Solving Equations by numerical methods.
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Solving Equations by numerical methods. The fact that the vast majority of the numbers on the real number line are irrational (i.e. cannot be put in the form p/q, where p and q are integers) has an associated consequence that the vast majority of equations involving powers of x (i.e. polynomials) are insoluble by closed analytical techniques. This coursework's solution implies finding values of x say c1, c2-...c3 where f(c1)=0; f(c2)=0 and so on. Further transcendental equations and non- linear equations require numerical methods for their solution. It is to be noted that in no way are numerical methods inferior to analytical solutions, they are indeed the only practical solutions available, however on the down side no exact solution is possible and error bounds have to be placed on the solution given. In this coursework three main methods of solution from one-dimensional equations are given, each of the methods could be extended to the multi-dimensional case. The aim of this coursework is to show example of the methods in action, where these are successful and where they are not. A comparison between the methods will also be attempted. Method 1: Decimal Search SOLVING AN EQUATION WITH THIS METHOD Fig1. Shows the function f(x) = 12 ln(x) - x3/2. (Where ln(x) represents the natural logarithm of x.) To find the roots, the equation is: 12 ln(x) - x3/2 = 0 This equation can not be solved numerically because x appears inside the function ln(x)
in the form x=g(x). A series of iterations attempts to improve on values of x; geometrically the method is relating on the intersection of the curve y=x with that of y=g(x). An initial value of x is chosen say x0. From this x value the value of the curve g(x) is obtained and a line drawn to the y=x curve, dropping down from this intersection point the value x1 is obtained from where the procedure continues as before. Eventually a point is reached simultaneous with the curve y=x and the curve g(x). In applying this method to the equation f(x) = 2ln(x-1) -x + 7.5 = 0, y = x f(x) = 2ln(x-1) - x + 7.5 x = 2ln(x - 1) + 7.5 = g(x) x = e^(0.5(x - 7.5)) + 1 = g(x) We immediately see the problem of this method. There are two choices for the iteration: 1. x n+1= 2ln(xn-1) + 7.5 2. xn+1= e^(0.5(xn-7.5)) + 1 1. Which of the choices of iteration for x should be used? Without a detailed analysis this question cannot be answered. However one criterion that could and should be used by the user is that for a successful iteration on an increasing function of x approaching the intersection with the curve y=x is that g'(x) should be less than 1 and greater than -1. For a decreasing function in this vicinity things are less clear cut.
A simple program (see appendix) can be programmed into the calculator that generates the x-values, or alternatively spreadsheet software such as Excel can be used. Overall Conclusion. Decimal search is largely reliable, if not slow. It required little programming, and would only fail in situations where the use was not careful. However even in taking care it is possible to miss roots in highly oscillatory local situations. Newton Raphson also can present problems; one often has to conduct an analysis to find the initial point, (not always possible with more complicated equations) on the range of values of x one can get away with in the initial trial for convergence to a root. The conclusion here must also be that further research is required into this method. Fixed point estimation iteration presents real problems for the user. By experience he must select from a number of options the most likely iterative method (normally it appears to be safest to choose an iteration with a (1/n)th root in it. Also one cannot be sure which root if any your chosen iterate will converge too, one can however for increasing functions use the criteria of the gradient of the iterative function should be less than 1 in the vicinity of the root. In other examples such as xn+1=K (1-xn) chaotic functions arise. Also in the literature using complex numbers Julia sets with beautiful and chaotic boundaries arise. The world of fractals may also be mentioned here as an area to explore if more time was available. Daniel Antelo Pure2 Coursework St Charles Sixth Form College ?? ?? ?? ??
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