Updated examples, function header, documentation, and readme.

Author: tamaskis

EXAMPLES.m

@@ -14,17 +14,18 @@ Calculates the fixed point of a univariate function using fixed-point iteration.
 
 ## Description
 
-`c = fixed_point_iteration(f,x0)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point. The default tolerance and maximum number of iterations are `TOL = 1e-12` and `imax = 1e6`, respectively.
+`c = fixed_point_iteration(f,x0)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?\inline&space;f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point. The default tolerance and maximum number of iterations are `TOL = 1e-12` and `imax = 1e6`, respectively.
 
-`c = fixed_point_iteration(f,x0,TOL)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point and `TOL` is the tolerance. The default maximum number of iterations is `imax = 1e6`.
+`c = fixed_point_iteration(f,x0,TOL)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?\inline&space;f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point and `TOL` is the tolerance. The default maximum number of iterations is `imax = 1e6`.
 
-`c = fixed_point_iteration(f,x0,[],imax)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point and `imax` is the maximum number of iterations. The default tolerance is `TOL = 1e-12`.
+`c = fixed_point_iteration(f,x0,[],imax)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?\inline&space;f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point and `imax` is the maximum number of iterations. The default tolerance is `TOL = 1e-12`.
 
-`c = fixed_point_iteration(f,x0,TOL,imax)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point, `TOL` is the tolerance, and `imax` is the maximum number of iterations.
+`c = fixed_point_iteration(f,x0,TOL,imax)` returns the fixed point of a function <img src="https://latex.codecogs.com/svg.latex?\inline&space;f(x)" title="f(x)" /> specified by the function handle `f`, where `x0` is an initial guess of the fixed point, `TOL` is the tolerance, and `imax` is the maximum number of iterations.
 
 `c = fixed_point_iteration(__,'all')` returns a vector, where the first element of this vector is the initial guess, all intermediate elements are the intermediate estimates of the fixed point, and the last element is the converged fixed point. This identifier 'all' may be appended to any of the syntaxes used above.
 
 
-## Additional Documentation and Examples
+## Examples and Additional Documentation
 
-See "DOCUMENTATION.pdf" for additional documentation and examples.
+   -  See "EXAMPLES.mlx" or the "Examples" tab on the File Exchange page for examples. 
+   -  See "DOCUMENTATION.pdf" (included with download, also available at https://github.com/tamaskis/fixed_point_iteration-MATLAB/blob/main/DOCUMENTATION.pdf) for additional documentation.

EXAMPLES.mlx

@@ -1,61 +1,51 @@
+%==========================================================================
+%
 % fixed_point_iteration  Calculates the fixed point of a univariate 
 % function using fixed-point iteration.
 %
-%   c = fixed_point_iteration(f,x0) returns the fixed point of a function 
-%   f(x) specified by the function handle f, where x0 is an initial guess 
-%   of the fixed point. The default tolerance and maximum number of 
-%   iterations are TOL = 1e-12 and imax = 1e6, respectively.
-%
-%   c = fixed_point_iteration(f,x0,TOL) returns the fixed point of a 
-%   function f(x) specified by the function handle f, where x0 is an 
-%   initial guess of the fixed point and TOL is the tolerance. The default 
-%   maximum number of iterations is imax = 1e6.
-%
-%   c = fixed_point_iteration(f,x0,[],imax) returns the fixed point of a 
-%   function f(x) specified by the function handle f, where x0 is an 
-%   initial guess of the fixed point and imax is the maximum number of 
-%   iterations. The default tolerance is TOL = 1e-12.
+%   c = fixed_point_iteration(f,x0)
+%   c = fixed_point_iteration(f,x0,TOL)
+%   c = fixed_point_iteration(f,x0,[],imax)
+%   c = fixed_point_iteration(f,x0,TOL,imax)
+%   c = fixed_point_iteration(__,'all')
 %
-%   c = fixed_point_iteration(f,x0,TOL,imax) returns the fixed point of a 
-%   function f(x) specified by the function handle f, where x0 is an 
-%   initial guess of the fixed point, TOL is the tolerance, and imax is the
-%   maximum number of iterations.
+% Copyright © 2021 Tamas Kis
+% Last Update: 2021-06-18
 %
-%   c = fixed_point_iteration(__,'all') returns a vector, where the first
-%   element of this vector is the initial guess, all intermediate elements
-%   are the intermediate estimates of the fixed point, and the last element
-%   is the converged fixed point. This identifier 'all' may be appended to 
-%   any of the syntaxes used above.
+%--------------------------------------------------------------------------
 %
 % MATLAB Central File Exchange: https://www.mathworks.com/matlabcentral/fileexchange/86992-fixed-point-iteration-fixed_point_iteration
 % GitHub: https://github.com/tamaskis/fixed_point_iteration-MATLAB
 %
-% See "DOCUMENTATION.pdf" for additional documentation and examples. 
-% Examples can also be found in EXAMPLES.m. Both of these files are 
-% included with the download.
+% See EXAMPLES.mlx for examples and "DOCUMENTATION.pdf" for additional 
+% documentation. Both of these files are included with the download.
 %
-% Copyright (c) 2021 Tamas Kis
-% Last Update: 2021-03-27
-
-
-
-%% FUNCTION
-
-% INPUT: f - function handle for f(x)
-%        x0 - initial guess for fixed point
-%        TOL - (OPTIONAL) tolerance
-%        imax - (OPTIONAL) maximum number of iterations
-%        output - if specified as 'all', function will returns all 
-%                 intermediate estimates of the fixed point; otherwise, a 
-%                 faster algorithm is used to only return the converged 
-%                 fixed point
-% OUTPUT: c - fixed point of f(x)
-%          --> if "output" is specified as 'all', then "c" will be a
-%              vector, where the first element is the initial guess, the 
-%              last element is the converged fixedd point, and the other 
-%              elements are intermediate estimates of the fixed point
-%          --> otherwise, "c" is a single number storing the converged
-%              fixed point
+%--------------------------------------------------------------------------
+%
+% -------
+% INPUTS:
+% -------
+%   f       - (function_handle) f(x)
+%   x0      - (1×1) initial guess for fixed point
+%   TOL     - (OPTIONAL) (1×1) tolerance
+%   imax    - (OPTIONAL) (1×1) maximum number of iterations
+%   output  - (OPTIONAL) (char) if specified as 'all', function will return
+%             all intermediate fixed point estimates; otherwise, a faster 
+%             algorithm is used to only return the converged fixed point
+%
+% --------
+% OUTPUTS:
+% --------
+%   c       - (1×1 or n×1) fixed point of f(x)
+%           	--> if "output" is specified as 'all', then "c" will be a
+%                   vector, where the first element is the initial guess,
+%                   the last element is the converged fixed point, and the 
+%                   other elements are intermediate estimates of the fixed
+%                   point
+%               --> otherwise, "c" is a single number storing the converged
+%                   fixed point
+%
+%==========================================================================
 function c = fixed_point_iteration(f,x0,TOL,imax,output)
 
     % sets default tolerance and maximum number of iterations if not
@@ -116,6 +106,9 @@
         % sets estimate of fixed point at the first iteration of the fixed
         % point iteration as the initial guess
         x_old = x0;
+        
+        % initializes x_new so its scope isn't limited to the while loop
+        x_new = 0;
 
         % fixed-point iteration
         i = 1;
@@ -127,16 +120,16 @@
             % calculates error
             err = abs(x_new-x_old);
 
-            % stores updated estimate of fixed point for next iteration
+            % stores current estimate of fixed point for next iteration
             x_old = x_new;
 
             % increments loop index
             i = i+1;
 
         end
 
-        % returns fixed point
-        c = x_old;
+        % returns converged fixed point
+        c = x_new;
 
     end