function [sol] = newt(er,varargin)
%NEWT A function that preforms Newton's method
% This function preforms Newton's method on a variable number of
% equations. Enter the variable name as a string, then an initial guess,
% then the functions in order.
% Written by Gabe Frangakis, 2010.
%
% The first input of newt must be the tolerance you desire. This must be
% a non-negative real number. The next inputs must be variables,
% equations, and guesses. There is no requirement for which order these
% are entered in, only that the order of variables and the order of
% guesses are the same. For example, if you enter your variables in the
% order y,z,x, your guesses must be entered as y guess, z guess, and x
% guess. Variables and equations must be entered as strings. Equations
% should be written explicitly i.e. instead of 2x^2, you write 2*x^2.
%% Initialize
guesscount=0;
eqcount=0;
optarg=size(varargin);
optarg=optarg(:,2);
varlist=[];
eqlist=[];
ber=0;
format long
%% Separate the Arguments
for k=1:optarg
mat=cell2mat(varargin(k));
str=isa(mat,'char');
dble=isa(mat,'double');
if str==1
l=length(mat);
if l==1
syms(mat)
varlist=[varlist;mat];%is a variable
else
eqcount=eqcount+1;
eqlist=[eqlist;sym(mat)];
end
end
if dble==1
l=length(mat);
if l==1
guesscount=guesscount+1;
guess(guesscount,1)=mat;
else
eqcount=eqcount+1;
var=varlist(eqcount);
str1=poly2str(mat,var);
eqlist=[eqlist;sym(str1)];
end
end
end
%% Make everything check out
% Number of variables must equal number of equations must equal number of
% guesses
if guesscount~=size(varlist,1)|guesscount~=eqcount
error('newt:argChk', 'Number of varibles, initial guesses, and equations must be equal')
end
%% Execute Newton's method
xo=guess;
%Evaluate the Jacobian and F(x).
while ber==0
jacob=jacobian(eqlist);
fxo=eqlist;
for j=1:eqcount
jacob=subs(jacob,varlist(j),xo(j)); %sub into the Jacobian
fxo=subs(fxo,varlist(j),xo(j)); %find F(xo)
end
h=linsolve(jacob,-fxo); %solve Jacob.h=-F(xo)
xo=xo+h;
if abs(fxo)<=er
ber=1;
end
end
sol=xo;
end
