Tham khảo tài liệu 'computational physics - m. jensen episode 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . ITERATION METHODS 89 Iteration methods To solve an equation of the type x 0 means mathematically to find all numbers s1 so that s 0. In all actual calculations we are always limited by a given precision when doing numerics. Through an iterative search of the solution the hope is that we can approach within a given tolerance 6 a value . V which is a solution to f s 0 if xo - s e and s 0. We could use other criteria as well like x s im -L e and xo e or a combination of these. However it is not given that the iterative process will converge and we would like to have some conditions on f which ensures a solution. This condition is provided by the so-called Lipschitz criterion. If the function f defined on the interval a b satisfies for all .r and . in the chosen interval the following condition . . with a constant then is continuous in the interval b . If f is continuous in the interval a b then the secant condition gives f xl -f x2 f xl-x2 with Xi X2 within a b and t within xi 2 . We have then 1 371 - 2 1 e kl - at2 The derivative can be used as the constant . We can now formulate the sufficient conditions for the convergence of the iterative search for solutions to f s 0. 1. We assume that is defined in the interval b . 2. f satisfies the Lipschitz condition with k 1. With these conditions the equation y 0 has only one solution in the interval a b and it coverges alien iterations towards the solution s irrespective of choice for 0 in the interval b . If we let be the value of X after iterations we have the condition g - xn ati - at2 . . . . . . The proof can be found in the text of Bulirsch and Stoer. Since it is difficult numerically to find exactly the point where s 0 in the actual numerical solution one implements three tests of the type 1In the following discussion the variable .s is reserved for the value of . where we have a solution. 90 CHAPTER 6. NON-LINEAR EQUATIONS AND ROOTS OF POLYNOMIALS 1. and K - 1 c 2. 3. and
