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Solving Equations Using Numerical Methods

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Introduction

Matt Coles        Centre Number – 40120        Candidate Number - 3135

Solving Equations Using Numerical Methods

For my coursework, I am going to solve equations that cannot be solved analytically. This is when the equation would have been solved by using the quadratic formula, completing the square or by factorising it. The equations used will be solved numerically and there are three methods that I will be using:

  • The Change of Sign method
  • The Newton-Raphson method
  • The Rearrangement method

Method 1: The Change of Sign Method

The first method that I will be using is the Change of Sign method. The equation that I will be solving is y=x3+3x2-2x-1. To do this, I will be drawing the function of                 f(x)=x3+3x2-2x-1.

Root α

Root β

Root γ

To solve this equation using the Change of Sign method, I will be using the Decimal Search on Microsoft Excel.Firstly, I started by entering the equation into Autograph to get a sketch of the graph.

...read more.

Middle

After that, I moved in closer, entering -3.5 to -3.49 (in 0.001’s) with the formula in the column next to it. This gave me the answer as to where the change of sign was – between -3.491 and -3.49.

Finally, I entered the numbers between -3.491 and -3.49 (in 0.0001’s) into Excel. This showed me that the change of sign was between         -3.4909 and -3.4908.

This is now up to 5 sig.fig. meaning that I have found the answer with the error bounds on the left hand side. The error bounds show that this works as there is a change of sign.

FAILURES

This method fails in two ways – when there is a repeated root or there are two roots within consecutive integers. To show how this method fails, I am using the equation x3+x2-5x+3=0. To do this, I will be drawing the function of f(x)= x3+x2-5x+3.

Method 2: The Newton-Raphson Method

Root α

Root β

...read more.

Conclusion

Root β. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents.The third root I have chosen to use is Root γ. Below is the Excel spread sheet with f(xn) and its derivative, f’(x) shown.

Root γ

On the left is the equation that shows the sliding tangent of the Newton-Raphson Method in a graphical form forRoot y. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents.  Root γ = 2.4023 (5 sig.fig.).This root has error bounds

FAILURES

y=2x3-3x2-6x+4

Root γ

Method 3: The Rearrangement Method

The third method that I am using is the Rearrangement Method. For this method, I will be using a different function to the other two. The equation I will be using for the Rearrangement method is y=x3+2x2-2x-2. This means that the function I will be using is f(x)= x3+2x2-2x-2.

...read more.

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