Analyse the use of three methods which are called the: change of sign, Newton-Raphson and the rearrangement method and use them to find roots of different equations.

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MATHS P2 COURSEWORK

There are many different kind of methods which can be used to find the roots of equations which can not be sold algebraically. In this coursework we are going to analyse the use of three of these methods which are called the: change of sign, Newton-Raphson and the rearrangement method and are going to use them to find roots of different equations.

Change of sign method

A root of an equation (where the graph crosses the x-axis) can be detected by finding where the solution of a formula changes sign from positive to negative. Where we find a change of sign on a graph using omnigraph we take the range of the numbers it is in and divide it by ten to find where it now changes sign. This procedure is then repeated to the required level of accuracy.

Here is a graphical representation of the systematic decimal search:

X 1

X 10-1

X10-2

Using change of sign method on excel

First you look up between 2 values (e.g. 1 and 2, or 4 and 5) where the graph crosses the x-axis. Then you take that as your range and divide it up into 10 equal segments. For example, if between 1 and 2.. you use 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, etc. you work out the formula for each of these and where the change of sign occurs you repeat the process between that range.

A

B

C

D

E

E

X1

f(A1)

A2+0.01

f(C1)

C1+0.001

f(C1)

2

A1+0.1

f(A2)

C1+0.01

f(C2)

E1+0.001

f(C2)

3

A2+0.1

f(A3)

C2+0.01

f(C3)

E2+0.001

f(C3)

4

A3+0.1

f(A4)

C3+0.01

f(C4)

E3+0.001

f(C4)

5

This process carries on to the required accuracy and this is how we try to find the root of an equation, using omnigraph and excel.

Example

I'm going to use the change of sign method to investigate when the method works and when it doesn't.

The equation I am going to find the root of is f(X)= x5+4x4-6x which looks like this.

As shown above there are 3 roots one which is at 0 and the others between -5 and -4, 1 and 2.

I'm going to find the root which lies between 1 and 2.

To find this root I'm going to use excel to see where the solution of the equation changes sign.

This shows the first few columns to how we came to find the root between 1 and 2.

FOR F(X)= x5+4x4-6x

x

f(x)

x

f(x)

x

f(x)

x

f(x)

x

f(x)

-4

24

-1

-1

.05

-0.16169

.058

-0.01045

-3

99

.1

0.86691

.01

-0.84657

.051

-0.14306

.0581

-0.00853

-2

44

.2

3.58272

.02

-0.68619

.052

-0.12434

.0582

-0.0066

-1

9

.3

7.33733

.03

-0.51869

.053

-0.10555

.0583

-0.00468

0

0

.4

2.34464

.04

-0.34391

.054

-0.08668

.0584

-0.00276

-1

.5

8.84375

.05

-0.16169

.055

-0.06774

.0585

-0.00083

2

84

.6

27.10016

.06

0.028133

.056

-0.04872

.0586

0.001093

3

549

.7

37.40697

.07

0.225736

.057

-0.02962

.0587

0.00302

4

2024

.8

50.08608

.08

0.431284

.058

-0.01045

.0588

0.004947

5

5595

.9

65.48939

.09

0.64495

.059

0.008804

.0589

0.006875

6

2924

2

84

.1

0.86691

.06

0.028133

.059

0.008804

The highlighted cells show where the solution to the function has a change of sign. The actual figure if we carry on this method comes to 1.0585432205 ± 0.0000000005. However, if we use 4.d.p as a sensible level of accuracy, we wud say the root is= 1.0585 < x < 1.0586

Where the change of sign method fails

However this method may not always be able to find the root of an equation. Here is an example of an equation that cannot be found using the systematic decimal search:

f(x)=x3+5.283x2-0.2144x-22.56371908

As shown there is a definite root between 1 and 2 and a possible root between -3 and

-4. Using systematic decimal search it is not clear whether a root lies between -3 and

-4. Here are the tables for f(x) =x3+5.283x2-0.2144x-22.56371908 produced on excel.

For: f(x)=x3+5.283x2-0.2144x-22.56371908

-5

-14.4167

-4

-1.17812

-3

-1.37352

-2

-9.00292

-1

-18.0663

0

-22.5637

-16.4951

2

6.139481

3

51.34008
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4

25.1067

5

233.4393

This table shows that a change of sign doesn't occur between -4 and -3 meaning there are no roots between these values. However, the graph shows that a change of sign does occur and has roots.

This proves that the change of sign method does not work for all equations, as the above shows there are two roots between -3.7 and -3.6 and -3.5 and -3.4 . The roots are very close together and this is why the roots don't show up, as a change of sign, in the ...

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