funfitxy.m

% FUNFITXY  Computes interpolation coefficients for d-dim function.
% USAGE
%   [c,B]=funfitxy(fspace,x,y);
% or
%   [c,B]=funfitxy(fspace,B,y);
% Approximates a function y=f(x) using a specified basis type.
% INPUTS
%   fspace  : a structure defining a family of functions (use FUNDEF to create this)
%   x       : input values (kxd matrix or 1xd cell array of vectors)
%   B       : a basis structure - if funfitxy is called multiple time
%               the basis information can be stored and reused (see below)
%   y       : kxp matrix of output values
% OUTPUTS
%   c   : a coefficient matrix
%   B   : the basis structure or matrix used to evaluate the function
%
% The approximating function can then be evaluated using 
%   funeval(g,fspace,X)
%
% Note: when evaluating on a multidimensional grid, pass x as a 
% 1 x d cell array of values. An efficient algorithm can be 
% used in this case.
%
% Example:
%   fspace=fundefx({'cheb',10,0,1},{'cheb',6,-1,1});
%   g=funfitxy(fspace,x,y);
% This produces a 2-d polynomial approximation to the data (x,y),
% where x must be a 2-column matrix or a 1x2 cell array.
% If the number of evaluation points is equal to prod(N) (60 in the
% example above) the approximation will interpolate those points.
% If there are more than this number of evaluation points, a least
% squares fit is provided.
%
% Reusing the basis information:
%   [c1,B]=funfitxy(fspace,x,y1);
%   c2=funfitxy(fspace,B,y2);
% produces more efficiently the same result as
%   c1=funfitxy(fspace,x,y1);
%   c2=funfitxy(fspace,x,y2);
% Note: B.order(1,:) must equal 0.
%
% To fit a function using a MATLAB function (M-file or inline function) use FUNFITF.
%
% USES: FUNBASX, CKRONXI
%
% See also: FUNFITF, FUNDEF, FUNEVAL, FUNNODE, FUNBAS.

% Copyright (c) 1997-2000, Paul L. Fackler & Mario J. Miranda
% paul_fackler@ncsu.edu, miranda.4@osu.edu

function [c,B]=funfitxy(fspace,x,y)

  if nargin~=3, error('Three parameters must be specified'); end 

  m=size(y,1);    
  % There cannot be more basis functions than data points
  if prod(fspace.n)>m
    error('Insufficient number of data points')
  end
  
  if isstruct(x)                 % Use precomputed basis structure
    B=x;
    switch B.format
    case 'tensor'
      if any(B.order(1,:)~=0)
        error('invalid basis structure - first elements must have order 0')
      end
      B.vals=B.vals(1,:);
      c=ckronxi(B.vals,y,fspace.d:-1:1);
    case 'direct'
      B=funbconv(B,zeros(1,size(B.vals,2)),'expanded');
      c=B.vals{1}\y;
    case 'expanded'
      c=B.vals{1}\y;
    end
  elseif iscell(x)               % evaluate at grid points
    mm=1;for i=1:size(x,2), mm=mm*size(x{i},1); end
    if mm~=m
      error('In FUNFITXY: x and y are incompatible')
    end
    B=funbasx(fspace,x,0);
    c=ckronxi(B.vals,y,fspace.d:-1:1);
  else                           % evaluate at arbitrary points
    % x and y must have the same number of rows
    if size(x,1)~=m
      error('In FUNFITXY: x and y are incompatible')
    end
    B=funbasx(fspace,x,0,'expanded');
    c=B.vals{1}\y;
  end