Efficient Methods for Computing Determinants of Large Matrices

[amcavoy]

How do computers evaluate determinants of large matrices? The cofactor method seems like it would be too time consuming. Does anyone know?

 

[matt grime]

my guess would be row operations. even a naive method takes n-1 operations with n and multipkciations additions for the first column, n-2 operations with n-1 additions so approximately order n^4 operations to work out a diagonal form, then multiply the n thingst together, not much more work...so quite cheap really, and that's just the naive version.

 

[TOKAMAK]

How would this fare computationally?

$$det(A) = \sum_{{i}_1, {i}_2,...,{i}_n = 1}^{N} \epsilon_{{i}_1, {i}_2,...,{i}_n} a_{{i}_1, 1} \cdot a_{{i}_2, 2} \cdot \cdot \cdot a_{{i}_n, N}$$

Cumbersome and inefficient for a computer algorithm?

 

[Hurkyl]

Well, how much work is it? You do n multiplications how many times?

 

[TOKAMAK]

Looks like if you pick out all the distinct terms (the ones in the summation that aren't equal to zero) you get (N!). On top of that, you would need to do N multiplications each time (so N multiplications N! times). Yikes

Yeah didn't work that one out before

 

[TheGinkgoNut]

well according to MATLAB's online documentation, determinants are normally computed using an LU decomposition (similar to matt's approach). If greater accuracy is required (such as when dealing with ill-conditioned matrices), the singular value decomposition is used. More details:

MATLAB's det function: http://www.mathworks.com/access/helpdesk/help/techdoc/ref/det.html

MATLAB's cond function: http://www.mathworks.com/access/helpdesk/help/techdoc/ref/cond.html

 

[matt grime]

I would think there are indeed many better ways than mine to find determinants. i would say mine is the best worst method or thee worst best method: the least complicated way of doing it that isn't naive. I've been told of other methods for speeding it up though i cant' recall them. of course when we think of a matrix we often think of something small with bige entries (ie integers) frequetly we need a clever method of finding it in the real world since we often have small entries, ie (numbers ivolved in the computation of) determinants so small that it may appear singular when it isn't.

 

[amcavoy]

So row-reduction (to get 0-entries) would be more efficient? Or would that take a long time in itself?