tailieunhanh - 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 | . 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 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 . 5. Modify the integration formula in Example 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 . 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 . Spline Functions 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. . The curvature of a spline y s x X a in the plane is given by