Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 116 CHAPTER 3 Interpolation and Polynomial Approximation This formula is called the Newton forward divided-difference formula. Another I form called the Newton forward-difference formula is constructed by making use of I the forward difference notation A introduced in Aitken s A2 method. With this notation . 1 _ 1 A . I 0 ---------- Í A W I XỊ - x0 h vl 1 AfOo 1 A2 . 1 x0 xb x2 7p A x0 4 2h _ n J 2n j and in general J 1 I Xo . . . xj AẢ x0 . 4 c d T T Consequently Eq. becomes the following formula. Newton Forward-Difference Formula Ệ n J PM ẳ u MfW ị A----0 K If the interpolating nodes are reordered as xn . . . x0 a formula similar to J Eq. results ỉị Pn x f xn-ỵ x x - x -1 n - n O H-1 p f x0. - J - n-l - 1 . Using equal spacing with X x sh andx Xj s 7Ĩ i h produces 7 F x p x sh . M shfix Xni s s l h2f xn X i_ x s s 1 j n V hnf Xfr Xỵ . x . . This form is called the Newton backward divided-difference formula. It is used to Ệ derive a more commonly applied formula known as the Newton backward-difference 7 formula. To discuss this formula we need the following definition. 7 Definition Given the sequence pn n o define the backward difference by Pn-Pn - Pn ỉ forn 1. A Higher powers are defined recursively by I pn V VÀ -1p for k 2. n B K -Ệ Definition implies that I AW x f xn-2 xn_ỵ xn T72V2f x JJ rt 2n T. Divided Differences ĨĨ7 er of Ì 1 -Ỉ 1 to to ice EXAMPLE 2 Table and in general f xn_k . . x 77TtVy x . k h Consequently M v2 x -r 5 5 1 s n 1 3 r-------------- w. nl Extending the binomial coefficient notation to include all real values of 5 we let i - A _ - -S ft 1 _ k s s- -if-fsrk - 1 k k k and p x W w -D -ộvyCQ -1 vyw to obtain the following formula. Newton Backward-Difference Wrmnla 3-14 J2 P w 2 1 7 vw Consider the table of data given in Example 1. The divided-difference table corresponding to these data is shown in Table . First Second divided divided differences differences Third Fourth divided divided differences differences