function v=rawNewton(F,DF,v0);
% RAWNEWTON: v=rawNewton(F,DF,v0)
%   A slow implementation of Newton--Raphson.  In practice
%   the gradient matrix could be approximated by Broyden's method
%   and a one-dimensional search for the solution would be used.
% Note the parameters [with default values]:
%   maxNewt [=20]    max number of iterations
%   Ftol    [=1e-8]  equations F=0 should be true to this accuracy
%   vtol    [=1e-5]  v should be determined to this accuracy
%   CONDtol [=1e10]  used to determine if DF is invertible.
%
% Try "F=inline('[x(1)^2+x(2)^2-4;exp(x(1))-x(2)]','x'),
%      DF=inline('[2*x(1), 2*x(2);exp(x(1)), -1]','x'),
%      rawNewton(F,DF,[1 2]', 10)"
% (ELB 4/27/02)

maxNewt=20; Ftol=1e-8; vtol=1e-5; CONDtol=1e10; 
v=v0;
for j=1:maxNewt
   FF=feval(F,v); DFF=feval(DF,v);
   if cond(DFF)>CONDtol % this way of testing for invertibility is slow
      error(['rawNewton error: Gradient matrix close to singular'...
            ' at Newton step' num2str(j)]); end;
   dv=-DFF\ FF; % note backslash
   v=v+dv;
   if (norm(dv,inf)<vtol)&(norm(FF,inf)<Ftol), return; end;
end;
error('rawNewton: Max number of iterations exceeded.');