%
% Written by:
% --
% John L. Weatherwax                2009-04-21
%
% email: wax@alum.mit.edu
%
% Please send comments and especially bug reports to the
% above email address.
%
% Problem on EPage 63; Solutions on EPage 161
%
%-----

close all; clc; clear all; 

addpath('../../BookCode');

% The problem is equivalent to solving:
% 
% x/5 - cos(x) = 2 
% 
% for x.  Then y = exp(x/10)
% 
fn = @(x) ( x/5 - cos(x) - 2 );
fn_prime = @(x) ( 1/5 + sin(x) );

% Lets get a good guess at the location of the root: 
% 
if( false )
  xs = linspace( 0, 10, 100 );
  ys = fn( xs );
  plot( xs, ys, '-b' ); grid('on');
end

%[x_guess,y_guess] = ginput(1);
%
% gives
% 
% x_guess ~ 8.2604
%
x_guess = 8.2604;

% we are told to try x_guess = 1 
x_guess = 1;

% Use the scalar version: 
% 
[ x_newton, n_newton ] = fnewton( fn, fn_prime, x_guess, 1.e-4 ) 
fn( x_newton ) 
y_newton = exp(x_newton/10.)

% Use the multivariate version: 
% 
fn_mv = @(x) ( [ exp(x(1)/10) - x(2); 2*log(x(2)) - cos(x(1)) - 2 ] );
fn_prime_mv = @(x) ( [ [ exp(x(1)/10)/10, -1 ]; [ sin(x(1)), 2/x(2) ] ] );
[ x_newton_mv, n_newton_mv ] = newtonmv( [1,1]', fn_mv, fn_prime_mv, 2, 1.e-4 )
fn_mv( x_newton_mv )