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In this piece of coursework, I will use three different methods (Change of Sign, Newton-Raphson, and the Rearrangement Method) to find the roots of a series of different equations. The number of roots found differs from method to method.

Extracts from this document...

Introduction

Colin Tebbett

Pure Mathematics 2: Coursework Assessment

Change of Sign Method

Equation:        x3 + x2 - 2x + 3 = 0

Using the above equation, the function f(x) = x3 + x2 - 2x + 3 may be generated.

This may be expressed in graphical form (see page 2)

This may be solved using the method of Interval Bisection (see page 3) <--------Interval-------> Mid Point Height at A Height at B Height at M Curve a b Mid Point f(a) f(b) f(m) y=x^3+x^2-2*x+3 -3 -2 -2.5 -9 3 -1.375 -2.5 -2 -2.25 -1.375 3 1.171875 -2.5 -2.25 -2.375 -1.375 1.171875 -0.005859375 -2.375 -2.25 -2.3125 -0.005859375 1.171875 0.606201172 -2.375 -2.3125 -2.34375 -0.005859375 0.606201172 0.306060791 -2.375 -2.34375 -2.359375 -0.005859375 0.306060791 0.151584625 -2.375 -2.359375 -2.3671875 -0.005859375 0.151584625 0.073235035 -2.375 -2.3671875 -2.37109375 -0.005859375 0.073235035 0.033781111 -2.375 -2.37109375 -2.373046875 -0.005859375 0.033781111 0.013984211 -2.375 -2.373046875 -2.374023438 -0.005859375 0.013984211 0.004068256 -2.375 -2.374023438 -2.374511719 -0.005859375 0.004068256 -8.94099E-04 -2.374511719 -2.374023438 -2.374267578 -8.94099E-04 0.004068256 0.001587443 -2.374511719 -2.374267578 -2.374389648 -8.94099E-04 0.001587443 3.46763E-04 The root 'x' can be seen to be in between -2.374511719 and -2.374267578 ie. -2.374511719 <    x    < -2.374267578 An estimate for the root is x= -2.374389648 ± 1.22070E-04 or -2.3744 (correct to 4 decimal places)

The Change of Sign Method can, however, fail when certain equations must be solved.

Middle

2.409090909

2.936232156

12.41115702

2.172510859

X3

2.172510859

0.39126991

9.159410291

2.129793053

X4

2.129793053

0.011815313

8.608055342

2.128420465

X5

2.128420465

1.20350E-05

8.590521025

2.128419064

X6

2.128419064

1.25322E-11

8.590503134

2.128419064

X7

2.128419064

-1.77636E-15

8.590503134

2.128419064

An estimate for root 'c' is

2.128419064

or

2.12842

(correct to 5 decimal places)

Error bounds for ‘c’

The value found in the spreadsheet above for root ‘c’ is 2.12842 (5 d.p.)

Because this value is to 5 decimal places, the error boundaries are as follows:

2.128415        <        c        <        2.128425

We can check that this is correct using the Change of Sign Method (Interval Bisection) as described above (see page 11).

 <--------Interval-------> Mid Point Height at A Height at B Height at M Curve a b Mid Point f(a) f(b) f(m) y=x^3-5*x+1 2 3 2.5 -1 13 4.125 2 2.5 2.25 -1 4.125 1.140625 2 2.25 2.125 -1 1.140625 -0.029296875 2.125 2.25 2.1875 -0.029296875 1.140625 0.530029297 2.125 2.1875 2.15625 -0.029296875 0.530029297 0.244049072 2.125 2.15625 2.140625 -0.029296875 0.244049072 0.105808258 2.125 2.140625 2.1328125 -0.029296875 0.105808258 0.037865162 2.125 2.1328125 2.12890625 -0.029296875 0.037865162 0.00418669 2.125 2.12890625 2.126953125 -0.029296875 0.00418669 -0.012579434 2.126953125 2.12890625 2.127929688 -0.012579434 0.00418669 -0.00420246 2.127929688 2.12890625 2.128417969 -0.00420246 0.00418669 -9.40741E-06 2.128417969 2.12890625 2.128662109 -9.40741E-06 0.00418669 0.002088261 2.128417969 2.128662109 2.128540039 -9.40741E-06 0.002088261 0.001039331 The root 'x' can be seen to be in between 2.128417969 and 2.128662109 ie. 2.128417969 <    x    < 2.128662109 An estimate for the root is x= 2.128540039 ± 1.22070E-04 or 2.12854 (correct to 5 decimal places)

The Newton-Raphson Method can, however, fail when certain equations must be solved.  Below is an example:

Equation:        5 – 1/((x – 1)2 + 0.2)) = 0

Function:        f(x) = 5 - 1/((x – 1)2 + 0.2))

This cannot be solved using the Newton-Raphson Method because the gradient of the tangent at the starting point is very shallow.  Therefore the method makes the tangent diverge away from the root.

The function expressed in graphical form is shown on page 13.

An attempt to solve the equation using the Newton-Raphson Method is shown below:

If y = 5 – 1/((x – 1)2 + 0.2))

Then dy/dx = (50 * (x - 1)) / (5x2 – 10x + 6)2

 Estimates:

Conclusion

The Newton-Raphson method, however, took 30 steps and still was not fully finished.  This is time-consuming and unnecessary.

The Rearrangement Method took 10 steps in order to prove success or express failure.  It also involved the use of manual calculations, which are relatively slow when compared to the use of a spreadsheet.

Thus, the Change of Sign method was the easiest and quickest method to use.

The Hardware available plays a large part in the ease of each method.  A computer helps greatly, as complex and repetitive calculations can be done easily.  This is helpful in all three methods, as may be seen above.

Graphical calculators can also make the methods easier to complete.  Methods that are graph-intensive, such as the Rearrangement Method, can either be completed or checked using a graphical calculator.

Other Hardware can also be used, however, these are generally large, specialised machines that are specifically designed to find the root to complex equations.

The Software available also can make different methods a lot easier.  Spreadsheets (using Microsoft Excel) and Graphs (predominantly drawn using Autograph) can be incorporated into text documents (Microsoft Word) quickly and clearly, and thus reference can be made to them throughout a report.  Difficult equations (such as that used in Rearrangement 2 in the Rearrangement Method) can be differentiated using Derive, which is helpful as it saves time and prevents mistakes.  Therefore, the software available plays a large part in the ease of which each method can be completed.

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