Numerical Methods for Chemical Engineering: Applications in MATLAB
Written in a pedagogic style, this textbook unites the applications of numerical mathematics and scientific computing to the practice of chemical engineering.
TABLE OF CONTENTS Numerical Methods for Chemical Engineering: Applications in MATLAB
Chapter 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
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,
with only a single real root,
or with no real roots at all,
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.
Iterative Methods and the Use...
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