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Lagrange Interpolator Polynomial

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Lagrange Interpolator Polynomial

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28 Nov 2006 (Updated )

Find the polynomial (defined by its coefficients) passing through a set of points.

lagrangepoly(X,Y,XX)
function [P,R,S] = lagrangepoly(X,Y,XX)
%LAGRANGEPOLY  Lagrange interpolation polynomial fitting a set of points
%   [P,R,S] = LAGRANGEPOLY(X,Y)  where X and Y are row vectors
%   defining a set of N points uses Lagrange's method to find 
%   the N-1th order polynomial in X that passes through these 
%   points.  P returns the N coefficients defining the polynomial, 
%   in the same order as used by POLY and POLYVAL (highest order first).
%   Then, polyval(P,X) = Y.  R returns the x-coordinates of the N-1
%   extrema of the resulting polynomial (roots of its derivative),
%   and S returns the y-values  at those extrema.
%
%   YY = LAGRANGEPOLY(X,Y,XX) returns the values of the polynomial
%   sampled at the points specified in XX -- the same as
%   YY = POLYVAL(LAGRANGEPOLY(X,Y)). 
%
%   Example:
%   To find the 4th-degree polynomial that oscillates between 
%   1 and 0 across 5 points around zero, then plot the interpolation
%   on a denser grid inbetween:
%     X = -2:2;  Y = [1 0 1 0 1];
%     P = lagrangepoly(X,Y);
%     xx = -2.5:.01:2.5;
%     plot(xx,polyval(P,xx),X,Y,'or');
%     grid;
%   Or simply:
%     plot(xx,lagrangepoly(X,Y,xx));
%
%   Note: if you are just looking for a smooth curve passing through 
%   a set of points, you can get a better fit with SPLINE, which 
%   fits piecewise polynomials rather than a single polynomial.
%
%   See also: POLY, POLYVAL, SPLINE

% 2006-11-20 Dan Ellis dpwe@ee.columbia.edu
% $Header: $

%  For more info on Lagrange interpolation, see Mathworld: 
%  http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

% Make sure that X and Y are row vectors
if size(X,1) > 1;  X = X'; end
if size(Y,1) > 1;  Y = Y'; end
if size(X,1) > 1 || size(Y,1) > 1 || size(X,2) ~= size(Y,2)
  error('both inputs must be equal-length vectors')
end

N = length(X);

pvals = zeros(N,N);

% Calculate the polynomial weights for each order
for i = 1:N
  % the polynomial whose roots are all the values of X except this one
  pp = poly(X( (1:N) ~= i));
  % scale so its value is exactly 1 at this X point (and zero
  % at others, of course)
  pvals(i,:) = pp ./ polyval(pp, X(i));
end

% Each row gives the polynomial that is 1 at the corresponding X 
% point and zero everywhere else, so weighting each row by the 
% desired row and summing (in this case the polycoeffs) gives 
% the final polynomial
P = Y*pvals;

if nargin==3
  % output is YY corresponding to input XX
  YY = polyval(P,XX);
  % assign to output
  P = YY;
end

if nargout > 1
  % Extra return arguments are values where dy/dx is zero
  % Solve for x s.t. dy/dx is zero i.e. roots of derivative polynomial
  % derivative of polynomial P scales each power by its power, downshifts
  R = roots( ((N-1):-1:1) .* P(1:(N-1)) );
  if nargout > 2
    % calculate the actual values at the points of zero derivative
    S = polyval(P,R);
  end
end

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