Efficient Methods for Computing Determinants of Large Matrices

In summary, MATLAB's det function is more efficient than row-reduction, but it is still cumbersome to use.
  • #1
amcavoy
665
0
How do computers evaluate determinants of large matrices? The cofactor method seems like it would be too time consuming. Does anyone know?
 
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  • #2
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.
 
  • #3
How would this fare computationally?

[tex] 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} [/tex]

Cumbersome and inefficient for a computer algorithm?
 
  • #4
Well, how much work is it? You do n multiplications how many times?
 
  • #5
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
 
  • #6
  • #7
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.
 
  • #8
So row-reduction (to get 0-entries) would be more efficient? Or would that take a long time in itself?
 

What is a computer?

A computer is an electronic device that is designed to process, store, and retrieve data. It consists of hardware components such as a processor, memory, and input/output devices, as well as software programs that allow it to perform various tasks.

What are determinants in computers?

Determinants in computers refer to the factors that influence the performance and behavior of a computer system. This can include hardware components, software programs, operating systems, and user behavior.

How do computers use determinants to function?

Computers use determinants in various ways to function effectively. For example, the type and speed of the processor can determine how quickly a computer can process data, while the amount of memory can affect its ability to store and access information.

What are the common determinants that affect a computer's performance?

Some common determinants that can impact a computer's performance include the processor speed, amount of memory and storage capacity, graphics capability, operating system, and software programs running on the computer.

How can I optimize a computer's determinants for better performance?

To optimize a computer's determinants for better performance, you can upgrade hardware components such as the processor, memory, and storage, as well as regularly maintain and update software programs. Additionally, minimizing the number of programs running at once and managing storage space can also help improve a computer's performance.

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