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Approximating co-efficents

  1. Nov 9, 2004 #1

    Zurtex

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    I've got a situation where I can approximate a line by the function:

    [tex]y = a_0 + a_1 \sin x + a_2 \sin 2x + b_1 \cos x + b_2 \cos 2x[/tex]

    From experimental data I will be able to find certain values for x and y, namely yn and xn for some value of n. Now I can easily write a computer program which could work out the co-efficients in this problem if I have 5 values for y and 5 values for x. However, being a particle situation and wanting as many results as possible or perhaps not being able to gain that many results due to constraints, how would I be able to approximate these co-efficient please?
     
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  3. Nov 9, 2004 #2

    shmoe

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    How about least squares approximation?

    Let the ith row of an n by 5 matrix [tex]M[/tex] be given by [tex][ 1\ \sin(x_i)\ \sin(2x_i)\ \cos(x_i)\ \cos(2x_i)][/tex], and let [tex]Y=[y_1,\ldots,y_n]^T[/tex].

    Then coefficients that will minimize the sum of the squares of the differences of your data points to [tex]y = a_0 + a_1 \sin x + a_2 \sin 2x + b_1 \cos x + b_2 \cos 2x[/tex] will be any column [itex]A=[a_0,a_1,a_2,b_1,b_2]^T[/itex] which satisfies the linear system [itex](M^{T}M)A=M^{T}Y[/itex]
     
  4. Nov 9, 2004 #3

    Zurtex

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    Thanks I haven't seen that method yet (well I've researched it but not been able to make sense of it). Thanks a lot, just need to try and make sense of it, but I think I can :smile:
     
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