tailieunhanh - Burden - Numerical Analysis 5e (PWS, 1993) Epside 1 Part 3

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The Bisection Method that the root in this interval is unique. The method calls for a repeated halving of subintervals of a. b and at each step locating the half containing p. To begin set úq a and z b and let J be the midpoint of a b . that is Pl i C I bi - Jỉf pS 0 then pi Pl if not then cpq has the same sign as either Cữ OTf bi . If f Pi and Cữi have the same sign thenp e pb bl and we set a2 Pl and b2 bỵ. If f Pi and íữỵ have opposite signs then p G aq Pl and we set a2 0-1 and b2 Pl-We then reapply the process to the interval a2 b-2 . This produces the method described in Algorithm . See Figure . I ị Figure 1 ALGORITHM n 0 JSt sd ng nd he nd ith P-itv I I I 2 Pi l 2 I------_J---------1 Ễ ỉ . 3 P - h . ị 1 iễắ --- ---------------- --- ---------- . . Bisection To find a solution to x 0 given the continuous function f on the interval a b where f a and Z0 have opposite signs INPUT endpoints a b tolerance TOL maximum number of iterations No. OUTPUT approximate solution p or message of failure. Step 1 Set i - 1. Step 2 While i Nq do Steps 3-6. Steps Set a b a fl. Compute Pi. Step 4 Iff p 0 or b - Ò fl TOL then OUTPUT Procedure completed successfully. STOP. - T s 42 CHAPTER 2 Solutions of Equations in One Variable EXAMPLE Step 5 Set ỉ i 1. Step 6 ỉĩf à f p 0 then set a p Compute ah bị. else set b p. Step 7 OUTPUT Method failed after 2V0 iterations j 0 - 2V0 Procedure completed unsuccessfully. STOP Listed next are some other stopping procedures that can be applied in Step 4 of Algorithm each of which applies to any iterative technique considered in this chapter. Select a tolerance 0 and generate Pi . . pN until one of the following conditions is met pN Pn-ì í Pn r j L pN 0 or iPvl I Pm - 1 Unfortunately difficulties can arise using any of these stopping criteria. For example there exist sequences p j with the property that the differences pn pn i converge to zero while the sequence itself diverges. See Exercise 13. It is

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