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numerical solutions-Comparison of the three methods and Newton Raphson

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Introduction

Numerical Solutions of equations

  1. Newton Raphson method

Equation to be solved is x³−5x−1=0  The function f(x)= x³−5x−1is shown below

image00.png

There are 3 roots. I will first find the root in the interval [2, 3]

I will do the first few lines of calculation manually.

The formula to use is:  xn+1= xn –f(xn)/ f’(xn)

Therefore I must first differentiate x³−5x−1 which is 3x2-5

Using x1=3

X2:   3 – [(33-5 x 3-1)/(3 x 32 -5)] = 2.5

X3:   2.5 – [(2.53-5 x 2.5-1)/(3 x 2.52 -5)] = 2.3455

I will now work out all 3 roots using autograph until 5 significant figures are guaranteedimage01.pngimage06.pngimage07.png

The 3 boxes above show how I obtained the 3 roots of the equation x³−5x−1=0    

Which are -0.20164, -2.1284 and 2.3301

Below is the function f(x)= x³−5x−1 showing where I applied the

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Middle

image10.png

image11.png

Error bounds for interval [2, 3]

2.3301 is the root in this interval to 5 significant figures.

Therefore the error is 2.3301 ± 0.00005 I will now perform the change of sign test to confirm it is within these limits.

Lower limit is 2.33005 then f (2.33005) = -0.000098648

Upper limit is 2.33015 then f (2.33015) = 0.0010302

There is a change of sign which confirms root in interval is 2.3301 ± 0.00005

When does this method fail?

The Newton Raphson method does not always work, I will show this Using the equation (5x+4)1/7 =0

This function is shown below with autographs attempt to find the root using tangents to the curve starting at x= -1

The function f(x) = (5x+4)1/7

image12.png

As you can see from the graph and the Values of x in the table, it was unable to find the root in the interval [-1, 0].image13.png

This is because of the gradient of the curve, where my starting value is very close to a turning point. Then rather than converging towards the root, it is diverging further and further away on the x axis, therefore overflows and is unable to find the root.

Comparison of methods

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Conclusion

Comparing Ease of use of Hardware and software

It’s very obvious that the hardware and software speeds up the process of working out the solutions dramatically, as manually is very time consuming and easier to make mistakes.

Autograph is very easy to use and therefore Newton Raphson and rearrangement methods are very easy to perform, as you simply have to add the equation and perform the functions required which only require very few steps and choosing the starting point.  However change of sign requires more knowledge of Excel, having to input the formula yourself and choosing appropriate values for x to converge to the root. However once you know how to use excel you are able to drag down and it copies the formula for each value of x without having to retype it. For the manual calculations I was able to use a calculator, although this was time consuming, setting the starting point to “ans” allowed me to quickly get all the values of x by simply pressing the equals button. Autograph also allowed me to zoom in on roots, changing axes, etc, this helped make the process of finding roots and showing failure much easier.

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