5-hw116

# 5-hw116 - notice that x t-p T,t is orthogonal to the entire subspace of polynomials Diﬀerentiate with respect to T which appears both in the

This preview shows page 1. Sign up to view the full content.

MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #5 Last modiﬁed: October 20, 2007 Due Thursday, Oct. 25 at class. 1. (a) Carry out one more step of the example of the Gram-Schmidt process for the space L 2 [ - 1 , 1]that was begun in lecture. (b) Verify that e i ( t ) for i = 0 , 1 , 2 (done in lecture) and i = 3 (which you just did) agree with the formula for the “well-known Legendre polynomials” on page 61 of Luenberger. (c) Let f ( t ) = 0 on [-1,0], f ( t ) = t on (0,1]. For k = 0,1,2,3, ﬁnd the polynomial of degree k that is closest to this f ( t ) in the space L 2 [ - 1 , 1]. 2. Luenberger, section 3.13, problem 3. 3. As a simple special case of the preceding problem, let m = 2 ,n = 1 and take Q = 5 3 3 5 . Find the vector in the subspace x +2 y = 0 that is closest to (1,0) with the inner product deﬁned by the preceding problem. 4. Luenberger, Section 3.13, problem 12. The secret of doing this problem is to
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: notice that x ( t )-p ( T,t ) is orthogonal to the entire subspace of polynomials. Diﬀerentiate with respect to T , which appears both in the upper limit of integration and in p [ T,t ]. It might be helpful to do the special cases n = 1 and n = 2 ﬁrst. 5. Luenberger, Section 3.13, problem 9. 6. Luenberger, Section 3.13, problem 6. 7. Luenberger, Section 3.13, problem 18. Here is a hint which you would quickly ﬁnd on the Internet. Let P n ( t ) be the n th orthogonal polynomial, and let t 1 , ··· t m be the points where it changes sign(this set includes simple roots, but not double roots). Deﬁne the polynomial S ( t ) = m Y j =1 ( t-t j ) . Show that the assumption m < n leads to a contradiction concerning the integral of w ( t ) P n ( t ) S t . 1...
View Full Document

## This document was uploaded on 08/30/2011.

Ask a homework question - tutors are online