# Solving Equations. Three numerical methods are discussed in this investigation. There are advantages and also disadvantage in these methods and they will be discussing throughout this investigation.

Numerical Solution of Equation

INTRODUCTION

In many situations, equation can be solving by algebra and by graphical analysis. However when equation involves terms which had a higher order than two then it became very difficult to solve by algebra. In this investigation we are focus of cubic functions – functions which involves term. It is possible to solve algebraically but it had certain level of difficulties because it involves imaginary number  and this increase the difficulties of solving cubic equations algebraically. A problem also occur because  was not introduce before early 16 century therefore mathematicians use numerical analysis instead of solving cubic equation algebraically. Three numerical methods are discussed in this investigation. There are advantages and also disadvantage in these methods and they will be discussing throughout this investigation.

In cubic functions, there is 1 solution which constitute by 3 or less distinct roots. All roots of the equations are laid on the x axis, in other words it is intercepting linear equation x = 0. Three methods we are discussing in this investigation is base on this concept and derive to find one target root in an equation.

1. Method of Bisection

This is an example of method of interval estimation; it was used in continuous function. In continuous function f(x) = 0, if it intercepts liner equation x = 0 then this implies that f(x) =y, must have at least one real root for solution. One observation can be made is variable y will have a ‘sign change’ on either side of roots, it means when the y value on the left hand side of root is negative then the right hand side of the root must be positive.

It can be illustrate graphically:

Example 1)

In this equation we define two point points, we call it a and b (they are 0 and 2 respectively in this case) on both sides of the root. The mid point, c is then found by formula. Then f(c) is calculate, the purpose of calculate f(c) is to define the position on c. If f(c) > 0 then it means the range  does not contain the roots and there fore the roots must between. We will then repeat the same process and reduce the range of root until the data approach to the solution, although we can never get the exact value of the root but we will be able to approach to the root to 5 decimal places (5d.p).

Microsoft excel can be set up in order to reduce human error and increase efficient of calculation.

A spread sheet is set up for

This is a sample I select form my actual table to show how a spread sheet can help. There are 8 columns to set up in order to use the method of bisection and they are as follows:

– The number of bisection done.

– Lower bound of the target root.

– Output of the function when x=a

– Upper bound of the target root.

– Output of the function when x=b

– The mid point of a and b

– Output of the function when x=c

– Possible error can occur in the result.

After 20th terms and approximation of the root  to is found. it had an error bound of, in other words it had a solution bound of  with a maximum error.

This method can also be shown graphically by computer:

This method can be drawn in software ‘autograph’. By using this program the equation can be enter and there is a function key call ‘bisection iteration’ then we can enter upper and lower bound and this program will be able to help us to find the root automatically.

However this method is not always working properly. There are two ...