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MathGroup Archive 2001

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Re: Sum of Squares

  • To: mathgroup at smc.vnet.net
  • Subject: [mg26615] Re: [mg26587] Sum of Squares
  • From: BobHanlon at aol.com
  • Date: Thu, 11 Jan 2001 10:39:22 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Here is an approach

var = {x, y, z};

expr1 = 5 x^2 + 8 x y + 5 y^2 + 2 x z - 2 y z + 2 z^2;

expr2 = Plus @@ ((Partition[coef = Table[Unique[s], {2*Length[var]}], 
                Length[var]].var)^2);

cl1 = Flatten[CoefficientList[expr1, var]];

cl2 = Flatten[CoefficientList[expr2, var]];

soln = Solve[Complement[Thread[cl1 == cl2],{True}], coef];

There are multiple solutions

ans = expr2 /. soln /. coef[[3]] -> 1

{(-2*x - y - z)^2 + (-x - 2*y + z)^2, 
  (-x - 2*y + z)^2 + (2*x + y + z)^2, 
  (x + 2*y - z)^2 + (2*x + y + z)^2, 
  (-x - 2*y + z)^2 + (2*x + y + z)^2}

And @@ ((expr1 == Expand[#])& /@ ans)

True


Bob Hanlon

In a message dated 2001/1/9 2:28:14 AM, wself at msubillings.edu writes:

>Is there, or could there be, a "sum of squares" function for
>multivariate polynomials, which would rewrite an expression as a sum of
>squares, when possible?
>
>It might work like this:
>
>SumOfSquares[5 x^2 + 8 x y + 5 y^2 + 2 x z - 2 y z + 2 z^2]
>
>--->
>
>(x + 2y - z)^2 + (2x + y + z)^2
>


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