Ban đầu, Python được phát triển để chạy trên nền Unix. Nhưng rồi theo thời gian, nó đã “bành trướng” sang mọi hệ điều hành từ MS-DOS đến Mac OS, OS/2, Windows, Linux và các hệ điều hành khác thuộc họ Unix. | 121 Interpolation with Cubic Spline Running the program produces the following result x y x y x Done. Press return to exit PROBLEM SET 1. Given the data points x y determine y at x 0 using a Neville s method and b Lagrange s method. 2. Find the zero of y x from the following data x 0 1 2 3 y Use Lagrange s interpolation over a three and b four nearest-neighbor data points. Hint After finishing part a part b can be computed with a relatively small effort. 3. The function y x represented by the data in Problem 2 has a maximum at x . Compute this maximum by Neville s interpolation over four nearest-neighbor data points. 4. Use Neville s method to compute y at x n 4 from the data points x 0 1 2 y 5. Given the data x 0 1 2 y find y at x n 4 and at n 2. Use the method that you consider to be most convenient. 6. The points x -2 1 4 -1 3 -4 y -1 2 59 4 24 -53 lie on a polynomial. Use the divided difference table of Newton s method to determine the degree of the polynomial. 122 Interpolation and Curve Fitting 7. Use Newton s method to find the polynomial that fits the following points x -3 2 -1 3 1 y 0 5 -4 12 0 8. Use Neville s method to determine the equation of the quadratic that passes through the points x -1 1 3 y 17 -7 -15 9. The density of air p varies with elevation h in the following manner h km 0 3 6 p kg m3 Express p h as a quadratic function using Lagrange s method. 10. Determine the natural cubic spline that passes through the data points x 0 1 2 y 0 2 1 Note that the interpolant consists of two cubics one valid in 0 x 1 the other in 1 x 2. Verify that these cubics have the same first and second derivatives at x 1. 11. Given the data points x 1 2 3 4 5 y 13 15 12 9 13 determine the natural cubic spline interpolant at x .
