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numerical mathematics and scientific computation volume 1 Episode 13

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 13', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6.2. Fixed-Point Iteration 417 Hence xn G J and a is proved. Repeated use of the inequality above gives xn a m xn-i m xo a and since m 1 the result b follows. Suppose finally that X ộ x has another root 3 G J 3 Ạ a. Then by 6.2.3 1 z Xa - a - M a contradiction thus c follows. Note that if j x exists then a sufficient condition for 6.2.3 to hold is that ự x rn 1 X e J. 6.2.4 since then by the mean value theorem we have for x y G J that I k - ệ y RCn ar - y x - y G J. On the other hand if a 1 then the iterative method 6.2.1 diverges. The four different cases that occur depending on the sign and magnitude of Ộ a were illustrated in Figures 1.2.1a-d. In Theorem 6.2.1 we assumed the existence of a fixed point a of ộ x . It is remarkable that the theorem can be modified so that it can be used to prove the existence of a fixed point and hence of a root of the equation X ộ x . Theorem 6.2.2. Let Xo be a starting point and Xn 1 ộ xn n 1 2 . Further let rn be a constant 0 rn 1 and J be a closed interval with Xo G J such that ộ x -ộ y m x-y x y e J m _ _. . . . and Xi xi Xo G J. Then a b and c of Theorem 6.2.1 are true. 1 m Proof. The theorem will be proved in a more general setting in Chapter 12 see Theorem 12.2.1 . Assume that ự x exists and is continuous in a neighborhood of a root a. Then it follows from the proof of Theorem 6.2.1 that if Xo is chosen sufficiently close to a then the sequence xn generated by 6.2.1 converges and it holds that lim a Ộ a . n- oo Xn-1 a If Ộ a Ạ 0 then we say that convergence is linear with rate ự a . The iteration method X 1 f xn is then a first order method. We now formally define the important concept of convergence order for a convergent sequence. 418 Chapter 6. Solving Scalar Nonlinear Equations Definition 6.2.3. A convergent sequence . with linifc-j.oo Xk a is said to have convergence order equal to p if it holds lim fc 1 c Ỷ 0. 6.2.5 c is called the asymptotic error constant. For p 1 it is necessary that C 1 and c is called the rate of