function [ x, ierror, k ] = jarratt ( x, abserr, kmax, f )

%*****************************************************************************80
%
%% jarratt() implements the Jarratt fourth order method.
%
%  Discussion:
%
%    Jarratt's method is of fourth order.
%
%    x(n+1) = x(n) - 1/2 * ( f(x(n) / f'(x(n)) )
%      + f(x(n)) / ( f'(x(n)) - 3*f'( x(n)-(2/3)*f(x(n))/f'(x(n)) ) )
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    15 May 2024
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    P Jarratt,
%    Some fourth-order multipoint iterative methods for solving equations,
%    Mathematics of Computation,
%    Volume 20, Number 95, 1966, pages 434 - 437.
%
%  Input:
%
%    real X: an estimate for the root of the equation.
%
%    real ABSERR, an error tolerance.
%
%    integer KMAX, the maximum number of iterations allowed.
%
%    real external F, the name of the routine that
%    evaluates the function or its derivatives, of the form
%      function f ( x, ider )
%
%  Output:
%
%    real X: an approximate root for which abs ( F(X) ) <= ABSERR.
%
%    integer IERROR, error indicator.
%    0, no error occurred.
%    nonzero, an error occurred, and the iteration was halted.
%
%    integer K, the number of steps taken.
%

%
%  Initialization.
%
  ierror = 0;
  k = 0;
  fx = f ( x, 0 );
  dfx = f ( x, 1 );
%
%  Iteration loop:
%
  while ( true )
%
%  If the error tolerance is satisfied, then exit.
%
    if ( abs ( fx ) <= abserr )
      break
    end

    k = k + 1;

    if ( kmax < k )
      ierror = 2;
      return
    end

    if ( dfx == 0.0 )
      ierror = 3;
      return
    end

    z = x - 2.0 * fx / ( 3.0 * dfx );
    dfz = f ( z, 1 );

    if ( dfx - 3.0 * dfz == 0.0 )
      ierror = 3;
      return
    end
%
%  Set the increment.
%
    dx = ( - 0.5 / dfx + 1.0 / ( dfx - 3.0 * dfz ) ) * fx;
%
%  Update the iterate and function values.
%
    x = x + dx;
    fx = f ( x, 0 );
    dfx = f ( x, 1 );

  end

  return
end
