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Least Square Minimisation and Partial Differentiation

  1. May 6, 2009 #1
    1. The problem statement, all variables and given/known data

    I am trying to understand how some equations are obtained in some lecture notes I have.

    This is my starting equation-http://i423.photobucket.com/albums/pp315/skaboy607/StartEquation.png [Broken]

    And I need to satisfy these conditions

    http://i423.photobucket.com/albums/pp315/skaboy607/Conditions.png [Broken]

    And somehow get to here

    http://i423.photobucket.com/albums/pp315/skaboy607/WhatIshouldendupwith.png [Broken]

    2. Relevant equations

    all above

    3. The attempt at a solution

    I've stared at it for a while now and just don't get it. Wouldn't the square come down in front of the brackets? and surely the other constants would dissapear?

    any help most appreciated..

    Thanks
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 6, 2009 #2
    Looking at only one term of the summation, if F=[R1*a - R2*b + R3 - c]^2 (here a, b, and c are constants), then partial of F with respect to R1 is, by the chain rule,

    2[R1*a - R2*b + R3 - c]^1 * a,

    where a is the derivative of R1*a - R2*b + R3 - c with respect to R1.

    But your 2 "disappears" when you set this equal to 0 and divide by 2.

    "a" would disappear too, if there were no summation, but in this case "a" is really a_i, so it can't be brought outside of the summation over i.
     
  4. May 6, 2009 #3
    Nice one, thanks very much.
     
  5. May 9, 2009 #4
    Hi, ive done this now. But I have just a quick question about it? The way I did it was to multiply the brackets then do partial differentiation with respect to R1, then divide 2 and take out cos(theta) as a common factor. This was quite time consuming, is there something i don't know about that will enable me to look at an equation like that, and jump to final stage like you did so I can miss out the bits in the middle?

    Thanks
     
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