tailieunhanh - Burden - Numerical Analysis 5e (PWS, 1993) Epside 1 Part 8

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1Ó8 CHAPTER 4 Numerical Differentiation and Integration 20. where Í is measured in seconds i is in amperes the inductance L is a constant henries and the resistance is ohms. Approximate the voltage at the values t and . All calculus students know that the derivative of a function f at X can be defined as x f x h f x f x lim ------------ h Choose your favorite function f nonzero number X and computer or calculator. Generate approximations ff x to x by _ f x 10 - f x fn x -------------------- Ĩ. 1 ÍỈ 21. 22. for n 1 2 . 20 and describe what happens. Derive a method for approximating f x0 whose error term is of order h2 by expanding the function f in a sixth Taylor polynomial about x0 and evaluating atx0 h and Xq 2h. Consider the function where M is a bound for I e h F M h 6 t 3 sin x dx3 Show that e h has a minimum at fy Ss M. on . Ỉ Richardson s Extrapolation 1 í ỉ A technique known as Richardson s Extrapolation is frequently employed to generate results of high accuracy by using low-order formulas. Although the name attached to the method refers to a paper written by L. F. Richardson and J. A. Gaunt 121 in 1927 the idea behind the technique is much older. An interesting article regarding the history and. application of extrapolation was written by Joyce 84 in SIAM Review Extrapolation can be applied when it is known that the approximation technique has 2 an error term with a predictable form one that depends on a parameter usually the step . size h. For example suppose that N h is a formula that approximates an unknown value Ạ M and that it is known that N h has O h truncation error in the form M N h Kỵh K2h2 K3h3 for some unspecified collection of constants Kỵ K2 K3 . We assume here that h 0 can be arbitrarily chosen and that improved approxima- u tions occur as h becomes small. The objective of extrapolation is to improve the formula ị from order O h to a formula of higher order. Do not be misled by the relative .

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