tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 8

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 242 Chapter 3. Series Operators and Continued Fractions In this example the theoretical criterion mentioned above is satisfied since one can show that sn a V3 1 Vsn Vsn-1 ộ à a as rỉ-4 OŨ. See the discussion of iteration in Sec. . Iterated Aitken yields with u 2-53 the results 2 . s l etc. and the errors e 2 -101 T3 e 5 Cg 7 e 4 9 after 2 4 6 8 evaluations of the function ự respectively. We shall see Problem 6 that the Aitken acceleration may converge even if the basic iteration diverges. When the basic sequence sn is as in this example produced by a convergent iterative process one can apply Aitken acceleration in a different way that is usually even much more efficient. We call this active Aitken acceleration since the result of an acceleration is actively used in the basic iterative process . one computes S 8 2 84 3 and makes an Aitken acceleration by means of 3 2 S3 84 that yields S4. This can evidently be repeated until some termination criterion is satisfied. See Problem 13. Example . Set an e-A n 1 n 0. As before the sn are the partial sums of an 8 lim sn and use the same notations as above. Note that Vsn Vsn-1 an an-4 1 n 1 so this series is slowly convergent. Computations with plain and iterated Aitken macheps 10 16 gave the results below j e2j e2j 0 1 1 1 2 1 1 3 1 2 2 1 3 5 1 4 6 1 4 7 1 5 8 1 5 T lio tíVỉiu ri ir is 111Í 11Í1 timiÍ Í 8 fhie I ie milrllv 4 lie ot v LLt llv t J lllCzllCz uCzlllV LI I lull J o. xlluvi ulllo I lữ lllllxllj A11 t i 11I t ill I nrcmnrl in flit lottRf until ĩ All mill till iliA im imwi iv e 1111 ll I 1II11 g 11 11 11 1 XVI 1 IX11U11 J X11VX III 111 1 11 1. V o 2 ý V lý are sometimes several powers of 10 smaller than the actual errors and are misleading as error estimates. The rounding errors have taken over and it is almost