TABLE OF CONTENTS When engineering systems are modeled, the mathematical description is frequently developed in terms of sets of algebraic simultaneous equations. Sometimes these equations are nonlinear and sometimes linear. In this chapter we discuss systems of simultaneous linear equations and describe the numerical methods for the approximate solutions of such systems. The solution of a system of simultaneous linear algebraic equations is probably one of the most important topics in engineering computation. Problems involving simultaneous linear equations arise in the areas of elasticity, electric-circuit analysis, heat transfer, vibrations, and so on. Also, the numerical integration of some types of ordinary and partial differential equations may be reduced to the solutions of such systems of equations. It has been estimated, for example, that about 75% of all scientific problems require the solution of a system of linear equations at one stage or another. It is therefore important to be able to solve linear problems efficiently and accurately.
A linear equation is an equation in which the highest exponent in a variable term is no more than one. The graph of such equation is a straight line.
A linear equation with two variables x and y is an equation that can be written in the form
where a, b, and c are real numbers. Note that this is the equation of a straight line in the plane. For example, the equations
are all linear equations in two variables.
A linear equation in n
