(x) are called Chebychev polynomials of the second kind.
(a) Show that {Sn(x)|n≥0}is an orthogonal family on [−1,1] with respect to the
weight function w(x) = √1−x2.
(b) Show that the family {Sn(x)}satisfies the same triple recursion formula as the
family {Tn(x)}.
(c) Given f∈C[−1,1] solve the problem: Minimize
Z1
−1p1−x2[f(x)−p(x)]2dx
where p(x) is allowed to range over all polynomials of degree ≤n.
3. Exercise 15, p.186, Sto er and Bulirsch.
4. Compute (by hand) the Q-R factorization of A where
A=5 9
12 7 .
5. Let f(x) = ex,0≤x≤1, π1= the set of all polynomials of degree ≤1.Find the
following.
(a) p∗
1, the minimax approximation to fin π1.
(b) f∗, the least squares approximation to fin π1.Here we use the norm
||f||2=qR1
0f(x)2dx.
(c) The function in π1interpolating fat the Chebychev points 1
2(1 ±1
√2).
Compare the three errors in the uniform (∞) norm and the 2-norm. Note: You
can use MATLAB to compute the integrals (either numerically or symbolically).
6. (MATLAB) Let xi=i−1
10 , i = 1,···11,yi=exi.Find the polynomial of degree ≤n
fitting this data in the sense of least squares for n= 1,2,3. Plot these polynomials
along with exfor 0 ≤x≤1.The MATLAB commands POLYFIT and POLYVAL
give you exactly what you need.