# Gaussian elimination

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## Gaussian elimination

[¦gau̇·sē·ən ə‚lim·ə′nā·shən]
(mathematics)
A method of solving a system of n linear equations in n unknowns, in which there are first n- 1 steps, the m th step of which consists of subtracting a multiple of the m th equation from each of the following ones so as to eliminate one variable, resulting in a triangular set of equations which can be solved by back substitution, computing the n th variable from the n th equation, the (n- 1)st variable from the (n- 1)st equation, and so forth.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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In addition, Gauss Elimination (GE) will be used for solving the linear algebraic equations produced by the discretization of the second-order LFIDEs.
Gauss Elimination method will be used in order to solve the system of linear algebraic equations.
In the first section they explain Gauss elimination, throwing in interesting tidbits to keep the interest of students who learned that in high school.
Authors like Turner  faced difficulty with Gauss Elimination approach because of round off errors and slow convergence for large systems of equations.
But it has few modifications over the Gauss elimination method.
The implementation of the Gauss Elimination algorithm on several parallel machines using shared memory design and message passing programming model is reported in .
(Gauss elimination for a dense linear system is a process of order [N.sup.3], with N being the number of unknowns.) This solve mechanism is very reliable and works for all types of problems, but the LU decomposition phase is expensive in terms of computation time because the process time grows with the cube of the number of unknowns in the system.
Examples of pivoting strategies which simultaneously control the conditioning of both triangular matrices L, U are the complete pivoting (of a computational cost of O([n.sup.3]) comparisons beyond the cost of Gauss elimination with no pivoting) and the rook pivoting (see ,  and ), of lower computational cost (in the worst case, O([n.sup.3]) comparisons beyond the cost of Gauss elimination with no pivoting).
Besides, the corresponding pivoting strategies associated to such decompositions have low computational cost: in the worst case, O([n.sup.2]) elementary operations beyond the cost of Gauss elimination with no pivoting.
FFT6, DCG4, and COLL refer to PELLPACK modules for sixth-order FFT nine-point differences, Dyakunov conjugate gradient with fourth-order accuracy, and collocation with band Gauss elimination, respectively.
The authors have organized the main body of their text in seven chapters devoted to basic linear algebra subprograms, basic concepts for matrix computations, Gauss elimination and LU decompositions of matrices, and a wide variety of other related subjects.

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