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maths pure

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Introduction

MEI PURE 2 COURSEWORK SOLUTION OF EQUATIONS BY NUMERICAL METHODS 1. Change of Sign Method This method makes use of the fact f(x) changes sign at a root of an equation. f(x) must be a continuous function i.e. it must not have any asymptotes or other breaks in it. Once an interval in which f(x) changes sign is located, we know that that interval contains a root. It is best to sketch the diagram of f(x) first so that we can see how many roots the equation has and their approximate positions. Decimal Search The equation that will be investigated here is f(x) = 4x3+5, a diagram of which is shown below. From the graph we can see that there is only one root. Zooming in, as shown below, we can also see that this root lies between X=-1 and X=-2 Taking increments in x of 0.1 within the interval [-2, -1] and working out the value of the function f(x) = 4x3+5 for each one and then seeing where the sign changes will enable the narrowing down of the interval. x -2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 f(x) -27 -22.4 -18.3 -14.7 -11.4 -8.5 -6.0 -3.8 -1.9 -0.3 1 The values calculated in the table above have only been given to 1 decimal place as we are only looking for a change in sign at this stage. ...read more.

Middle

will be very small. This will result in x2 not being close to the root. The values that are being computed may converge but it may be that they are converging towards a root other than the one we are trying to locate. 3. Rearranging f(x) = 0 in the form x = g(x) When trying to solve an equation f(x) = 0 by an iterative method, we first rearrange f(x) = 0 into a form x=g(x). The iteration formula is then? xn+1 = g(xn) The equation that will be looked at here is x3-3x-5=0, the graph of which is illustrated below. As can be seen above there is a root close to x=2. Rearranging the equation x3-3x-5=0 gives x = 3?(3x+5) Using the formula? xn+1 = 3?(3x+5) and starting with x0 = 2, the results for successive iterations are as follows: X0 2 X1 2.223980091 X2 2.268372388 X3 2.276967161 X4 2.278623713 X5 2.278942719 From this we can conclude that the root of the equation is 2.279 to 3 d.p. The convergence of a root such as this is often described as a staircase approach; why this is can be seen from the diagram below: The successive steps taken to approach the root is like that of a staircase with the values of xn approaching from one side. ...read more.

Conclusion

The rearranging method can be tedious in the sense that once the equation is rearranged, it has to be checked each time by differentiating and using an approximate value for the root. Again, similar to the Newton-Raphson method, if the rearranged function is a large, complex one a significant amount of time may be spent on the calculus part and with this method there is the added uncertainty that even when the function is differentiated it may not be suitable i.e. it may produce a diverging sequence. The use of SPA Software Omnigraph, Microsoft Excel and my Casio graphical calculator was beneficial for all three methods. Graphs could be easily plotted so that approximate values for boundaries could be seen and calculations were speeded up by the use of formulae functions. Although the use of software and hardware benefited all three methods it is perhaps the change of sign method that was speeded up most as virtually all the calculation with regards to this technique could be done by Excel/Graphical calculator. The other two techniques required differentiation which had to be done by hand. However, if the equation is a simple one that can be differentiated easily, the Newton-Raphson method, once programmed onto a computer is undoubtedly the quickest and reaches a high degree of accuracy in a relatively short amount of time. ...read more.

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