1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Lagrange Polynomial Interpolation

  1. Oct 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Find the polynomial p(x) of degree 20 satisfying:
    p(-10) =p(-9) = p(-8) = .......=p(-1) = 0
    p(0) = 1
    p(1) = p(2) = p(3) = .....p(10) = 0

    2. Relevant equations

    L(x) := \sum_{j=0}^{k} y_j \ell_j(x)

    3. The attempt at a solution

    i tried using the formula above:

    a = p(-10) / (-1)(-2)(-3)....(-20) = 0

    and got zero for all the coefficients excluding the 20th coefficient, which i got 1/0

    then i thought about it graphically - it looks like a cos graph so i tried using the maclaurin series expansion but realised that it only works from n to infinity.

    any tips?
  2. jcsd
  3. Oct 2, 2011 #2
    A few suggestions. Throw some itex and tex tags around your latex to make them render. Next, could you define some of things you're using? What are [itex] y_j, \ell_j,a [/itex]? It is impossible for us to guess what these are supposed to be.

    Now assuming I've guess your notation correctly, the Lagrange polynomial is given by

    [tex] L(x) = \sum_{j=0}^{20} p(x_j) \ell_j(x) [/tex]
    [tex] \ell_j(x) = \prod_{i\neq j, i=0}^{20} \frac{ (x -x_j) }{(x_j-x_i)} [/tex]

    in which case you are correct, all terms except the j=0 term disappear. So what is [itex] \ell_0 [/itex]?
  4. Oct 2, 2011 #3
    so to find the nth coefficient for the x^n-1 term

    a(n) = p(n)/(x-x1)(x-x2)(x-x3).....

    x1, x2 terms are the x values of the data points provided from ascending order excluding the x(n) point if that makes sense

    but what i'm saying is, i get zero or 1/0 (for P(0) = 1) point for all the coefficients i calculate which is where i'm stuck :/
  5. Oct 2, 2011 #4
    Okay, I understand the x(n) notation but you still haven't defined what [itex] y_j [/itex] is, what [itex] \ell_j [/itex] is, or how a(n) fits into the definition of the Lagrange polynomial. It's impossible to help you unless you make these things clear.
  6. Oct 2, 2011 #5
    yj is the y co-ordinate of the data point...lj is the coefficient of x^n, where n is the varying degrees of the polynomial (in this example, 0-20 because there are 21 data points)

    a(n) was just my way of explaining lj.
  7. Oct 2, 2011 #6
    Ah, okay.

    Well, the [itex] \ell_j [/itex] are never zero. Take a look at how I defined [itex] \ell_j [/itex] in one of my previous posts and you'll see that this is true. What is the formula you are using to calculate them?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook