# Numerical Differentiation

## Numerical Differentiation

Introduction

When finding the roots of an equation, the first thing you would try to do is factorise it. However sometimes factorizing does not find the root, therefore there are Numerical Methods that can be used to approximate the root, and the three different methods that I am using in this coursework are: Decimal search, Fixed Point Iteration, and Newton-Raphson.

These are the three main curves that I am using:

## Change of Sign

Change of sign works by taking increments between two values, then substituting the incremented values into the equation to get the y values. The intersection of the axis is found by the change of sign, because the line must have gone through the x-axis because it has changed from positive to negative or vice versa. The positive and negative values are then used as the bounds for the increments, and so the process is continued as you refine your search to the required degree of accuracy.

Finding the root of the equation between x=2 and x=3.

### Error Bounds

The change of sign method gives bounds between where the root could be. On the number line below you can see that the root alpha lies anywhere between the two bounds given by sign change.

This can be written as [2.392149, 2.392150]

Another way of writing the error bounds is that assuming the root lies exactly in-between the two bounds (midpoint).

1. Firstly finding the midpoint of the two gives: 2.3921495
2. Then taking the lower bound away from the midpoint gives 0.0000005
3. Therefore you can write α= 2.3921495 ± 0.0000005

Therefore you can write α= 2.392145 (6s.f.)

Or using the error formula: Error = X – x

α= 2.3921495 the error in this approximation is 5.0x107

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