Analysis
Gaussian elimination to solve a system of n equations for n unknowns requires n(n-1) / 2 divisions, (2n3 + 3n2 − 5n)/6 multiplications, and (2n3 + 3n2 − 5n)/6 subtractions, for a total of approximately 2n3 / 3 operations. Thus it has arithmetic complexity of O(n3). However, the intermediate entries can grow exponentially large, so it has exponential bit complexity.
This algorithm can be used on a computer for systems with thousands of equations and unknowns. However, the cost becomes prohibitive for systems with millions of equations. These large systems are generally solved using iterative methods. Specific methods exist for systems whose coefficients follow a regular pattern (see system of linear equations).
The Gaussian elimination can be performed over any field.
Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices. For general matrices, Gaussian elimination is usually considered to be stable in practice if you use partial pivoting as described below, even though there are examples for which it is unstable.
Read more about this topic: Gaussian Elimination
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