Question

I'm working on the newton's method on matlab, could someone help me and show what two lines are needed to be added in the codes in order to make this function work?

function sample_newton(f, xc, tol, max_iter)

% sample_newton(f, xc, tol, max_iter)

% finds a root of the given function f over the interval [a, b] using Newton-Raphson method

% IN:

% f - the target function to find a root of

% function handle with two outputs, f(x), f'(x)

% e.g., f=@(x) [sin(x);cos(x)];

% xc - the initial guess to start the Newton method

% tol - the tolerance to stop the iterative method

% max_iter - the maximum iterations allowed

% OUT:

% none

% if there is no user given tolerance, the code will pick one

if nargin < 2

xc = 0;

tol = 1.0e-5;

max_iter = 1000;

elseif nargin < 3

tol = 1.0e-5;

max_iter = 1000;

elseif nargin < 4

max_iter = 1000;

end

% check to see if we can start the code

fc = f(xc);

if length(fc)~=2

error('sample_newton:exception','Incorrect function handle type');

end

if abs(fc(1)) < tol

fprintf('We have found an (approximated) root: x_c = % 10.4e and f(x_c) = % 10.4e.\n', xc, fc);

else if abs(fc(2)) < 1e-8

fprintf('derivative of f at x_c is very close to zero, the code will now abort.\n');

error('sample_newton:exception', 'Bad initial guess!');

   end

end

% start the bisection method

ind = 1; % iteration counter

while abs(fc(1))>tol && ind < max_iter

% complete the code to evalute the function and derivative at xk

% complete the code to update xc

if abs(fc(2))<1e-8 % check derivative isn't nearly zero

fprintf('Derivative of f at x_c is very close to zero, the code will now abort.\n');

error('sample_newton:exception', 'Bad initial guess!');

end

if abs(xc)>1e8 % check x_c isn't heading out to infinity

fprintf('x_c appears to be diverging, the code will now abort.\n');

error('sample_newton:exception', 'Bad initial guess!');

end

fprintf('At Iter. # = %4d, x_c = %10.4e and f(x_c) = % 10.4e.\n', ind, xc, fc(1));

ind = ind + 1; % increase the iteration counter

end

end

Answer 1

function sample_newton(f, xc, tol, max_iter)

% sample_newton(f, xc, tol, max_iter)

% finds a root of the given function f over the interval [a, b] using Newton-Raphson method

% IN:

% f - the target function to find a root of

% function handle with two outputs, f(x), f'(x)

% e.g., f=@(x) [sin(x);cos(x)];

% xc - the initial guess to start the Newton method

% tol - the tolerance to stop the iterative method

% max_iter - the maximum iterations allowed

% OUT:

% none

% if there is no user given tolerance, the code will pick one

if nargin < 2
  
xc = 0;
  
tol = 1.0e-5;
  
max_iter = 1000;
  
elseif nargin < 3
  
tol = 1.0e-5;
  
max_iter = 1000;
  
elseif nargin < 4
  
max_iter = 1000;
  
end

% check to see if we can start the code

fc = f(xc);

if length(fc)~=2
  
error('sample_newton:exception','Incorrect function handle type');
  
end

if abs(fc(1)) < tol
  
fprintf('We have found an (approximated) root: x_c = % 10.4e and f(x_c) = % 10.4e.\n', xc, fc);
  
else
if abs(fc(2)) < 1e-8
  
fprintf('derivative of f at x_c is very close to zero, the code will now abort.\n');
  
error('sample_newton:exception', 'Bad initial guess!');
  
end
  
end

% start the bisection method

ind = 1; % iteration counter

while abs(fc(1))>tol && ind < max_iter
  
% complete the code to evalute the function and derivative at xk
fc = f(xc);
% complete the code to update xc
xc = xc - fc(1)/fc(2);
if abs(fc(2))<1e-8 % check derivative isn't nearly zero
  
fprintf('Derivative of f at x_c is very close to zero, the code will now abort.\n');
  
error('sample_newton:exception', 'Bad initial guess!');
  
end
  
if abs(xc)>1e8 % check x_c isn't heading out to infinity
  
fprintf('x_c appears to be diverging, the code will now abort.\n');
  
error('sample_newton:exception', 'Bad initial guess!');
  
end
  
fprintf('At Iter. # = %4d, x_c = %10.4e and f(x_c) = % 10.4e.\n', ind, xc, fc(1));
  
ind = ind + 1; % increase the iteration counter
  
end

end