tailieunhanh - 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 | . 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 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. where Kk-i DỳKk i 1 2 . as long as Kk-i is integrable. The lower order kernels are useful . 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. Applications of Operator Techniques for Finding Approximation Formulas Example . 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 offers no difficulties and we omit the details. Truncate the expansion before V fc. By the theorem we obtain