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Topic: Iterative numerical algorithm to find canonical form of 3D quartic
Replies: 6   Last Post: Aug 14, 2012 10:00 AM

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Robert H. Lewis

Posts: 58
From: New York
Registered: 12/4/04
Iterative numerical algorithm to find canonical form of 3D quartic
Posted: Aug 11, 2012 12:25 PM
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I am engaged this summer with some undergraduate students on a project that comes from chemistry. Part of it boils down to converting a general 3 variable quartic

Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J=0

to a canonical form (no cross terms xy, xz, yz) via rotations. (Don't worry about the linear terms Gx + Hy + Iz + J yet.)

In 2D this is quite easy. The well known formula for the angle to effect this is tan(2*theta) = D/(A-B). One could certainly do that in 3D, killing the xy coefficient. Then one could similarly kill, say, the xz coefficient, but that would reintroduce an xy coefficient. It seems that nothing would be accomplished.

To my astonishment, we found that continuing this naive strategy works! That is, iterating the idea, killing all three cross terms again and again in sequence, all three cross terms quickly converge to 0, so that a small number of repetitions of this cycle can yield a canonical form to any desired tolerance.

I emphasize that this is numerical computing. I am not an expert in numerical computing, but I don't recall ever hearing of this. We can prove why it works. Is this known? Any references?

Robert H. Lewis
Fordham University

P.S. There are other ways to solve the general problem. There is a rather sophisticated way to compute the three rotation angles (one for each axis) from the original coefficents of the quartic polynomial, but that's not the point here.

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