inear systems of equations arise in many computational tasks in many different disciplines.
They arise naturally when a continuous mathematical model is converted to a discrete numerical algorithm. However, there are also a huge number of initially discrete models where they also arise. As Wassily Leontief wrote in Scientific American in 1951 [153]:
This article is concerned with a new effort to combine economic facts and theory known as “interindustry” or “input-output” analysis. Essentially it is a method of analysis that takes advantage of the relatively stable pattern of the flow of goods and services among the elements of our economy to bring a much more detailed statistical picture of the system into the range of manipulation by economic theory. As such, the method has had to await the modern high-speed computing machine as well as the present propensity of government and private agencies to accumulate mountains of data. It is now advancing from the phase of academic investigation and experimental trial to a broadining sphere of application in grand-scale problems of national economic policy.1
Gaussian elimination is the oldest and the simplest — but not always the fastest — algorithm for solving matrix equations. The title of this chapter is quite long because a matrix equation can be solved by many different algorithms. The only ones we discuss are Gaussian elimination and a variant which is faster in certain circumstances. Frequently in physical systems, the matrix is sparse, that is, most of its elements are zero. Then the solution of the matrix equation might be faster if an iterative method is used (MATLAB has a number of functions to use such methods) — but it might not be. Even in this case, the sparse Gaussian elimination algorithm used in MATLAB will be “reasonably fast”.
Solving linear systems is not usually an end in itself; instead, it is just a small piece in a much larger code. Thus, this will be a rather technical chapter with few examples. However, it is a very important one. The solution of linear systems of equations by Gaussian elimination has been discussed analytically in Section 1.5 and numerically in Section 3.5. Now for the rest of the story.
Warning: Always keep in mind that the matrix A might be singular. It is possible that a mistake has been made — one or more of the elements were entered incorrectly — but maybe