Engineering Optimization

Appendix BRoot Finding

B.1 Introduction

Root finding is a basic tool for optimization as well as for determining values in mathematical relationships. I’ll first develop it as a method to determine dependent variable values in a relation.

Explicit Equation: In an explicit equation, y(x), given a value of x you can arrange the relation so that y is isolated on the LHS. Then given a value for x, you can execute the RHS procedure to explicitly calculate the value for y. For example,

(B1)

Given a value of θ, you can explicitly solve for the value of y. The equation looks like an assignment statement in computer programming. Calculate the RHS value and then assign it to the y storage location.

Implicit Equation: By contrast in an implicit relation, you cannot isolate the dependent variable. For instance, if z is a function of x,

(B2)

Given an x‐value you cannot explicitly isolate z to LHS. The value of z is required to calculate the value of z.

However, either implicit or explicit relations can be rearranged to describe a function that equals zero by subtracting the RHS from both sides of the equation. The RHS becomes zero:

(B3)

(B4)

Now you have a function of the two variables, and the right ...