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divided_difference.m

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  • divided_difference.m 1.20 KiB
    function TDD = divided_difference(X, Y)
    %
    % TDD = divdiff(X, Y)
    %
    % DIVDIFF
    %   Newton's Method for Divided Differences.
    %
    %   The following formula is solved:
    %       Pn(x) = f(x0) + f[x0,x1](x-x0) + f[x0,x1,x2](x-x0)(x-x1) + ...
    %           + f[x0,x1,..,xn](x-x0)(x-x1)..(x-x[n-1])
    %       where f[x0,x1] = (f(x1-f(x0))/(x1-x0)
    %             f[x0,x1,..,xn] = (f[x1,..,xn]-f[x0,..,x_[n-1]])/(xn-x0)
    %
    % NOTE: f^(n+1)(csi)/(n+1)! aprox. = f[x0,x1,..,xn,x_[n+1]]
    %
    % Input::
    %	X = [ x0 x1 .. xn ] - object vector
    %	Y = [ y0 y1 .. yn ] - image vector
    %
    % Output:
    %   TDD - table of divided differences
    %
    % Example:
    %   TDD = divdiff( [ 1.35 1.37 1.40 1.45 ], [ .1303 .1367 .1461 .1614 ])
    %
    % Author:	Tashi Ravach
    % Version:	1.0
    % Date:     22/05/2006
    %
    
        if nargin ~= 2
            error('divdiff: invalid input parameters'); 
        end
    
        [ p, m ] = size(X); % m points, polynomial order <= m-1
        if p ~= 1 || p ~=size(Y, 1) || m ~= size(Y, 2)
            error('divdiff: input vectors must have the same dimension'); 
        end
    
        TDD = zeros(m, m);
        TDD(:, 1) = Y';
        for j = 2 : m
            for i = 1 : (m - j + 1)
                TDD(i,j) = (TDD(i + 1, j - 1) - TDD(i, j - 1)) / (X(i + j - 1) - X(i));
            end
        end
    
    end