多项式插值或拟合

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1. function [p,S] = polyfit(x,y,n)
   
2. %p=polyfit(x,y,k)用k次多项式拟合向量数据(x,y)
   
3. %p返回多项式的降幂系数.当k>=n-1时,polyfit实现多项式插值.  
   
4. %例如 用二次多项式拟合数据
   
5. %   x | 0.1  0.2  0.15 0.0  -0.2 0.3
   
6. %   --|------------------------------
   
7. %   y | 0.95 0.84 0.86 1.06 1.50 0.72
   
8. % 求解
   
9. %    x=[0.1,0.2,0.15,0,-0.2,0.3];
   
10. %    y=[0.95,0.84,0.86,1.06,1.50,0.72];
    
11. %    p=polyfit(x,y,2)
    
12. %    xi=-0.2:0.01:0.3;
    
13. %    yi=polyval(p,xi);
    
14. %    plot(x,y,'o',xi,yi,'k');
    
15. %    title('polyfit');
    
16. 
    
17. %POLYFIT Fit polynomial to data.
    
18. %   POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
    
19. %   degree N that fits the data, P(X(I))~=Y(I), in a least-squares sense.
    
20. %
    
21. %   [P,S] = POLYFIT(X,Y,N) returns the polynomial coefficients P and a
    
22. %   structure S for use with POLYVAL to obtain error estimates on
    
23. %   predictions.  If the errors in the data, Y, are independent normal
    
24. %   with constant variance, POLYVAL will produce error bounds which
    
25. %   contain at least 50% of the predictions.
    
26. %
    
27. %   The structure S contains the Cholesky factor of the Vandermonde
    
28. %   matrix (R), the degrees of freedom (df), and the norm of the
    
29. %   residuals (normr) as fields.   
    
30. %
    
31. %   See also POLY, POLYVAL, ROOTS.
    
32. 
    
33. %   J.N. Little 4-21-85, 8-23-86; CBM, 12-27-91 BAJ, 5-7-93.
    
34. %   Copyright (c) 1984-98 by The MathWorks, Inc.
    
35. %   $Revision: 5.9 $  $Date: 1997/11/21 23:40:57 $
    
36. 
    
37. % The regression problem is formulated in matrix format as:
    
38. %
    
39. %    y = V*p    or
    
40. %
    
41. %          3  2
    
42. %    y = [x  x  x  1] [p3
    
43. %                      p2
    
44. %                      p1
    
45. %                      p0]
    
46. %
    
47. % where the vector p contains the coefficients to be found.  For a
    
48. % 7th order polynomial, matrix V would be:
    
49. %
    
50. % V = [x.^7 x.^6 x.^5 x.^4 x.^3 x.^2 x ones(size(x))];
    
51. 
    
52. if ~isequal(size(x),size(y))
    
53.     error('X and Y vectors must be the same size.')
    
54. end
    
55. 
    
56. x = x(:);
    
57. y = y(:);
    
58. 
    
59. % Construct Vandermonde matrix.
    
60. V(:,n+1) = ones(length(x),1);
    
61. for j = n:-1:1
    
62.     V(:,j) = x.*V(:,j+1);
    
63. end
    
64. 
    
65. % Solve least squares problem, and save the Cholesky factor.
    
66. [Q,R] = qr(V,0);
    
67. p = R\(Q'*y);    % Same as p = V\y;
    
68. r = y - V*p;
    
69. p = p.';          % Polynomial coefficients are row vectors by convention.
    
70. 
    
71. % S is a structure containing three elements: the Cholesky factor of the
    
72. % Vandermonde matrix, the degrees of freedom and the norm of the residuals.
    
73. 
    
74. S.R = R;
    
75. S.df = length(y) - (n+1);
    
76. S.normr = norm(r);
    

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