function Newton1(x,nmax,ep,de)
% Cheney/Kincaid 5th Ed., Page 106.
% Newton's Method with a maximum number of iterations and
% a fixed value for epsilon and delta.
% x = initial guess x_0
% nmax = max iterations
% ep = max error for convergence of d
% de = max error for convergence of fprime
% I used:
% Newton1(4,100,1e-10,1e-10)
tic
% We will use the example on p. 107.
% f(x) is really x^3 - 2*x^2 + x - 3, but the authors remind us
% that the NESTED form will increase efficiency.  Thus, we have
% f(x)  = ((x - 2)*x + 1)*x - 3    and
% f'(x) = (3*x - 4)*x + 1.
% Our initial value of x will be 4.
% nmax = maximum number of iterations.
f = @(x) ((x - 2)*x + 1)*x -3;
fprime = @(x) (3*x - 4)*x + 1;
% Initialize f(x_0).
fx = f(x);
% I guess we can start a table.
% I counted out all of the field spaces.
fprintf('  n   x_n                   f(x_n)\n');
fprintf('  0  %20.17f  %20.17f\n',x,fx);
% We will run nmax iterations (or fewer).
for i=1:nmax
    % Calculate f'(x).
    fp = fprime(x);
    % If f'(x) is small enough, then stop the algorithm.
    if abs(fp) < de
        fprintf('Near root; fprime < delta\n');
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        return;
    end
    % Calculate the new value of x for this iteration.
    d = fx/fp;
    x = x - d;
    fx = f(x);
    fprintf('%3d  %20.17f  %20.17f\n',i,x,fx);
    % 
    if abs(d) < ep
        fprintf('Convergence; abs(d) < epsilon\n');
        toc
        return;
    end
end
% We've run out of iterations!
fprintf('Did not converge!\n');
toc