Some improvements of the Gaussian elimination method for solving simultaneous linear equations

Although it is known that Gaussian elimination method for solving simultaneous linear equations is not asymptotically optimal, it is still one of the most useful methods for solving systems of moderate size. This paper proposes some ideas how to speed-up the standard method. First, the trick which takes the advantage of the eventual symmetry of the system is presented, which speeds up the calculation by the factor slightly less than 2. Second, it is shown that by using some rearrangement of the calculation, it is possible to get additional speed-up, no matter whether the system is symmetric or not, although the eventual symmetry additionally doubles the execution speed. This rearrangement is performed using similar approach as in LU factorization, but retaining basic features of the Gaussian elimination method, like producing the triangular form of the system. As the required modifications in the original method are quite simple, the improved method may be used in all engineering applications where the original Gaussian elimination is used.