How to Solve Equations Using Fixed Point Iteration Method: Theory and Practice with Free PDF
Fixed Point Iteration Method: A Simple and Effective Way to Solve Equations
Have you ever encountered an equation that is too difficult or tedious to solve by hand? Have you ever wondered how computers can find solutions to equations quickly and accurately? If so, then you might be interested in learning about the fixed point iteration method, a simple and effective way to solve equations using numerical analysis.
Fixed Point Iteration Method Pdf Free
In this article, we will explain what a fixed point iteration method is, why you should use it, how to implement it, and how to download a free PDF on the topic. By the end of this article, you will have a better understanding of this powerful and versatile method that can help you solve equations in various fields of science, engineering, and mathematics.
What is a fixed point iteration method?
A fixed point iteration method is a numerical method that uses repeated function applications to find an approximate solution to an equation. More specifically, given a function f(x) defined on the real numbers with real values and given a point x0 in the domain of f(x), the fixed point iteration method is:
xn+1 = g(xn), n = 0, 1, 2, ...
which gives rise to the sequence x0, x1, x2, ... of iterated function applications x0, g(x0), g(g(x0)), ... which is hoped to converge to a point xfix such that:
f(xfix) = 0.
A fixed point is a point in the domain of a function g(x) such that g(x) = x. In other words, a fixed point is a point that does not change when the function is applied to it. The fixed point iteration method uses the concept of a fixed point in a repeated manner to compute the solution of the given equation.
Definition and examples of fixed points
Let us look at some examples of fixed points to illustrate the idea. Consider the function g(x) = cos(x), which is defined on the real numbers with real values. The graph of this function looks like this:
We can see that there are some points on the graph where the function value is equal to the input value. These points are called fixed points. For example, one such point is x = 0.7390851332151607..., which is approximately equal to 0.74. We can verify that this is a fixed point by plugging it into the function:
g(0.7390851332151607...) = cos(0.7390851332151607...) = 0.7390851332151607...
This means that applying the function g(x) to this point does not change its value. This point is also called the Dottie number, named after the mathematician Dorothy Lewis Bernstein.
Another example of a fixed point is the function g(x) = x^2 - 2, which is defined on the real numbers with real values. The graph of this function looks like this:
We can see that there are two points on the graph where the function value is equal to the input value. These points are called fixed points. For example, one such point is x = -1.414213562373095..., which is approximately equal to -1.41. We can verify that this is a fixed point by plugging it into the function:
g(-1.414213562373095...) = (-1.414213562373095...)^2 - 2 = -1.414213562373095...
This means that applying the function g(x) to this point does not change its value. This point is also called the negative square root of 2.
How to convert an equation into a fixed point form
Now that we have seen some examples of fixed points, how can we use them to solve equations? The key idea is to convert the given equation into a fixed point form, that is, an equation of the form x = g(x). This way, we can use the fixed point iteration method to find an approximate solution.
There are many ways to convert an equation into a fixed point form, but we have to be careful about choosing a suitable function g(x) that satisfies some conditions for convergence. We will discuss these conditions later, but for now, let us look at some examples of converting equations into fixed point forms.
Example 1: Consider the equation 2x^3 - 2x - 5 = 0. We can convert this equation into a fixed point form by isolating x on one side and applying some algebraic manipulations. For example, one possible way is:
2x^3 - 2x - 5 = 0
2x^3 - 5 = 2x
x^3 - (5/2) = x
x^3 = x + (5/2)
x = (x + (5/2))^(1/3)
Therefore, we can choose g(x) = (x + (5/2))^(1/3) as our fixed point form.
Example 2: Consider the equation cos(x) = x. We can see that this equation is already in a fixed point form, where g(x) = cos(x). Therefore, we do not need to do any conversion for this equation.
How to choose an initial guess and a function g(x)
Once we have converted the equation into a fixed point form, we need to choose an initial guess and a function g(x) for the fixed point iteration method. The initial guess is a point x0 in the domain of f(x) that is close to the actual solution. The function g(x) is the function that defines the fixed point form.
The choice of the initial guess and the function g(x) can affect the convergence and accuracy of the method. Therefore, we have to be careful about choosing them wisely. Here are some tips and guidelines for choosing them:
Choose an initial guess that is close to the actual solution. This can reduce the number of iterations and improve the accuracy of the method.
Choose a function g(x) that satisfies some conditions for convergence. These conditions are related to the derivative and magnitude of g(x). We will discuss these conditions in detail later.
Choose a function g(x) that has a small derivative at the fixed point. This can increase the rate of convergence and make the method faster.
Choose a function g(x) that has only one fixed point in the interval of interest. This can avoid confusion and ambiguity about which fixed point is the actual solution.
If there are multiple ways to convert an equation into a fixed point form, choose the one that has the best properties for convergence and accuracy.
Why use a fixed point iteration method?
A fixed point iteration method is a simple and effective way to solve equations using numerical analysis. But why should we use it instead of other methods? What are its advantages and disadvantages? What are its applications and use cases? How does it compare with other numerical methods? Let us answer these questions in this section.
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