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numerical mathematics and scientific computation volume 1 Episode 7

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 3.2. Difference Operators and Operator Expansions 207 For example suppose that X e a where b a is of the order of magnitude of a step size parameter h and that is analytic in a . By Taylor s formula x p x x a k x - a fc 1 . w a Ạ 0 1. K r 1 where p G pk hence Rp 0. Most of the common functionals can be applied term by term. Then Rf 0 ỉ- - Rx x - a k - a k 1 . n K 1 Assume that for some c Rx x a k O hk c for k 1 2 3 . This is often the case. Then 3.2.29 becomes an asymptotic error estimate as h t 0. It was mentioned above that for formulas derived by operator methods an asymptotic error estimate is directly available anyway but if a formula is derived by other means see Chapter 4 this error estimate is important. If Rp 0 for p ePk then a fortiori Rp 0 for p e pk-i i 0 k. We may thus obtain a Peano kernel for each i which is temporarily denoted by Kk-i u . They are obtained by integration by parts Rkf l Kk u f u du l Kk_1 u f k-1Hu du I Kk_2 u f k-2Hu du. 3.2.30 where Kk-i DỳKk i 1 2 . as long as Kk-i is integrable. The lower order kernels are useful e.g. if the actual function f is not as smooth as the usual remainder formula requires. For the trapezoidal rule we obtain Ki u ệuị ậ u ệ u fe ị. A second integration by parts can only be performed within the framework of Dirac s delta functions distributions Kq is not integrable. A reader who is familiar with these generalized functions may enjoy the following formula Rf ị Kữ ù Ị ù du Ị - ổ u 1 - ổ u - ỉ. u du. This is for one step of the trapezoidal rule but many functionals can be expressed analogously. 3.2.4 Applications of Operator Techniques for Finding Approximation Formulas Example 3.2.9. Finding interpolation formulas by operator methods. Consider the operator expansion - yh E-y b 1 - V W s j 0 v -V W . 208 Chapter 3. Series Operators and Continued Prob The verification of the assumptions of Theorem 3.2.6 offers no difficulties and we omit the details. Truncate the expansion before V fc. By the theorem we obtain