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C3 Numerical Solutions to Equations

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Introduction

Adam Arstall

Numerical Solutions to Equations Coursework

Change of sign method

The change of sign method involves finding the interval in which a root of an equation lies by taking two values of x and showing that the root lies between them as the value of f(x) for each case has a different sign. A change of sign will always indicate a root if the function is continuous.

This method will be used to find a root of the equation f(x)=x⁵+x⁴−2x³+5x²−7x−2=0. As the function f(x) is continuous, a change of sign will always indicate a root.

image00.png

The method will be used to find the root which lies between -2 and -3

image01.png

As the root lies in the interval [-2.7211575,-2.7211574], x=-2.72115745 ± 0.000000005

image09.png

Here it is shown that f(x) changes from negative to positive between -2.7211575 and-2.7211574 and the root to f(x)=0 is between these x values.

This method can fail to find roots in some cases. For example if the equation has a repeated root as shown below for the equation x³−0.96x²−5.

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Middle

Therefore x=-1.53407020 ± 0.000000005

To confirm this root there must be a change of sign. f(-1.5340703) = 1.16*10^-6.

f(-1.5340701) = -1.08*10^-6. Therefore there is a root as the function is continuous.

image13.png

image14.png

These two diagrams show the convergence of the iterations at different magnifications. It is shown that the root lies between -1.53407035 and -1.53407025.

Taking 1 as the first guess gives the following results

image15.png

Therefore x=0.48269595 ± 0.000000005

f( 0.48269594) = -1.08*10^-7. f(0.48269596) = 0.0113

Therefore there is a root as the function is continuous.

Taking 4 as the first guess gives

Therefore x=4.05137424 ± 0.000000005image16.png

f(4.05137423) = 2.34*10^-7  f(4.05137425) = -1.65*10^-7

Therefore there is a root as the function is continuous.

This method can fail for some starting points such as ones where the gradient of the curve is very small which can lead to the iterations converging on the wrong root. For the equation ¼x³−½x²−2x+3=0, an initial guess of 2.35 to find the root between 0 and 3 makes the iteration converge on the root between -4 and -2 as shown below.

image02.png

Rearranging f(x)=0 in the form x=g(x)

In this method the equation f(x)

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Conclusion

The change of sign method was fairly easy to use as it requires little extra hardware or software and can be carried out manually if necessary using only a calculator. This makes it particularly useful if computer software is not available. However the need for much manual computation can make the process quite laborious and time consuming. The Newton-Raphson and x=g(x) methods are relatively similar in terms of ease of use with hardware and software. Both make good use of Autograph software visually interpret equations before using an Excel spreadsheet to carry out the calculations to find each root. Compared to the change of sign method both are generally more easy to use as once the initial formula has been entered it is very quick and simple to do the iterations many times.

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