This chapter discusses the solution of sets of linear algebraic equations and defines basic vector/matrix operations. The focus is upon elimination methods such as Gaussian elimination, and the related LU and Cholesky factorizations. Following a discussion of these methods, the existence and uniqueness of solutions are considered. Example applications include the modeling of a separation system and the solution of a fluid mechanics boundary value problem. The latter example introduces the need for sparse-matrix methods and the computational advantages of banded matrices. Because linear algebraic systems have, under well-defined conditions, a unique solution, they serve as fundamental building blocks in more-complex algorithms. Thus, linear systems are treated here at a high level of detail, as they will be used often throughout the remainder of the text.

Linear Systems of Algebraic Equations

We wish to solve a system of N simultaneous linear algebraic equations for the N unknowns x ₁, x ₂, , x _N, that are expressed in the general form

a _ij is the constant coefficient (assumed real) that multiplies the unknown x _j in equation i. b _i is the constant right-hand-side coefficient for equation i, also assumed real. As a particular example, consider the system

for which

It is common to write linear systems in matrix/vector form as

where

Row i of A contains the values a _{i 1}, a _{i 2}, , a _iN that...