C3 Coursework

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  • Topic: Numerical analysis, Newton's method, Root-finding algorithm
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  • Published : March 6, 2012
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C3 Coursework

Numerical Methods

In this coursework I am going to investigate numerical methods of solving equations. The methods I will use are:

1. Change of sign method, for which I am going to use decimal search

2. Fixed point iteration using x = g(x) method

3. Fixed point iteration using Newton-Raphson method

I will then compare the methods in terms of speed of convergence and ease of use with hardware/software

Contents

|Change of sign | |Decimal search |3 | |Failure of change of sign method |7 | | | | x = g(x) | |x = g(x) method |8 | |Failure of x = g(x) method |12 | | | |Newton-Raphson | |Newton-Raphson method |13 | |Failure of Newton-Raphson method |16 | | | |Comparison of methods |17 |

Change of sign method

I have chosen the function of x, f(x) = 4x³ + 5x² + 4x – 1 and I am going to find one root using decimal search. I have plotted the graph y = 4x³ + 5x² + 4x – 1 below, and it can be seen that the root lies between 0 and 1.

Firstly, I am going to increase x by 0.1 each time between 0 and 1 and work out the value of the function for each value of x.

|x |f(x) |
|0 |-1 |
|0.1 |-0.546 |
|0.2 |0.032 |
|0.3 |0.758 |
|0.4 |1.656 |
|0.5 |2.75 |
|0.6 |4.064 |
|0.7 |5.622 |
|0.8 |7.448 |
|0.9 |9.566 |
|1 |12 |

We can see from the table that there is a change of sign between 0.1 and 0.2; therefore the root must lie between 0.1 and 0.2. I have plotted the graph below to illustrate this.

Since we now know that the root lies between 0.1 and 0.2, I am going to use this as my new interval and find the value of the function for each value of x.

|x |f(x) |
|0.1 |-0.546 |
|0.11 |-0.494176 |
|0.12 |-0.441088 |
|0.13 |-0.386712 |
|0.14 |-0.331024 |
|0.15 |-0.274 |
|0.16 |-0.215616 |
|0.17 |-0.155848 |
|0.18 |-0.094672 |
|0.19 |-0.032064 |
|0.2 |0.032 |

We can see that there is a change of sign between 0.19 and 0.2. I am going to illustrate this using a graph once more.

Since we now know that the root lies between 0.19 and...
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