The Practical Solution of Linear Equations

Abstract

The solution of the linear simultaneous equations

$$$$

(4.1)

where A is an nth order nonsingular matrix and b an arbitrary vector, is one of the most common tasks encountered when performing scientific, engineering or economics calculations. Such equations stem naturally from linear problems. They also arise in the solution of nonlinear problems, since most methods of solving a nonlinear problem involve breaking it down into a sequence of linear problems, where each linear problem requires the solution of one or more sets of linear equations for its own resolution. It is thus highly desirable to have at our disposal a good method for solving a general set of linear equations, with perhaps one or two special methods for use when the matrix of coefficients A exhibits special properties.