tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 12

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 12', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 382 Chapter 4. Interpolation and Related Subjects From this we obtain d d f xữ h y0 k fo o pỉ-i- 7- 0 0 V ơx ơyj - cP rỉ2 I __L Ọhb lĩ2 1 í h2 k2l I I I o 9 I -f- tv I Jo 0 17 ự i I tv . or oxoy oyz Ỉ An interpolation formula valid for all quadratic functions can be obtained by replacing in Taylor s formula the derivatives by difference approximations valid for quadratic polynomials f x0 ph y0 qh 0 0 2P i 0 - -1 0 2 o i - 0 -1 2 1 0 - 2 0 0 -1 0 w i i - 1 -1 - -1 1 -1 -1 2 2 o i - 2 0 0 0 -1 - This formula uses function values in nine points. The proof of the expression for . . a2 . . approximating the mixed derivative 0 0 is left as an exercise Problem 2. oxoy Cubic Hermite interpolation treating one variable at a time see Example f x0 ph y0 qh 1 - ợ Mỉ o ỉ i ợ i - q 1 - ợ M4 yo - Aự y0 - ợ M4 yi - A Mỉ o Xy 1 -p o y p xi y p l-p 1 -P Wx x0 y - A x0 y -p h. xi y - A x0 y . This formula requires that the quantities df dx df dy and Ỡ2 fdxdy are given at the four points Xi yj for 0 i j 1. Hence 16 quantities are needed to specify the bicubic polynomial. Review Questions 1. How is bilinear interpolation performed What is the order of accuracy Problems 1. Compute by bilinear interpolation when 0 0 1 1 0 2 0 1 3 1 1 5. M. Examples of Interpolation in Nonlinear Function Spaces 383 2. Derive a formula for f y 0 0 using fij I j 1 1 j 1 which is exact for all quadratic functions. Examples of Interpolation in Nonlinear Function Spaces A little about interpolation by rational functions and by a sum of exponentials. Not yet written. Notes and References The basic idea of scaled divided differences is due to F. Krogh 16 . The notation and presentation used in Sec. is due to L. o. Eriksson and G. Dahlquist 9 Runge s example is from 1901 see the bibliography in Cheney 12 . 1 c. de Boor. On calculating with B-splines. J. Approx. Theory 6 50-62 1972. 2 c. de Boor. Spline Toolbox for use with Matlab. The Math. Works South Natick 1990. 3 c. de