function [p, err] = polyfitn(varargin)

% polyfitn(f, order) -- ND polynomial fit.
%  polyfitn('demo') demonstrates itself.
%  polyfitn(N, order) demonstrates itself with N random
%   points f(x, y) = (x - y).^order.  Defaults:
%   N = 10, order = 1.
%  polyfitn(f, order) fits the data, a matrix arranged
%   column-wise as [x y z ... f(x, y, z, ...)], to
%   the given maximum order (default = 1).  The
%   result is an ND array.  Each array index, minus
%   one (1), represents the powers of the independent
%   variables.  For example, the index [1 2 3] would
%   denote the x^0 + y^1 + z^2 term.  Use "polyvaln"
%   to evaluate the polynomial.
%  polyfitn(x, y, ..., f, order) is an alternative
%   syntax.
%  [p, err] = ... also returns the error-norm
%   of the fit.
%
% Also see: polyvaln.
 
% Copyright (C) 2003 Dr. Charles R. Denham, ZYDECO.
%  All Rights Reserved.
%   Disclosure without explicit written consent from the
%    copyright owner does not constitute publication.
 
% Version of 22-Apr-2003 10:47:46.
% Updated    29-Apr-2003 13:15:41.

% Strange behavior: polyfit(100, 9) does not work,
%  whereas (99, 9) and (101, 9) do work.

if nargin < 1
    help(mfilename)
    varargin = {'demo'};
end

if nargin < 2 & isequal(varargin{1}, 'demo')
    varargin{1} = 20;
end

if ischar(varargin{1})
    varargin{1} = eval(varargin{1});
end

if nargin < 3 & length(varargin{1}) == 1
    N = varargin{1};
    x = rand(N, 1);
    y = rand(size(x));
    order = 1;
    if length(varargin) > 1
        order = varargin{2};
    end
    if ischar(order), order = eval(order); end
    f = (x - y).^order;
    [pp, err] = feval(mfilename, x, y, f, order);
    if nargout > 0, p = pp; end
    xi = linspace(0, 1, 11);
    yi = xi;
    [xi, yi] = meshgrid(xi, yi);
    g = zeros(size(xi));
    g(:) = polyvaln(pp, xi(:), yi(:));
    hold off
    surf(xi, yi, g, ...
        'FaceColor', 'y', 'EraseMode', 'xor')
    hold on
    plot3(x, y, f, 'k*', 'EraseMode', 'xor')
    hold off
    xlabel X
    ylabel Y
    zlabel Z
    title(['Error Norm: ' num2str(err)])
    figure(gcf)
    set(gcf, 'Name', ...
        [mfilename ' ' int2str(N) ' ' int2str(order)])
    return
end

if length(varargin) > 2
    if length(varargin{end}) == 1
        order = varargin{end};
        data = [varargin{1:end-1}];
    else
        order = 1;
        data = [varargin{:}];
    end
    if nargout > 1
        [p, err] = feval(mfilename, data, order);
    else
        p = feval(mfilename, data, order);
    end
    return
end

data = varargin{1};
if nargin > 1
    order = varargin{2};
else
    order = 0;
end

[m, n] = size(data);

sz = [order+1 order+1];

% Exponents for Vandermonde matrix.

sub = cell(1, n-1);
[sub{:}] = ind2sub(sz, 1:prod(sz));
for i = 1:length(sub)
    sub{i} = sub{i}(:);
end
expon = [sub{:}] - 1;

% Trim exponents to requested order.

for k = 1:length(expon)
    if sum(expon(k, :)) > order
        expon(k, :) = NaN;
    end
end

% Vandermonde matrix.

vander = zeros(m, prod(sz));

k = 0;
for i = 1:prod(sz)
    x = data(:, 1:n-1);
    k = k+1;
    ex = expon(k, :);
    if all(isfinite(ex))
        ex = repmat(ex, [m 1]);
        x = data(:, 1:n-1) .^ ex;
        vander(:, k) = prod(x, 2);
    end
end

% Solve.

p = zeros([order+1 order+1]);

warning off   % Vander is habitually rank-deficient.
p(:) = vander \ data(:, end);
warning on

if order == 0, p = p(1); end

% Error-norm.

if nargout > 1
    err = norm(polyvaln(p, data(:, 1:end-1)) - data(:, end));
end