Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Function f(x)=x^3-c-bx-ax^2

  1. Nov 18, 2009 #1

    MathematicalPhysicist

    User Avatar
    Gold Member

    Assume I have the next function f(x)=x^3-c-bx-ax^2 and I am asked to find the coefficients a,b,c which minimizes the norm of f under L_2[-1,1].

    All I need to do here is equate [tex]f=\sum_k <f,\phi_k>\phi_k[/tex] where the phis are orthonormal functions, in this case simply 1,x,x^2,x^3, I am not sure this correct cause I found the next coefficients:
    <f,1>=sqrt(-2a-2c/3)
    <f,x>=sqrt(2/5-2/3 b)
    <f,x^2>=sqrt(-2a/3-2c/5)
    <f,x^3>=sqrt(2/7-2b/5)

    But when equation I find two different solutions to b, so I suspect this is the wrong to solve this problem, any hints as to how to minimize this functional.
     
  2. jcsd
  3. Nov 18, 2009 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    Re: Minimization.

    OK I think I know why I didn't get it right, I should be using Legendre polynomial cause they are defined on this interval [-1,1].

    Have I got it right this time?
     
  4. Nov 18, 2009 #3

    mathman

    User Avatar
    Science Advisor
    Gold Member

    Re: Minimization.

    Your approach seems unduly complicated. Why not simply integrate f2 and find the values of a, b, c which gives a minimum? I tried it myself (no guarantee - I am lousy in arithmetic) and got a=c=0 and b=3/5.
     
    Last edited: Nov 18, 2009
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Function f(x)=x^3-c-bx-ax^2
Loading...