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Change of sign method - Finding a root by using change of sign method

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Introduction

Pure 2 coursework Change of sign method Finding a root by using change of sign method I will divide interval into a number to apply the change of sign method by using decimal search.. The two values are the new interval's two sides on the number axis and the roots must lie between these intervals. i am going to use the following equation to find out a root. it is clear that we can see the first interval lies on [1, 2]. Then i will use Excel. Value 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 In this method, I take increments in size 0.1 within the interval [1 , 2] Work out each value of and see whether the value is positive or negative. In this case, I will use Microsoft Exel to solve it. The diagram decimal research above shows how the sign change between [1.5,1.6]. Because of this, the interval of this equation will be [1.5,1.6]. The following graph can prove whether the interval is correct. I decide to use Autograph. This graph has been zoom in from the first graph. It is very clear to see that the result is correct Which is between[1.5,1.6]. I use the same method to keep doing decimal research. to work out a more accurate answer. Take increments in size 0.1 within the interval [1.5,1.6] From the above we can see that must lie between [1.52,1.53].It can be very clear if I use graph. ...read more.

Middle

Poor choice of starting point. If your initial value is close enough to a root, the method will nearly always give convergence to it. However if the initial point is not close to the root or is near a turning point of y=f(x),the iteration may diverge, or converge to another root. This is the equation that I am going to use, As we can see there are 3 roots lie on the graph. I start by using Newton-Raphson method to do it. Firstly, I estimate x=3 to see whether it can work. It is converge to 0. If I try another equation which is Now I am going to investigate it by using the same method. I take x=1.5 In this case I found that after one step the value is converging rapidly, but it converges to another root. Rearrangement f(x)=0 in the form x=g(x) Find a root of the equation I am going to investigate the following equation: let the equation then rearrange it into the form. The first step, with an equation f(x)=0,is to rearrange it into the form x=g(x).Any value of x for which x=g(x) is clearly a root of the original equation. Can be written in this way: The graphs of y=x and y=g(x) in this case: From the graph above, the red line is , the blue line is y=x. ...read more.

Conclusion

Merits of the three methods: Change of sign method is a good method because the whole process and set up is the simplest in the three methods. Its error bounds are also very easy to find out. The failure only would happen when some several roots are close together, the curve touches the x-Axis or there is a discontinuity. I think Newton-Raphson method is the best method of these three. It can find the root with a easy set up and fast iteration. But it is not easy to find out a very accurate error bounds. Rearranging method is the most difficult way to solve question. With a lot of requirements. Such as we need the gradient and the starting point. However, this method is the fastest method that can converge to the root. It is good to iterate a lot to get a good solution. The failure only would happen, which the point of the tangent is not between and the Comparison of hardware and software It is always need to draw the graph and then check carefully because all of these three methods have possibilities to make a failure. I have used the Microsoft Excel, Change of sign method; the Autograph, Microsoft Excel for Newton-Raphson method; Autograph for Rearranging method;Microsoft Excel for Comparison of method. I found the Autograph is the easiest method to use. It can quickly be set up and easy to use.The graph can also be shown very clearly. ...read more.

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