Page 72 - Numerical Methods for Chemical Engineering

2 Nonlinear algebraic systems

When a set of algebraic equations is nonlinear, there are no general uniqueness and existence
criteria, and solution can be quite difficult, even for sets of equations that appear simple.
This chapter discusses iterative techniques, in which we make an initial guess of the solution
that is refined by solving successive sets of linear equations. Hopefully, this sequence of
estimates converges to a solution. These methods are first introduced for a single nonlinear
algebraic equation, and then extended to systems of multiple nonlinear equations. The use
of MATLAB nonlinear algebraic solvers is demonstrated.

Existence and uniqueness of solutions to a nonlinear
algebraic equation

A single linear algebraic equation, ax = b, is easily solved, and the condition for existence
and uniqueness of the solution x = b/a, a = 0, is trivial. For a single nonlinear algebraic
equation

f (x) = 0 (2.1)

there is, in general, no way to tell a priori whether a solution exists, and if so, whether it is
unique. It is easy to identify nonlinear algebraic equations with multiple real roots,

3 2
f (x) = (x − 3)(x − 2)(x − 1) = x − 6x + 11x − 6 (2.2)
with only a single real root,

2
3
f (x) = (x − 3)(x − i)(x + i) = x − 3x + x − 3 (2.3)
or with no real roots at all,

4
2
f (x) = 3x + 2x + 1 (2.4)
Typically, we are presented with a nonlinear function that is not simple to factorize, and
so we know nothing about the number of real solutions. The methods described in this
chapter are designed to search for a real solution starting from an initial guess and will be
demonstrated on systems with varying numbers of solutions.

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