NR_Transcendental()

% Name           : Chandan Kumar
% Title          : Solution of Transcendental equation by Newton Raphson Method
% Date           : 24th July 2009
% Last Modified  : 26th july 2009
% Disclaimer     : This code has been specially designed to overcome the
%                  common faults of the Newton-Raphson method  for example
%                  Infinite cycle,Zero derivative at root or non-existance 
%                  of derivative at root etc.
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------

function NR_Transcendental()
clc, clear all                              % Clears the command window and variables
syms x 			                            % variable x declared symbol

F_x = input('Input the function f(x): ');
D_F = diff(F_x);                            % Obtains differentiation of function.

x_initial = input('Give an initial iteration point: ');  % User input of initial guess
F_Accuracy = input('Input the tolerance needed[Default 1e-6]: ');
if isempty(F_Accuracy)
    F_Accuracy = 1e-6;                       % No of digit for termination
end
clc
disp('Transcendental equation to be solved  is:')
disp(' ');
disp(F_x);
x = x_initial;                               % Passing argument for eval function
for i = 1:1000                               % Initialization loop for iteration
    x_prev(i) = x;                           % Assigning previous value to variable
    eval(F_x);
    eval(D_F);
    if (eval(F_x)~=0)&& (isempty(eval(D_F))) % Condition when First derivative DNE
        [f_left,f_right] = eval(F_x(x-0.5, x+0.5)); % Calculates value of F away from point
        if (abs(F_x) > abs(f_left))          % Proceeds in the direction of minimum value
            x = x - 0.5;                     % towards left
        elseif (abs(F_x) > abs(f_right))     % Towards right
            x = x+0.5;
        else 
            disp('First Derivative DOESNOT exists. Solution NOT found!!!');
            break;
        end
    elseif (eval(F_x)~=0)&& (abs(eval(D_F))<F_Accuracy)    % Derivative is close to zero
        disp('First Derivative is Zero.Solution NOT found!!!');
        break;
    elseif (abs(eval(F_x))< F_Accuracy)&& (isempty(eval(D_F)))% Derivative DNE at root point.
        disp('Root of equation found.');
        disp(['Root is: ' num2str(x) '']);
        disp(['But function' num2str(F_x) ' DOESNOT have a continuous derivative at root']);
        break;
    elseif (abs(eval(F_x))< F_Accuracy)&& (eval(D_F)==0)   % Derivative is zero at root point
        disp('Root of equation found.');
        disp(['Root is: ' num2str(x) '']);
        disp(['Function' num2str(F_x) ' has optima at root point']);
        break;
    elseif(abs(eval(F_x)/eval(D_F)) < F_Accuracy)          % Desired accuracy for root achieved
        disp('Root of equation found.');
        disp(['Root is: ' num2str(x) '']);
        disp(['No of iteration required is:' num2str(i) '']);
        break;
    elseif (i > 2)&&(x_prev(i-2) == x)       % Case when solution oscillates between two values
        disp('Starting point has entered in a cycle. Convergence NOT possible');
        index = input('Search for solution using other starting point Y/N or y/n,[y]: ','s');
        if isempty(index)||(index== 'Y')||(index=='y')     % Validating the request
            x = x + 0.5;                     % Taking starting point outside of interval of cycle
        end  
    else
        x = x - (eval(F_x)/eval(D_F));       % Used for next iteration 
    end         
end