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Numerical Methods/Equation Solving An equation of the type is either algebraic or transcendental. E.g, these equations are algebraic. and these are transcendental While roots can be found directly for algebraic equations of fourth order or lower, and for a few special transcendental equations, in practice we need to solve equations of higher order and also arbitrary transcendental equations. As analytic solutions are often either too cumbersome or simply do not exist, we need to find an approximate method of solution. This is where numerical analysis comes into the picture. Some Useful Observations The total number of roots an algebraic equation can have is the same as its degree. An algebraic equation can have at most as many positive roots as the number of changes of sign in . An algebraic equation can have at most as many negative roots as the number of changes of sign in . In an algebraic equation with real coefficients, complex roots occur in conjugate pairs If with roots then the following hold good: o o o If is continuous in the interval the interval and then a root must exist in Initial Approximation The last point about the interval is one of the most useful properties numerical methods use to find the roots. All of them have in common the requirement that we need to make an initial guess for the root. Practically, this is easy to do graphically. Simply plot the equation and make a rough estimate of the solution. Analytically, we can usually choose any point in an interval where a change of sign takes place. However, this is subject to certain conditions that vary from method to method. Convergence A numerical method to solve equations will be a long process. We would like to know, if the method will lead to a solution (close to the exact solution) or will lead us away from the solution. If the method, leads to the solution, then we say that the method is convergent. Otherwise, the method is said to be divergent. Rate of Convergence Various methods converge to the root at different rates. That is, some methods are slow to converge and it takes a long time to arrive at the root, while other methods can lead us to the root faster. This is in general a compromise between ease of calculation and time. For a computer program however, it is generally better to look at methods which converge quickly. The rate of convergence could be linear or of some higher order. The higher the order, the faster the method converges. If is the magnitude of the error in the th iteration, ignoring sign, then the order is approximately constant. It is also important to note that the chosen method will converge only if if is . Bisection Method This is one of the simplest methods and is strongly based on the property of intervals. To find a root using this method, the first thing to do is to find an interval such that . Bisect this interval to get a point . Choose one of or so that the sign of is opposite to the ordinate at that point. Use this as the new interval and proceed until you get the root within desired accuracy. Example Solve correct up to 2 decimal places. Error Analysis The maximum error after the th iteration using this process will be given as As the interval at each iteration is halved, we have linearly. . Thus this method converges If we are interested in the number of iterations the Bisection Method needs to converge to a root within a certain tolerance than we can use the formula for the maximum error. Example How many iterations do you need to get the root if you start with a = 1 and b = 2 and the tolerance is 10−4? The error needs to be smaller than 10−4. Use the formula for the maximum error: Solve for i using log rules