Solving Linear Equations in Computer Programs

Date: 02/13/2003 at 01:38:46
From: Venkatasubramani
Subject: Best method to solve linear equations in computer programs?

Dear Dr. Math,

Could you please tell me the best and simplest method in computer 
programs, among the following, for solving linear equations of the 
form
   
        x + y = 3
        2x + y = 4

1) Cramers method
2) Gaussian Elimination method
3) Inverse matrix method

Thank you.

Date: 02/13/2003 at 02:26:02
From: Doctor Jacques
Subject: Re: Best method to solve linear equations in computer 
programs?

Hello and thank you for writing to Dr Math.

The "best" choice depends somewhat on what you want to be able to do.

If you only want to solve 2*2 systems like the one above, I would 
recommend Cramer.

For larger systems you should, in general, use Gaussian elimination. 
Note that, in a "real" computer program (to be used for real-world 
applications), there are various things to consider in order to 
minimise the amount of rounding error.

You should only use the inverse matrix if you have to solve many 
different systems with the same left-hand side; in this case, you 
need to invert the matrix only once (and save the inverse), and this 
allows you to compute solutions with different right-hand sides by 
using only multiplications and additions. Inverting an n*n matrix is 
equivalent to solving n systems of n equations - the additional work 
only pays off if you can re-use the inverse. In this case, you should 
also note that the problems of rounding errors can be even more 
serious.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/