tailieunhanh - Burden - Numerical Analysis 5e (PWS, 1993) Epside 1 Part 7

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Cubic Spline Interpolation 141 Ỉ-3 25 Table j Xj ữj bj q dj 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 of the interval. In the case where the nodes are equally spaced near both endpoints approximations can be obtained by using Eq. or any of the other appropriate formulas given in Sections and . In the case of unequally spaced nodes the problem is considerably more difficult. 142 CHAPTER 3 interpolation and Polynomial Approximation Figure To conclude this section we list an error-bound formula for the cubic spline with clamped boundary conditions . The proof of this result can be found in Schultz 131 pp. 57-58. A fourth-order error-bound result also holds in the case of free boundary conditions but it is more difficult to express. See Birkhoff and de Boor 14 pp. 827-835. Theorem 3 13 Let G c4 a b with maxaSj. fo i 4 U I M. If S is the unique clamped cubic spline interpolant to f with respect to the nodes a XQ xl xn b then . . 5M . max f x - SỌc Ị max xJ 1 - Xj . m B a -V b -J 84 0 j n 1 EXERCISE SET 1. Construct the free cubic spline for the following data a. X f x b. X f x c. X f x d. X f x 0 3-4 Cubic Spline Interpolation 143 e. X fCx f. X f x g- X f x h. X A