# Numberical method

Extracts from this document...

Introduction

### In this coursework I am going to use knowledge of numerical methods to produce an approximation root which cannot be solved using analytical methods.

Some particular equations Iwill be using during the coursework because they’re cannot be integrated by any analytical method therefore approximation method need to be use, because of this numerical method is a suitable method to find out the solution of these function.

## 1 Change of sign method

## Consider this equation

## From the graph we can see point A lies between -2 and -3

## When x= -2 given

Y=1.6

## When x= -3 given

## Y= -28.1

## There is a change in sign between – 2 and -3 means that there is a root between the two points.

Decimal research

## For function f(x) =

Middle

- I now repeat the process by change the increment from 0.1 to 0.01 between -2.2 and -2.1 given that

To work out x I’ll use the formula r+1= r+0.01 to give the function of x

➔ f(x)+1=

x | -2.2 | -2.19 | -2.18 | -2.17 | -2.16 | -2.15 | -2.14 | -2.13 | -2.12 | -2.11 | -2.1 |

f(x) | -0.20726 | -0.0849 | 0.03381 | 0.14885 | 0.2603 | 0.36822 | 0.472669 | 0.5736954 | 0.6713563 | 0.76570596 | 0.856798 |

From the table there is a sign change from -2.19 to -2.18 which means this become the new interval in which the root is between.

- The interval had narrow down to -2.18 and -2.19, I’ll then now change the increment size to 0.001 which will zoom in the interval even more

Conclusion

In other cases where this method would fails to work is when there’re roots are very much close together because they are likely to be in the same interval result a missing of change in sign.

And also when a function given a discontinuous curve the whole fraction has an asymptote between some specific intervals that make the change in size think there is a root present but there is no real root present.

## Fixed point Iteration using method

## Given the equation = 0

This can be rearrange into the form of x=g(x)

= 0

=>

=>

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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