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numerical mathematics and scientific computation volume 1 Episode 10

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 312 Chapter 3. Series Operators and Continued Fractions ữi i 0 Oj 2 fi-i o z a - fo i-i z for j 2 2 i 2 Lifj I ữị-I j 1 H 1 V O i I j - end end Results fm n z am n i 2n 2 m 0 n 0 rn n 1 r . The above program sketch must be improved for practical use e.g. something should be done about the risk for a division by zero. Example 3.4.6. Not yet written The typical convergence rates of the -algorithm observed in numerical experiments are about the same as for GCA Sec. 3.3.7. 3.4.4 More about Continued Fractions and the Padé Table. A plan Toeplitz look at Recipes Sec. 6.2 and Hankel Matrices related to a power series Hankel determinant Hjy Hankel matrix. rr n l rr n o- n n m-l n Tj n o- n l o- n Tr n 1 1 lr I m Xlm quotient-difference algorithm qd algorithm bf rhombus rules refer to Henrici vol I 7.6. n _ l _ n n . I 9. n n. 1.2. ztm m-1 ll L9 9 9 11 9 x9 - 9 9 n im n D. m l _ n with initial conditions Cq 0 Ợ1 n Cn 1 cn n 0 1 2 . Correspondence of formal power series Co C _z C2Z2 . Co Ạ 0 and a continued fraction 00 0 0 __ 0 _ 0 z J-i tU t-1 tU tU Cọ tU CjXi co - j 0 Quote Henrici vol II p. 518 Thm 12.4c qd-algorithm see 7.6 and determinant formulas Thm 7.6a . Example ex34.dise Henrici vol I p. 610 and qdalgl.dia. Series in powers of z-1 Henrici vol II p. 525. incomplete gamma different type of cf s fraction Henr.II p. 561. Ref. to odd functions 3.4.9 in the text divide by z and put z2 X. Example ex34.logcf Henrici vol II p. 534 and logcf.dia. The convergents of the corresponding continued fractions are equal to the sequence of Fade approximants with m n 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 . Review Questions 313 Review Questions 1. Define a continued fraction and show how the convergents can be evaluated backwards and forwards. 2. Show how any positive number can be expanded into a continued fraction with integer elements. In what sense are the convergents the best approximations How accurate are they 3. What is the Fade table Describe how the Fade approximants can be .