Introduction
Numerical methods are a set of computational techniques used to solve complex mathematical problems, particularly differential and algebraic equations that lack analytical solutions. These methods allow us to obtain accurate approximate solutions. MATLAB is a powerful numerical computing software widely used in engineering, science, and other fields due to its ease of use and extensive libraries. One of MATLAB’s key applications is the implementation of various numerical methods.
The Newton-Raphson Method
The Newton-Raphson method is a root-finding algorithm that provides good approximations to the roots of a function. For a function f(x) with derivative f'(x) and an initial guess x0, if the function and initial guess are sufficiently accurate, then x1 is a better approximation than x0. Geometrically, (x1, 0) is the point where the x-axis intersects the tangent line of the function f at the point (x0, f(x0)). The general form of the Newton’s algorithm is:
which is basically obtained from the following relationship:
We know that at the point where the function meets the axis, the value of the function will be zero, therefore:
Finally, by dividing by , the equation can be rewritten in the following form:
As it is evident, the Newton-Raphson method utilizes a truncated Taylor series of the assumed function as a linear approximation around the initial guess point. It is called an approximation because there is no need to write the series of the function up to higher orders, and only the first two terms are sufficient, which is also a reason for the linear approximation of the Newton method.
Also, since this method simplifies the equation of a function to the equation of a linear function, regardless of how many roots the function has, the algorithm ultimately finds only one solution.
Secant Method:
One of the methods of finding the root of an equation. The reason for naming this method is that in step n, the point x(n+1) is obtained from the intersection of the line with the graph. The advantages and features of this method compared to the Newton method is that it does not need the derivative of the function. Also, compared to the fixed-point method, it is not necessary for the two initial guess points to be on both sides of the root of the function. This method does not have guaranteed convergence. However, if it converges, it converges to the root quickly.
Fixed-Point Iteration Method:
In the discussion of calculating equilibrium concentrations in equilibrium reactions, it is also good to be familiar with the discussion of solving equations using numerical methods. As we have seen so far, in most cases, to obtain equilibrium concentrations, we need to solve equations and find the unknowns considered.
In general, two types of methods are considered for solving equations: analytical methods and numerical methods or equivalently direct methods and iterative methods. Direct methods give the exact solution of an equation based on a specific algorithm and by going through a certain number of steps. For example, the delta method is a direct method for finding the roots or solutions of quadratic equations. Unfortunately, there is no direct method for solving many equations, and for solving most equations, we are forced to resort to iterative or numerical methods. There are various numerical methods, and many software programs in computers or calculators use these methods to find solutions. Here, we present a numerical method for solving equations called the fixed-point iteration method in MATLAB software.
The following numerical methods are available in MATLAB code: