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Thread: sum of squares

  1. #1 sum of squares 
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    x_1, x_2, x_3, ..., x_n are n positive real numbers whose sum S is constant

    the sum of their squares shuld be as smallas possible

    what are the n numbers?


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  3. #2  
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    We can write the n numbers as








    where .

    The sum of their squares is







    Hence the sum of squares is minimized when , i.e. when .

    Thus the n numbers are .


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  4. #3  
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    Quote Originally Posted by JaneBennet
    We can write the n numbers as








    where .

    The sum of their squares is







    Hence the sum of squares is minimized when , i.e. when .

    Thus the n numbers are .
    That is nice tight reasoning. I like it.

    Here is a way to think about the problem geometrically. The problem is scalable, so take S =1. Mininizing the sum of the squares is equivalent to minimizing the square root of the sum of the squares.

    Now, first think about the case n=2. The locus of the points whose absolute value sums to 1 is a square with vertices at (1.0), (0.1), (-1,0), and (-1.1). We are only interested in what happens in the upper right hand quadrant, but think about the whole square. Now the square root of the sum of the squares is the ordinary Eudlidean distance to the origin. Geometrically you can see that this is minimized for points on the square at the points that also lie on lines with slope 1 or -1, i.e. where , If you do this same exercise in higher dimensions you find that the locus of points whose absolute value sums to a constant is just a hyper-cube and the solution is still in the center point of each hyper-surface, which is where the absolute values of the invividual coordinates are equal.

    Your solution is tighter and more rigorously argued, but I think the intuition from the geometric version is helpful in figuring out what the answer ought to be.
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  5. #4  
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    Well, I solved a problem similar to this very recently, so I was able to use the same method as before.

    But itís always nice to be able to generalize a solution, to understand why a problem is a particular case of a more general situation.
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