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text-dude

Posts: 74
Registered: 4/30/08

Matrix inversion - linear algebra - higher accuracy for some matrix
rows ? (Tikhonov regularization???)
Posted: Sep 10, 2012 3:55 PM

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Here's the deal:

A-matrix is approx. 90x90, mostly diagonal but also quite some
offdiagonal elements here and there.

Some rows in the matrix are equations that is harder to satisfy than
others because something is rotating at different speeds - it means that
those matrix equations (the lower rows in A) that has a physical
connection to something that rotates really fast, causes some severe
oscillations (it gives oscillating accelerations and wrong forces)
because the timestep is very high compared to the rotation speed for the
last rows in A... Got it?

Ok, is there any way to make: x = A\b more accurate for the lower
rows/equations or ???

Another possibility I've been thinking about - in order to avoid these
oscillations I talked about before - is to use regularization, maybe? I
did a project some years ago using Tikhonov regularization, but here
AFAIR - the idea was that all the elements in the solution-vector x
could not change too much... This will allow me to use higher timesteps
and avoid the oscillations... Understand it ?

ELABORATION FOR THOSE WHO DON'T UNDERSTAND THE PHYSICS:
Here's an elaboration of why I have a problem with high angular
velocities: If some part of the problem (top rows in A) rotates very
slow, i.e. at 1 rad/s and your timestep is 0.01 sec - it means that you
have 100 timesteps per rotation. That's ok... However, the lower rows in
A rotates much faster - then I only get maybe 5-8 timesteps per
rotation. This gives some VERY annoying oscillations and forces going in
"random" directions because the motion is not smoothly captured.

I want my program to solve x = A\b quickly and efficent - any ideas that
might help me (and keep the computation time reasonable) ? Any tricks /
tips ?

I really hope to hear from some clever experts here... Thanks...