added for verification

Anomaly Detection and Recommender Systems/checkCostFunction.m

+function checkCostFunction(lambda)
+%CHECKCOSTFUNCTION Creates a collaborative filering problem 
+%to check your cost function and gradients
+%   CHECKCOSTFUNCTION(lambda) Creates a collaborative filering problem 
+%   to check your cost function and gradients, it will output the 
+%   analytical gradients produced by your code and the numerical gradients 
+%   (computed using computeNumericalGradient). These two gradient 
+%   computations should result in very similar values.
+
+% Set lambda
+if ~exist('lambda', 'var') || isempty(lambda)
+    lambda = 0;
+end
+
+%% Create small problem
+X_t = rand(4, 3);
+Theta_t = rand(5, 3);
+
+% Zap out most entries
+Y = X_t * Theta_t';
+Y(rand(size(Y)) > 0.5) = 0;
+R = zeros(size(Y));
+R(Y ~= 0) = 1;
+
+%% Run Gradient Checking
+X = randn(size(X_t));
+Theta = randn(size(Theta_t));
+num_users = size(Y, 2);
+num_movies = size(Y, 1);
+num_features = size(Theta_t, 2);
+
+numgrad = computeNumericalGradient( ...
+                @(t) cofiCostFunc(t, Y, R, num_users, num_movies, ...
+                                num_features, lambda), [X(:); Theta(:)]);
+
+[cost, grad] = cofiCostFunc([X(:); Theta(:)],  Y, R, num_users, ...
+                          num_movies, num_features, lambda);
+
+disp([numgrad grad]);
+fprintf(['The above two columns you get should be very similar.\n' ...
+         '(Left-Your Numerical Gradient, Right-Analytical Gradient)\n\n']);
+
+diff = norm(numgrad-grad)/norm(numgrad+grad);
+fprintf(['If your cost function implementation is correct, then \n' ...
+         'the relative difference will be small (less than 1e-9). \n' ...
+         '\nRelative Difference: %g\n'], diff);
+
+end
\ No newline at end of file

Anomaly Detection and Recommender Systems/cofiCostFunc.m

+function [J, grad] = cofiCostFunc(params, Y, R, num_users, num_movies, ...
+                                  num_features, lambda)
+%COFICOSTFUNC Collaborative filtering cost function
+%   [J, grad] = COFICOSTFUNC(params, Y, R, num_users, num_movies, ...
+%   num_features, lambda) returns the cost and gradient for the
+%   collaborative filtering problem.
+%
+
+% Unfold the U and W matrices from params
+X = reshape(params(1:num_movies*num_features), num_movies, num_features);
+Theta = reshape(params(num_movies*num_features+1:end), ...
+                num_users, num_features);
+
+            
+% You need to return the following values correctly
+J = 0;
+X_grad = zeros(size(X));
+Theta_grad = zeros(size(Theta));
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the cost function and gradient for collaborative
+%               filtering. Concretely, you should first implement the cost
+%               function (without regularization) and make sure it is
+%               matches our costs. After that, you should implement the 
+%               gradient and use the checkCostFunction routine to check
+%               that the gradient is correct. Finally, you should implement
+%               regularization.
+%
+% Notes: X - num_movies  x num_features matrix of movie features
+%        Theta - num_users  x num_features matrix of user features
+%        Y - num_movies x num_users matrix of user ratings of movies
+%        R - num_movies x num_users matrix, where R(i, j) = 1 if the 
+%            i-th movie was rated by the j-th user
+%
+% You should set the following variables correctly:
+%
+%        X_grad - num_movies x num_features matrix, containing the 
+%                 partial derivatives w.r.t. to each element of X
+%        Theta_grad - num_users x num_features matrix, containing the 
+%                     partial derivatives w.r.t. to each element of Theta
+%
+
+
+J=sum(sum(((X*(Theta')-Y).^2).*R))/2+lambda*(sum(sum(Theta.^2)))/2 +lambda*(sum(sum(X.^2)))/2;
+for i=1:num_movies
+idx = find(R(i, :)==1);
+Thetatemp = Theta(idx, :);
+Ytemp = Y(i, idx);
+
+X_grad(i, :) = (X(i, :) * (Thetatemp') - Ytemp) * Thetatemp;
+end
+for i=1:num_users
+idx = find(R(:,i)==1);
+Xtemp = X(idx, :);
+Ytemp = Y(idx, i);
+
+Theta_grad(i, :) = (Theta(i, :) * (Xtemp') - Ytemp') * Xtemp;
+end
+
+
+X_grad=X_grad+lambda.*X;
+Theta_grad=Theta_grad+lambda.*Theta;
+
+
+
+
+
+
+% =============================================================
+
+grad = [X_grad(:); Theta_grad(:)];
+
+end

Anomaly Detection and Recommender Systems/computeNumericalGradient.m

+function numgrad = computeNumericalGradient(J, theta)
+%COMPUTENUMERICALGRADIENT Computes the gradient using "finite differences"
+%and gives us a numerical estimate of the gradient.
+%   numgrad = COMPUTENUMERICALGRADIENT(J, theta) computes the numerical
+%   gradient of the function J around theta. Calling y = J(theta) should
+%   return the function value at theta.
+
+% Notes: The following code implements numerical gradient checking, and 
+%        returns the numerical gradient.It sets numgrad(i) to (a numerical 
+%        approximation of) the partial derivative of J with respect to the 
+%        i-th input argument, evaluated at theta. (i.e., numgrad(i) should 
+%        be the (approximately) the partial derivative of J with respect 
+%        to theta(i).)
+%                
+
+numgrad = zeros(size(theta));
+perturb = zeros(size(theta));
+e = 1e-4;
+for p = 1:numel(theta)
+    % Set perturbation vector
+    perturb(p) = e;
+    loss1 = J(theta - perturb);
+    loss2 = J(theta + perturb);
+    % Compute Numerical Gradient
+    numgrad(p) = (loss2 - loss1) / (2*e);
+    perturb(p) = 0;
+end
+
+end

Anomaly Detection and Recommender Systems/estimateGaussian.m

+function [mu sigma2] = estimateGaussian(X)
+%ESTIMATEGAUSSIAN This function estimates the parameters of a 
+%Gaussian distribution using the data in X
+%   [mu sigma2] = estimateGaussian(X), 
+%   The input X is the dataset with each n-dimensional data point in one row
+%   The output is an n-dimensional vector mu, the mean of the data set
+%   and the variances sigma^2, an n x 1 vector
+% 
+
+% Useful variables
+[m, n] = size(X);
+
+% You should return these values correctly
+mu = zeros(n, 1);
+sigma2 = zeros(n, 1);
+
+% ====================== YOUR CODE HERE ======================
+% Instructions: Compute the mean of the data and the variances
+%               In particular, mu(i) should contain the mean of
+%               the data for the i-th feature and sigma2(i)
+%               should contain variance of the i-th feature.
+%
+
+mu=(sum(X))'/m;
+for i=1:n
+    for j=1:m
+sigma2(i)=sigma2(i)+ (X(j,i)-mu(i)).^2;
+    end
+end
+sigma2=sigma2/m;
+
+
+
+
+
+
+
+
+% =============================================================
+
+
+end

Anomaly Detection and Recommender Systems/ex8.m

+%% Machine Learning Online Class
+%  Exercise 8 | Anomaly Detection and Collaborative Filtering
+%
+%  Instructions
+%  ------------
+%
+%  This file contains code that helps you get started on the
+%  exercise. You will need to complete the following functions:
+%
+%     estimateGaussian.m
+%     selectThreshold.m
+%     cofiCostFunc.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% Initialization
+clear ; close all; clc
+
+%% ================== Part 1: Load Example Dataset  ===================
+%  We start this exercise by using a small dataset that is easy to
+%  visualize.
+%
+%  Our example case consists of 2 network server statistics across
+%  several machines: the latency and throughput of each machine.
+%  This exercise will help us find possibly faulty (or very fast) machines.
+%
+
+fprintf('Visualizing example dataset for outlier detection.\n\n');
+
+%  The following command loads the dataset. You should now have the
+%  variables X, Xval, yval in your environment
+load('ex8data1.mat');
+
+%  Visualize the example dataset
+plot(X(:, 1), X(:, 2), 'bx');
+axis([0 30 0 30]);
+xlabel('Latency (ms)');
+ylabel('Throughput (mb/s)');
+
+fprintf('Program paused. Press enter to continue.\n');
+pause
+
+
+%% ================== Part 2: Estimate the dataset statistics ===================
+%  For this exercise, we assume a Gaussian distribution for the dataset.
+%
+%  We first estimate the parameters of our assumed Gaussian distribution, 
+%  then compute the probabilities for each of the points and then visualize 
+%  both the overall distribution and where each of the points falls in 
+%  terms of that distribution.
+%
+fprintf('Visualizing Gaussian fit.\n\n');
+
+%  Estimate my and sigma2
+[mu sigma2] = estimateGaussian(X);
+
+%  Returns the density of the multivariate normal at each data point (row) 
+%  of X
+p = multivariateGaussian(X, mu, sigma2);
+
+%  Visualize the fit
+visualizeFit(X,  mu, sigma2);
+xlabel('Latency (ms)');
+ylabel('Throughput (mb/s)');
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 3: Find Outliers ===================
+%  Now you will find a good epsilon threshold using a cross-validation set
+%  probabilities given the estimated Gaussian distribution
+% 
+
+pval = multivariateGaussian(Xval, mu, sigma2);
+
+[epsilon F1] = selectThreshold(yval, pval);
+fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
+fprintf('Best F1 on Cross Validation Set:  %f\n', F1);
+fprintf('   (you should see a value epsilon of about 8.99e-05)\n');
+fprintf('   (you should see a Best F1 value of  0.875000)\n\n');
+
+%  Find the outliers in the training set and plot the
+outliers = find(p < epsilon);
+
+%  Draw a red circle around those outliers
+hold on
+plot(X(outliers, 1), X(outliers, 2), 'ro', 'LineWidth', 2, 'MarkerSize', 10);
+hold off
+
+fprintf('Program paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 4: Multidimensional Outliers ===================
+%  We will now use the code from the previous part and apply it to a 
+%  harder problem in which more features describe each datapoint and only 
+%  some features indicate whether a point is an outlier.
+%
+
+%  Loads the second dataset. You should now have the
+%  variables X, Xval, yval in your environment
+load('ex8data2.mat');
+
+%  Apply the same steps to the larger dataset
+[mu sigma2] = estimateGaussian(X);
+
+%  Training set 
+p = multivariateGaussian(X, mu, sigma2);
+
+%  Cross-validation set
+pval = multivariateGaussian(Xval, mu, sigma2);
+
+%  Find the best threshold
+[epsilon F1] = selectThreshold(yval, pval);
+
+fprintf('Best epsilon found using cross-validation: %e\n', epsilon);
+fprintf('Best F1 on Cross Validation Set:  %f\n', F1);
+fprintf('   (you should see a value epsilon of about 1.38e-18)\n');
+fprintf('   (you should see a Best F1 value of 0.615385)\n');
+fprintf('# Outliers found: %d\n\n', sum(p < epsilon));

Anomaly Detection and Recommender Systems/ex8_cofi.m

+%% Machine Learning Online Class
+%  Exercise 8 | Anomaly Detection and Collaborative Filtering
+%
+%  Instructions
+%  ------------
+%
+%  This file contains code that helps you get started on the
+%  exercise. You will need to complete the following functions:
+%
+%     estimateGaussian.m
+%     selectThreshold.m
+%     cofiCostFunc.m
+%
+%  For this exercise, you will not need to change any code in this file,
+%  or any other files other than those mentioned above.
+%
+
+%% =============== Part 1: Loading movie ratings dataset ================
+%  You will start by loading the movie ratings dataset to understand the
+%  structure of the data.
+%  
+fprintf('Loading movie ratings dataset.\n\n');
+
+%  Load data
+load ('ex8_movies.mat');
+
+%  Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies on 
+%  943 users
+%
+%  R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
+%  rating to movie i
+
+%  From the matrix, we can compute statistics like average rating.
+fprintf('Average rating for movie 1 (Toy Story): %f / 5\n\n', ...
+        mean(Y(1, R(1, :))));
+
+%  We can "visualize" the ratings matrix by plotting it with imagesc
+imagesc(Y);
+ylabel('Movies');
+xlabel('Users');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ============ Part 2: Collaborative Filtering Cost Function ===========
+%  You will now implement the cost function for collaborative filtering.
+%  To help you debug your cost function, we have included set of weights
+%  that we trained on that. Specifically, you should complete the code in 
+%  cofiCostFunc.m to return J.
+
+%  Load pre-trained weights (X, Theta, num_users, num_movies, num_features)
+load ('ex8_movieParams.mat');
+
+%  Reduce the data set size so that this runs faster
+num_users = 4; num_movies = 5; num_features = 3;
+X = X(1:num_movies, 1:num_features);
+Theta = Theta(1:num_users, 1:num_features);
+Y = Y(1:num_movies, 1:num_users);
+R = R(1:num_movies, 1:num_users);
+
+%  Evaluate cost function
+J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
+               num_features, 0);
+           
+fprintf(['Cost at loaded parameters: %f '...
+         '\n(this value should be about 22.22)\n'], J);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============== Part 3: Collaborative Filtering Gradient ==============
+%  Once your cost function matches up with ours, you should now implement 
+%  the collaborative filtering gradient function. Specifically, you should 
+%  complete the code in cofiCostFunc.m to return the grad argument.
+%  
+fprintf('\nChecking Gradients (without regularization) ... \n');
+
+%  Check gradients by running checkNNGradients
+checkCostFunction;
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ========= Part 4: Collaborative Filtering Cost Regularization ========
+%  Now, you should implement regularization for the cost function for 
+%  collaborative filtering. You can implement it by adding the cost of
+%  regularization to the original cost computation.
+%  
+
+%  Evaluate cost function
+J = cofiCostFunc([X(:) ; Theta(:)], Y, R, num_users, num_movies, ...
+               num_features, 1.5);
+           
+fprintf(['Cost at loaded parameters (lambda = 1.5): %f '...
+         '\n(this value should be about 31.34)\n'], J);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ======= Part 5: Collaborative Filtering Gradient Regularization ======
+%  Once your cost matches up with ours, you should proceed to implement 
+%  regularization for the gradient. 
+%
+
+%  
+fprintf('\nChecking Gradients (with regularization) ... \n');
+
+%  Check gradients by running checkNNGradients
+checkCostFunction(1.5);
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ============== Part 6: Entering ratings for a new user ===============
+%  Before we will train the collaborative filtering model, we will first
+%  add ratings that correspond to a new user that we just observed. This
+%  part of the code will also allow you to put in your own ratings for the
+%  movies in our dataset!
+%
+movieList = loadMovieList();
+
+%  Initialize my ratings
+my_ratings = zeros(1682, 1);
+
+% Check the file movie_idx.txt for id of each movie in our dataset
+% For example, Toy Story (1995) has ID 1, so to rate it "4", you can set
+my_ratings(1) = 4;
+
+% Or suppose did not enjoy Silence of the Lambs (1991), you can set
+my_ratings(98) = 2;
+
+% We have selected a few movies we liked / did not like and the ratings we
+% gave are as follows:
+my_ratings(7) = 3;
+my_ratings(12)= 5;
+my_ratings(54) = 4;
+my_ratings(64)= 5;
+my_ratings(66)= 3;
+my_ratings(69) = 5;
+my_ratings(183) = 4;
+my_ratings(226) = 5;
+my_ratings(355)= 5;
+
+fprintf('\n\nNew user ratings:\n');
+for i = 1:length(my_ratings)
+    if my_ratings(i) > 0 
+        fprintf('Rated %d for %s\n', my_ratings(i), ...
+                 movieList{i});
+    end
+end
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+
+%% ================== Part 7: Learning Movie Ratings ====================
+%  Now, you will train the collaborative filtering model on a movie rating 
+%  dataset of 1682 movies and 943 users
+%
+
+fprintf('\nTraining collaborative filtering...\n');
+
+%  Load data
+load('ex8_movies.mat');
+
+%  Y is a 1682x943 matrix, containing ratings (1-5) of 1682 movies by 
+%  943 users
+%
+%  R is a 1682x943 matrix, where R(i,j) = 1 if and only if user j gave a
+%  rating to movie i
+
+%  Add our own ratings to the data matrix
+Y = [my_ratings Y];
+R = [(my_ratings ~= 0) R];
+
+%  Normalize Ratings
+[Ynorm, Ymean] = normalizeRatings(Y, R);
+
+%  Useful Values
+num_users = size(Y, 2);
+num_movies = size(Y, 1);
+num_features = 10;
+
+% Set Initial Parameters (Theta, X)
+X = randn(num_movies, num_features);
+Theta = randn(num_users, num_features);
+
+initial_parameters = [X(:); Theta(:)];
+
+% Set options for fmincg
+options = optimset('GradObj', 'on', 'MaxIter', 100);
+
+% Set Regularization
+lambda = 10;
+theta = fmincg (@(t)(cofiCostFunc(t, Ynorm, R, num_users, num_movies, ...
+                                num_features, lambda)), ...
+                initial_parameters, options);
+
+% Unfold the returned theta back into U and W
+X = reshape(theta(1:num_movies*num_features), num_movies, num_features);
+Theta = reshape(theta(num_movies*num_features+1:end), ...
+                num_users, num_features);
+
+fprintf('Recommender system learning completed.\n');
+
+fprintf('\nProgram paused. Press enter to continue.\n');
+pause;
+
+%% ================== Part 8: Recommendation for you ====================
+%  After training the model, you can now make recommendations by computing
+%  the predictions matrix.
+%
+
+p = X * Theta';
+my_predictions = p(:,1) + Ymean;
+
+movieList = loadMovieList();
+
+[r, ix] = sort(my_predictions, 'descend');
+fprintf('\nTop recommendations for you:\n');
+for i=1:10
+    j = ix(i);
+    fprintf('Predicting rating %.1f for movie %s\n', my_predictions(j), ...
+            movieList{j});
+end
+
+fprintf('\n\nOriginal ratings provided:\n');
+for i = 1:length(my_ratings)
+    if my_ratings(i) > 0 
+        fprintf('Rated %d for %s\n', my_ratings(i), ...
+                 movieList{i});
+    end
+end

Anomaly Detection and Recommender Systems/fmincg.m

+function [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+% Minimize a continuous differentialble multivariate function. Starting point
+% is given by "X" (D by 1), and the function named in the string "f", must
+% return a function value and a vector of partial derivatives. The Polack-
+% Ribiere flavour of conjugate gradients is used to compute search directions,
+% and a line search using quadratic and cubic polynomial approximations and the
+% Wolfe-Powell stopping criteria is used together with the slope ratio method
+% for guessing initial step sizes. Additionally a bunch of checks are made to
+% make sure that exploration is taking place and that extrapolation will not
+% be unboundedly large. The "length" gives the length of the run: if it is
+% positive, it gives the maximum number of line searches, if negative its
+% absolute gives the maximum allowed number of function evaluations. You can
+% (optionally) give "length" a second component, which will indicate the
+% reduction in function value to be expected in the first line-search (defaults
+% to 1.0). The function returns when either its length is up, or if no further
+% progress can be made (ie, we are at a minimum, or so close that due to
+% numerical problems, we cannot get any closer). If the function terminates
+% within a few iterations, it could be an indication that the function value
+% and derivatives are not consistent (ie, there may be a bug in the
+% implementation of your "f" function). The function returns the found
+% solution "X", a vector of function values "fX" indicating the progress made
+% and "i" the number of iterations (line searches or function evaluations,
+% depending on the sign of "length") used.
+%
+% Usage: [X, fX, i] = fmincg(f, X, options, P1, P2, P3, P4, P5)
+%
+% See also: checkgrad 
+%
+% Copyright (C) 2001 and 2002 by Carl Edward Rasmussen. Date 2002-02-13
+%
+%
+% (C) Copyright 1999, 2000 & 2001, Carl Edward Rasmussen
+% 
+% Permission is granted for anyone to copy, use, or modify these
+% programs and accompanying documents for purposes of research or
+% education, provided this copyright notice is retained, and note is
+% made of any changes that have been made.
+% 
+% These programs and documents are distributed without any warranty,
+% express or implied.  As the programs were written for research
+% purposes only, they have not been tested to the degree that would be
+% advisable in any important application.  All use of these programs is
+% entirely at the user's own risk.
+%
+% [ml-class] Changes Made:
+% 1) Function name and argument specifications
+% 2) Output display
+%
+
+% Read options
+if exist('options', 'var') && ~isempty(options) && isfield(options, 'MaxIter')
+    length = options.MaxIter;
+else
+    length = 100;
+end
+
+
+RHO = 0.01;                            % a bunch of constants for line searches
+SIG = 0.5;       % RHO and SIG are the constants in the Wolfe-Powell conditions
+INT = 0.1;    % don't reevaluate within 0.1 of the limit of the current bracket
+EXT = 3.0;                    % extrapolate maximum 3 times the current bracket
+MAX = 20;                         % max 20 function evaluations per line search
+RATIO = 100;                                      % maximum allowed slope ratio
+
+argstr = ['feval(f, X'];                      % compose string used to call function
+for i = 1:(nargin - 3)
+  argstr = [argstr, ',P', int2str(i)];
+end
+argstr = [argstr, ')'];
+
+if max(size(length)) == 2, red=length(2); length=length(1); else red=1; end
+S=['Iteration '];
+
+i = 0;                                            % zero the run length counter
+ls_failed = 0;                             % no previous line search has failed
+fX = [];
+[f1 df1] = eval(argstr);                      % get function value and gradient
+i = i + (length<0);                                            % count epochs?!
+s = -df1;                                        % search direction is steepest
+d1 = -s'*s;                                                 % this is the slope
+z1 = red/(1-d1);                                  % initial step is red/(|s|+1)
+
+while i < abs(length)                                      % while not finished
+  i = i + (length>0);                                      % count iterations?!
+
+  X0 = X; f0 = f1; df0 = df1;                   % make a copy of current values
+  X = X + z1*s;                                             % begin line search
+  [f2 df2] = eval(argstr);
+  i = i + (length<0);                                          % count epochs?!
+  d2 = df2'*s;
+  f3 = f1; d3 = d1; z3 = -z1;             % initialize point 3 equal to point 1
+  if length>0, M = MAX; else M = min(MAX, -length-i); end
+  success = 0; limit = -1;                     % initialize quanteties
+  while 1
+    while ((f2 > f1+z1*RHO*d1) || (d2 > -SIG*d1)) && (M > 0) 
+      limit = z1;                                         % tighten the bracket
+      if f2 > f1
+        z2 = z3 - (0.5*d3*z3*z3)/(d3*z3+f2-f3);                 % quadratic fit
+      else
+        A = 6*(f2-f3)/z3+3*(d2+d3);                                 % cubic fit
+        B = 3*(f3-f2)-z3*(d3+2*d2);
+        z2 = (sqrt(B*B-A*d2*z3*z3)-B)/A;       % numerical error possible - ok!
+      end
+      if isnan(z2) || isinf(z2)
+        z2 = z3/2;                  % if we had a numerical problem then bisect
+      end
+      z2 = max(min(z2, INT*z3),(1-INT)*z3);  % don't accept too close to limits
+      z1 = z1 + z2;                                           % update the step
+      X = X + z2*s;
+      [f2 df2] = eval(argstr);
+      M = M - 1; i = i + (length<0);                           % count epochs?!
+      d2 = df2'*s;
+      z3 = z3-z2;                    % z3 is now relative to the location of z2
+    end
+    if f2 > f1+z1*RHO*d1 || d2 > -SIG*d1
+      break;                                                % this is a failure
+    elseif d2 > SIG*d1
+      success = 1; break;                                             % success
+    elseif M == 0
+      break;                                                          % failure
+    end
+    A = 6*(f2-f3)/z3+3*(d2+d3);                      % make cubic extrapolation
+    B = 3*(f3-f2)-z3*(d3+2*d2);
+    z2 = -d2*z3*z3/(B+sqrt(B*B-A*d2*z3*z3));        % num. error possible - ok!
+    if ~isreal(z2) || isnan(z2) || isinf(z2) || z2 < 0 % num prob or wrong sign?
+      if limit < -0.5                               % if we have no upper limit