In this coursework, I am going to solve equations by using the Numerical Methods. Numerical methods are used to solve equation that cannot be solved algebraically

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Pure mathematic 2 coursework

Introduction:

In this coursework, I am going to solve equations by using the Numerical Methods. Numerical methods are used to solve equation that cannot be solved algebraically e.g. quadratic equations ax²+bx+c=0 can be solved using this formula:

x= -b± √ b² - 4ac

2a

Therefore numerical methods would not be used for quadratic equations. I will be working with equation which don’t have a formula to solve it. There are three methods, which I will be using:

· Change of sign method

· Newton-Raphson method

· Rearranging f(x) = 0 in the form x = g(x)

Objective:

Our object is to investigate the solution of equations using the three different methods.  To solve an equation we must find all its roots; a method which misses one or more roots will fail to solve the equation.

Change of sign method:

There are three ways in which we can do this method which are:

  • Decimal searcher
  • Interval bisection
  • Linear interpolation

Description:

This method works when a function crosses the x-axis. If we are looking for the root of the equation f(x) = 0. The point at which the curve crosses x-axis is the root. Once an interval where f(x) changes sign then the root must be in the interval.                                      

Decimal search:

The reason why I choose decimal search is because that is the easiest to use between the 3 different ways to find change is sign; this process can be continued to the required degree of accuracy. Decimal search is much faster then bisection. All the roots that I am going to find by using this method will be to 4 decimal places. By finding the root to 4 decimal place I can get an accurate and precise result for the roots.

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When decimal search method works:

I will try solving: F(x) = 0.2x5–x4+ x3+10  

I made this function up, the reason why I choose this equation is because this equation has just 1 root. We know if this method does not find all the roots then it will fail so therefore to show it does not fail I choose an equation which just has 1 root. Also there is no formula to solve this kind of equation.

Figure 1 shows the graph of F(x) = 0.2x5–x4+ x3+10.  

Figure 1

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