Page 12 - Numerical Methods for Chemical Engineering

1 Linear algebra

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 elim-
ination, 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 11 x 1 + a 12 x 2 +· · · + a 1N x N = b 1
a 21 x 1 + a 22 x 2 +· · · + a 2N x N = b 2 (1.1)
.
.
.
a N1 x 1 + a N2 x 2 +· · · + a NN x N = b N
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

x 1 + x 2 + x 3 = 4
2x 1 + x 2 + 3x 3 = 7 (1.2)
3x 1 + x 2 + 6x 3 = 2
for which

a 11 = 1 a 12 = 1 a 13 = 1 b 1 = 4
a 21 = 2 a 22 = 1 a 23 = 3 b 2 = 7 (1.3)
a 31 = 3 a 32 = 1 a 33 = 6 b 3 = 2

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