Iterative Methods for Solving Equations - Revision Notes

In these revision notes for Iterative Methods for Solving Equations, we cover the following key points:

• What are iterative methods?
• When do we use iterative methods for solving equations?
• What does the 'change of sign method' for solving equations consist of?
• What are the two specific 'change of sign methods' for solving equations? Which is the best method to use?
• When is an equation that is solved by iterative methods considered as completed?
• What are some applications of iterative methods in geometry?
• What is recursive iteration? How to use it to solve equations?
• What is an iteration machine? What is it used for?

Iterative Methods for Solving Equations Revision Notes

We apply iterative methods when the exact solution of an equation is impossible to find because the roots are real (infinite) numbers. They are approximate solutions to more complex equations through repeating procedures. The term "iteration" means repeatedly carrying on a process.

In iterative methods, we first make a plan and choose to calculate the root values to a certain degree of accuracy (for example, to the nearest integer, tenth, hundredth, etc.) and then, we apply the procedure as many times as needed to "isolate" the possible solution to the desired order of accuracy.

The general method used for solving an equation is called the change of sign method. Thus, according to this method, we must look for the solutions (roots) where the expression contained in the left part of an equation is about to change the sign. The values of an equation's variables for which the corresponding expressions are exactly zero represent the roots (solutions) of this equation. The closer to zero the value of the expression, the closer to the correct value of the variable is the number chosen to represent the variable in calculations.

There are two 'change of sign' methods we can use when solving an equation through iterative methods. They are:

Going through values in order until there is a change in sign.

In this method, we choose a starting value and then we go through values in order (ascending or descending) until we obtain a change in the sign of the expression.

Using the half - interval division to localize the root.

In this method, after identifying an interval where is a change in sign, one must take half of that interval and make a new calculation. Then, consider again the half interval where there is a change in sign; then consider the half of this half (one - quarter of the original interval), and so on. This procedure is carried out until we obtain the desired level of accuracy.

Iterative methods are useful in many fields of science but they are particularly important in geometry, where we often deal with measurements. Thus, we often encounter situations, where the quantities measured, are irrational because there may be an irrational constant such as Archimedes' constant π involved in the calculations of the area of a circle, the volume of a sphere, etc.

An exercise involving iteration is considered as completed when after applying the iteration (no matter which of the two methods explained earlier you are using) a number of times, the variable is isolated in the desired range.

Sometimes, you will be given a formula to use in questions involving iteration. Such a formula - known as "iteration machine" - is applied a number of times for variables unit until you obtain two same results in a row after rounding it to the desired number of decimal places. According to this procedure, if x_n + 1 = x_n when the result is rounded at the desired number of decimal places, then this result represents the root of the original equation. This method is known as the recursive iteration.