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15 Solutions of Nonlinear Equations

15 Solutions of Nonlinear Equations - 15 Iterative Methods...

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Solution of Equations - 1 15 - Iterative Methods for the Solution of Non-linear Algebraic Equations in One Variable Motivation For any given y , it is easy to find an x such that y=35*x . For more complicated equations, the solution is not so simple. Consider finding x such that ) 13 , ( 10 * 4 ( 8288 . 3 18 x i abs β + = where β (x,13) is a Bessel function of the first kind. (The answer is 42). In this section, we discuss certain simple iterative methods for the solution of non-linear algebraic equations of the form f ( x ) = 0, where f is a continuous function (i.e., it does not exhibit "jumps" for any values of x ). They can easily be modified to an equation of the form f(x)=y (where y is a known number) by simply solving f(x)-y=0 . The answer is called the root of the equation. In the iterative methods discussed below, we solve equations by starting with a guess for x (which could be a very bad guess). Then we calculate what f(x) would be for that guess. We see how far off we were (i.e. by how much f(x) doesn’t equal zero), adjust our guess and try again. Eventually, we reason, we should get to a pretty good guess which will be our answer. Iterative Methods Convergence All iterative methods start with an initial guess, x 0 , and produce a sequence { x n } = x 1 , x 2 , ... , which, hopefully, converges to a true value of x , called x’ which solves the equation.. We say that the sequence { x n }, which converges to x’ , converges with order α when the error e n =x’-x n is such that C e e n n n = + α 1 lim C is a constant which is known as the asymptotic error constant .
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