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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]
    where
    [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?
     
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