多项式插值或拟合
function = polyfit(x,y,n)%p=polyfit(x,y,k)用k次多项式拟合向量数据(x,y)
%p返回多项式的降幂系数.当k>=n-1时,polyfit实现多项式插值.
%例如 用二次多项式拟合数据
% x | 0.10.20.15 0.0-0.2 0.3
% --|------------------------------
% y | 0.95 0.84 0.86 1.06 1.50 0.72
% 求解
% x=;
% y=;
% p=polyfit(x,y,2)
% xi=-0.2:0.01:0.3;
% yi=polyval(p,xi);
% plot(x,y,'o',xi,yi,'k');
% title('polyfit');
%POLYFIT Fit polynomial to data.
% POLYFIT(X,Y,N) finds the coefficients of a polynomial P(X) of
% degree N that fits the data, P(X(I))~=Y(I), in a least-squares sense.
%
% = POLYFIT(X,Y,N) returns the polynomial coefficients P and a
% structure S for use with POLYVAL to obtain error estimates on
% predictions.If the errors in the data, Y, are independent normal
% with constant variance, POLYVAL will produce error bounds which
% contain at least 50% of the predictions.
%
% The structure S contains the Cholesky factor of the Vandermonde
% matrix (R), the degrees of freedom (df), and the norm of the
% residuals (normr) as fields.
%
% See also POLY, POLYVAL, ROOTS.
% J.N. Little 4-21-85, 8-23-86; CBM, 12-27-91 BAJ, 5-7-93.
% Copyright (c) 1984-98 by The MathWorks, Inc.
% $Revision: 5.9 $$Date: 1997/11/21 23:40:57 $
% The regression problem is formulated in matrix format as:
%
% y = V*p or
%
% 32
% y = [p3
% p2
% p1
% p0]
%
% where the vector p contains the coefficients to be found.For a
% 7th order polynomial, matrix V would be:
%
% V = ;
if ~isequal(size(x),size(y))
error('X and Y vectors must be the same size.')
end
x = x(:);
y = y(:);
% Construct Vandermonde matrix.
V(:,n+1) = ones(length(x),1);
for j = n:-1:1
V(:,j) = x.*V(:,j+1);
end
% Solve least squares problem, and save the Cholesky factor.
= qr(V,0);
p = R\(Q'*y); % Same as p = V\y;
r = y - V*p;
p = p.'; % Polynomial coefficients are row vectors by convention.
% S is a structure containing three elements: the Cholesky factor of the
% Vandermonde matrix, the degrees of freedom and the norm of the residuals.
S.R = R;
S.df = length(y) - (n+1);
S.normr = norm(r);
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