Engineering Optimization

Appendix CGaussian Elimination

C.1 Linear Equation Sets

Often optimization procedures result in a set of N linear equations with N unknown variable values. Gaussian elimination is one method for determining the variable values.

There are diverse ways to present a system of linear equations, but these are simply notational variations. One is a classic reveal of each equation. Equation set (C1) is an example with three unknowns, x, y, and z. The coefficients a, b, and c, and the right‐hand side elements, r, all have known values. Note that the unknowns x, y, and z each appear linearly (to the first power and independent of other unknowns) in the equation set. Remove the third equation and the cz elements, and it is a system of two equations and two unknowns. In general it could be a system of N equations linear in N unknowns. If a variable does not appear in a particular equation, insert it with a coefficient value of zero:

(C1)

Note that the structure of the abovementioned equation set has all of the unknown variables and coefficients on the left‐hand side, in the same order, and the constant on the right‐hand side. This makes it convenient to represent in matrix–vector notation:

(C2)

Although now there is only one equal sign, for the LHS matrix–vector product (a vector) to equal ...