math problem: fuzzy system of linear equations

[lighthousekeeper]

i've got a math problem and am struggling with coming up with an answer. i have a system of linear equations like:

ax + by + cz = 123456...
dx + ey + fz = 7890123...
gx + hy + iz = 456789...

(In my case, instead of a 3 x 3 matrix like the one above, it's an 8 by 8 matrix.) 8 equations, 8 unknowns. I know that if the 8 equations are "consistent", then an exact solution could be determined using linear algebra row reduction, Cramer's rule, or similar techniques.

BUt my problem is that the 8 equations are known to be inconsistent. In other words, ax+by+cz doesn't exactly equal 123456, but rather 123456 is some apprximation with an unknown accuracy.

I don't expect or want an exact answer to the 8 unknowns, but rather a range of values, or a statistical probability of answers. Any thoughts?

[gstelmack]

Matlab? Maple? Mathcad?

[johnnyshaka]

Nothing to add, I'll move along.

[Dr. Sak]

Do you know how much the approximate values range?

[twothree]

I think you would want to use Maximum likelihood estimation.

Maximum likelihood - Wikipedia, the free encyclopedia

[lighthousekeeper]

Originally Posted by twothree

I think you would want to use Maximum likelihood estimation.

Maximum likelihood - Wikipedia, the free encyclopedia

thanks - i think this might be what i'm looking for (now i just need a couple years of advanced statistics training)

[Toddzilla]

Maximum Likelihood - snicker....

[RedHawk00]

if you have matlab or one of the free knockoffs, look up SVD (Singular Value Decomposition), this gives you a best estimation of the 8 unknown. Note: you can also feed it 8+ equations to give you best guess on the 8 unknowns