function [ x, x1, ierror, k ] = regula ( x, x1, abserr, kmax, f )

%*****************************************************************************80
%
%% regula() implements the Regula Falsi method.
%
%  Discussion:
%
%    The algorithm was modified to guarantee that f(x) <= f(x1).
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    16 May 2024
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    real X, X1: two distinct points that start the method.
%
%    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, X1: X is an approximate root satisfying
%    abs ( F(X) ) < ABSERR, and X1 is the previous estimate.
%
%    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 );
  fx1 = f ( x1, 0 );

  if ( fx < 0.0  )
    t = x;
    x = x1;
    x1 = t;
    t = fx;
    fx = fx1;
    fx1 = t;
  end
%
%  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
%
%  Set the increment.
%
    dx = - fx1 * ( x - x1 ) / ( fx - fx1 );
%
%  Update the iterate and function values.
%
    x2 = x1 + dx;
    fx2 = f ( x2, 0 );

    if ( 0.0 <= fx2 )
      x = x2;
      fx = fx2;
    else
      x1 = x2;
      fx1 = fx2;
    end
%
%  Let x,fx be minimal.
%
    if ( abs ( fx1 ) < abs ( fx ) )
      t = x;
      x = x1;
      x1 = t;
      t = fx;
      fx = fx1;
      fx1 = t;
    end

  end

  return
end
