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The Comparison Of Guassian Elimination And Cholesky Composition Methods To Linear System Of Equation

THE COMPARISON OF GUASSIAN ELIMINATION AND CHOLESKY COMPOSITION METHODS TO LINEAR SYSTEM OF EQUATION

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    1. INTRODUCTION

Gaussian elimination method, from the computational point of view is the simplest of all methods and is the most widely used standard method for solving a general system of linear simultaneous system. This method for solving systems of linear equation is a systematic process of elimination that reduces any given system of the form AX=b to “triangular form” because then, the system can be exactly solved by backward substitution.
We shall restrict ourselves to solving system of linear equations such as to solving system of linear equations such as 4 x 4, 5 x 5 and proceed to generalize for an n x n system.
Consider the system of equations

This can be written in matrix form:- AX=b.
Where

We are interested at this point in solving system of linear equation for the case of n=4, n=5, and we shall derive the general case.
2.2 DERIVATIVE OF GAUSSIAN ELIMINATION METHOD
We consider the linear system
AX = b, for a 4 x 4 system as shown below

PROCEDURE
STEP ONE
The first equation can be used to eliminate X1 from each of the remaining n-1 equation in the system of a110. in this case we call the first equation the pivotal equation and a11 the pivotal element, as simply the pivot that is:

And subtraction  from row 2 to obtain

Next is to eliminate X1 from equation (3)
Subtract  from row 3

Eliminate X1 from equation (4) subtract from Row 4

Hence we have

The Comparison Of Guassian Elimination And Cholesky Composition Methods To Linear System Of Equation

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