HW03_short.pdf - Math 104A Homework#3 Instructor Lihui Chai...

This preview shows page 1 - 2 out of 2 pages.

Math 104A Homework #3 * Instructor: Lihui Chai General Instructions: Please write your homework papers neatly. You need to turn in both full printouts of your codes and the appropriate runs you made. Write your own code, individually. Do not copy codes! 1. (a) Let f C 2 [ x 0 , x 1 ] and P 1 its interpolation linear polynomial at x 0 and x 1 . Prove that k f - P 1 k 1 8 ( x 1 - x 0 ) 2 M 2 , (1) where | f 00 ( x ) | ≤ M 2 for all x [ x 0 , x 1 ] and k f - P 1 k = max x [ x 0 ,x 1 ] | f ( x ) - P 1 ( x ) | . (b) Let P 1 ( x ) be the linear polynomial that interpolates f ( x ) = sin x at x 0 = 0 and x 1 = π/ 2. Using (a) find a bound for the maximum error k f - P 1 k and compare this bound with the actual error at x = π/ 4. 2. (a) Equating the leading coefficient of in the Lagrange form of the interpolation polynomial P n ( x ) with that of the Newton’s form deduce that f [ x 0 , x 1 , ..., x n ] = n X j =0 f ( x j ) n Q k =0 k 6 = j ( x j - x k ) . (2) (b) Use (a) to conclude that divided differences are symmetric functions of their arguments, i.e. any permutation of x 0 , x 1 , ..., x n
Image of page 1

Subscribe to view the full document.

Image of page 2
  • Fall '08
  • Staff
  • Math, Polynomial interpolation, interpolation polynomial

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern