Question

Newton's method for finding the root in MATLAB. I am stuck on the two lines of codes which are required to complete. can anyone give me a hint on this?

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

% (c) Z. Lubberts (JHU)

% v1 by M. Zhong

% Demo. code for EN.553.281

% 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

% (c) Z. Lubberts (JHU)

% v1 by M. Zhong

% Demo. code for EN.553.281

% 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
  
    fc = f(xc);
    % complete the code to evalute the function and derivative at xk
    xc=xc-fc(1)/fc(2);
    % 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