{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw3 - Homework 3 Foundations of Computational Math 2 Spring...

This preview shows pages 1–3. Sign up to view the full content.

Homework 3 Foundations of Computational Math 2 Spring 2011 Solutions will be posted Wednesday, 2/2/11 Problem 3.1 Show that given a set of points x 0 , x 1 , . . . , x n a Leja ordering can be computed in O ( n 2 ) operations. Problem 3.2 Consider a polynomial p n ( x ) = α 0 + α 1 x + · · · + α n x n p n ( x ) can be evaluated using Horner’s rule (written here with the dependence on the formal argument x more explicitly shown) c n ( x ) = α n for i = n - 1 : - 1 : 0 c i ( x ) = xc i +1 ( x ) + α i end p n ( x ) = c 0 ( x ) If the roots of the polynomial are known we can use a recurrence based on p n ( x ) = α n ( x - ρ 1 ) · · · ( x - ρ n ) given by: d 0 = α n for i = 1 : n d i = d i - 1 * ( x - ρ i ) end p n ( x ) = d n This algorithm can be shown to compute p n ( x ) to high relative accuracy. Specifically, d n = p n ( x )(1 + μ ) , | μ | ≤ γ 2 n u where γ k = ku/ (1 - ku ) and u is the unit roundoff of the floating point system used. 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
3.2.a An error analysis of Horner’s rule shows that the computed value of the polynomial satisfies ˆ c 0 = (1 + θ 1 ) α 0 + (1 + θ 3 ) α 1 x + · · · + (1 + θ 2 n ) α n x n where | θ k | ≤ γ k .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern