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numerical mathematics and scientific computation volume 1 Episode 11

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.4. Spline Functions 347 2. Given Xi y xi y xi Xi Xo ih i 1 2 3. Let p Vữ be the Hermite interpolation polynomial to these data. a Find the remainder term and show that the interpolation error for X iCi Xs does not exceed h in magnitude. b Write a program that computes p xi 2jh k j 0 k. Comment This is one of several possible procedures for starting a multistep method for an ordinary differential equation y f x y . Two steps with an accurate one-step method provide values of y y and this program then produces starting values y only for the multistep method. 3. Derive the usual formula of Leibniz for the fc th derivative from 4.3.7 by a passage to the limit. 4. Give a short and complete proof of the uniqueness of the interpolation polynomial for distinct points by the use of the ideas of the proof of Theorem 4.3.1. 5. Modify the integration formula in Example 4.1.1 to a formula for jJ1 x-1 2 f x dx and derive an asymptotic error estimate h 0 by means of the technique of Example 4.3.4. 6. a Derive an asymptotic error estimate for one step of length h with the midpoint rule f f x dx hf o . Derive also a strict local error bound by integrating a Taylor expansion of x with remainder on the assumption that I f x M. b Derive an asymptotic global error estimate for the trapezoidal rule over the interval a b with step size h b a n n 00. Hint xi ằ f x dx etc. c Derive also a strict global error bound on the assumption that f x I M for X a . Compare these results with results that can be derived from the analysis of the Euler-Maclaurin formula. Hint Recall the relation of the midpoint rule rectangle rule to the trapezoidal rule that was mentioned in Sec.3.3. 4.4 Spline Functions 4.4.1 Introduction Before the computer age ship builders and others in engineering design used a spline to draw smooth curves. A spline is a thin elastic ruler which can be bent so that it passes trough a given set of points see Fig. 4.4.1. The curvature of a spline y s x X a in the plane is given by