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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 4. Gaussian Elimination In this part, our focus will be on the most basic method for solving linear algebraic systems, known as Gaussian Elimination in honor of one of the all-time mathematical greats the early nineteenth century German mathematician Carl Friedrich Gauss. As the father of linear algebra, his name will occur repeatedly throughout this text. Gaus- sian Elimination is quite elementary, but remains one of the most important algorithms in applied (as well as theoretical) mathematics. Our initial focus will be on the most important class of systems: those involving the same number of equations as unknowns although we will eventually develop techniques for handling completely general linear systems. While the former typically have a unique solution, general linear systems may have either no solutions or infinitely many solutions. Since physical models require exis- tence and uniqueness of their solution, the systems arising in applications often (but not always) involve the same number of equations as unknowns. Nevertheless, the ability to confidently handle all types of linear systems is a basic prerequisite for further progress in the subject. In contemporary applications, particularly those arising in numerical solu- tions of differential equations, in signal and image processing, and elsewhere, the governing linear systems can be huge, sometimes involving millions of equations in millions of un- knowns, challenging even the most powerful supercomputer. So, a systematic and careful development of solution techniques is essential. Section 4.5 discusses some of the practical issues and limitations in computer implementations of the Gaussian Elimination method for large systems arising in applications. 4.1. Solution of Linear Systems. Gaussian Elimination is a simple, systematic algorithm to solve systems of linear equations. It is the workhorse of linear algebra, and, as such, of absolutely fundamental importance in applied mathematics. In this section, we review the method in the most important case, in which there are the same number of equations as unknowns. To illustrate, consider an elementary system of three linear equations x + 2 y + z = 2 , 2 x + 6 y + z = 7 , x + y + 4 z = 3 , (4 . 1) 3/15/06 45 c 2006 Peter J. Olver in three unknowns x, y, z . Linearity refers to the fact that the unknowns only appear to the first power, and there are no product terms like xy or xy z . The basic solution method is to systematically employ the following fundamental operation: Linear System Operation #1: Add a multiple of one equation to another equation. Before continuing, you might try to convince yourself that this operation doesnt change the solutions to the system. Our goal is to judiciously apply the operation and so be led to a much simpler linear system that is easy to solve, and, moreover has the same solutions as the original. Any linear system that is derived from the original system by successiveas the original....
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