Fill in the gaps in the code function [xi,fi,n] = newton(f,fp,x0,tol) % Descript

Question

Fill in the gaps in the code

function [xi,fi,n] = newton(f,fp,x0,tol)
% Description: Approximate the root of f(x) using Newton's method.
%
%
% Inputs
%
% f: function handle, function whose root we seek
%
% fp: function handle, derivative of function f
%
% x0: double, initial guess for x
%
% tol: double, tolerance level (tol > 0)
%
%
% Outputs
%
% xi: double, (n+1,1) vector listing the iterates x0,x1,...xn
%
% fi: double, (n+1,1) vector listing the function values f(xi)
%
% n: int, number of iterations the algorithm took to converge
%
%
%
%
% -------------------------------------------------------------------------
n_max = 100; % maximum number of iterations
count = 0; % iteration counter

% Initialize xi
xi = zeros(n_max,1);
xi(1) = x0;

% Initialize fi
fi = zeros(n_max,1);
fi(1) = f(x0);

xnew = x0;
error = 2*tol; % initialize the error to be larger than the tolerance
while error > tol && count < n_max
count = count + 1;
  
% ---------------------------------------------------------------------
% Check if derivative is zero
% ---------------------------------------------------------------------
% If the derivative is zero at x = x0, then assign xi = nan,
% fi = nan, and n = nan, print a warning and exit.
  

% !! Your code here !!
  
  
% ---------------------------------------------------------------------
% Update the x-value
% ---------------------------------------------------------------------
% Update the previous iterate
x0 = xnew;

xnew = % !! your code here !!
  
  
% Store iterate and function value in vectors
xi(count+1) = xnew;
fi(count+1) = % !! your code here !!

% ---------------------------------------------------------------------
% Compute the error
% ---------------------------------------------------------------------
error = % !! your code here !!
  
end

% Discard unwanted entries
n = count;
xi = xi(1:n+1);
fi = fi(1:n+1);

Explanation / Answer

Answer:

function [xi,fi,n] = newton(f,fp,x0,tol)

n_max = 100; % maximum number of iterations

count = 0; % iteration counter

% Initialize xi

xi = zeros(n_max,1);

xi(1) = x0;

% Initialize fi

fi = zeros(n_max,1);

fi(1) = f(x0);

xnew = x0;

error = 2*tol; % initialize the error to be larger than the tolerance

while error > tol && count < n_max

     count = count + 1;

     % ---------------------------------------------------------------------

     % Check if derivative is zero

     % ---------------------------------------------------------------------

     % If the derivative is zero at x = x0, then assign xi = nan,

     % fi = nan, and n = nan, print a warning and exit.

     if (feval(fp,x0)==0)

        fi = nan;

        n=nan;

        xi=nan;

        error('Derivative of function is 0')

    end

     % ---------------------------------------------------------------------

     % Update the x-value

     % ---------------------------------------------------------------------

     % Update the previous iterate

     x0 = xnew;

     xnew = count;

     % Store iterate and function value in vectors

     xi(count+1) = xnew;

     fi(count+1) = x0-feval(f,x0)/feval(fp,x0);

    % ---------------------------------------------------------------------

     % Compute the error

     % ---------------------------------------------------------------------

     error = abs(fi(count+1));

end

% Discard unwanted entries

n = count;

xi = xi(1:n+1);

fi = fi(1:n+1);

end

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