Page 216 - MATLAB an introduction with applications

CHAPTER 4 4
4

Numerical Methods

4.1 INTRODUCTION

In this chapter, we introduce the solution of system of linear algebraic equations using such methods as
the Gauss elimination method, LU decomposition method, Choleski’s decomposition, Gauss-Seidel method,
Gauss-Jordan method and Jacobi method. A procedure based on Jacobi rotations, the Householder
factorization, symmetric matrix eigenvalue problems, Jacobi method, Householder reduction to tridiagonal
form, Sturn sequence and QR method are presented for the treatment of algebraic eigenvalue problems.
Numerical examples using MATLAB are provided to illustrate the procedures.

4.2 SYSTEM OF LINEAR ALGEBRAIC EQUATIONS

Here, we consider the solution of n linear, algebraic equations in n unknowns. A system of algebraic equations
has the form
 A 11 A 12 " A 1n   x 1  b  1 
 A A " A   x   
b
 21 22 2n   2  =  
2
#  # # # #     ...(4.1)
# 
     
  A 1 n A n 2 " A nn     x n   b   n  
where the coefficients A and the constants b are known, and x represents the unknowns.
j
i
ij
Equation (4.1) is simply written as
Ax = b
4.3 GAUSS ELIMINATION METHOD

Consider the equations at some instant during the elimination phase.