Switch to a polymorphic method invokation

Work 1/scripts/GivenEnv.m

@@ -1,16 +1,17 @@
-%
-% Select the given interval: [-1,3]
-a_0 = -1;
-b_0 = 3;
-
-% Setup the functions under test
-f_1 = @(x) (x-2)^2 + x*log(x+3);
-f_2 = @(x) exp(-2*x) + (x-2)^2;
-f_3 = @(x) exp(x)*(x^3 - 1) + (x-1)*sin(x);
-funs = {f_1, f_2, f_3};
-
-% Setup the function titles
-title_f1 = "$f_1(x) = (x - 2)^2 + x \cdot \ln(x + 3)$";
-title_f2 = "$f_2(x) = e^{-2x} + (x - 2)^2$";
-title_f3 = "$f_3(x) = e^x \cdot (x^3 - 1) + (x - 1) \cdot \sin(x)$";
-titles = [title_f1; title_f2; title_f3];
+%
+% Select the given interval: [-1,3]
+a_0 = -1;
+b_0 = 3;
+
+% Setup the functions under test
+syms x;
+f_1 = (x-2)^2 + x*log(x+3);
+f_2 = exp(-2*x) + (x-2)^2;
+f_3 = exp(x)*(x^3 - 1) + (x-1)*sin(x);
+funs = [f_1; f_2; f_3];
+
+% Setup the function titles
+title_f1 = "$f_1(x) = (x - 2)^2 + x \cdot \ln(x + 3)$";
+title_f2 = "$f_2(x) = e^{-2x} + (x - 2)^2$";
+title_f3 = "$f_3(x) = e^x \cdot (x^3 - 1) + (x - 1) \cdot \sin(x)$";
+titles = [title_f1; title_f2; title_f3];

Work 1/scripts/Work1.m

@@ -0,0 +1,37 @@
+%
+%
+%
+%
+%
+%
+
+
+disp (" ");
+disp (" ");
+disp ('1. Number of iterations for different epsilon values for min_bisection');
+disp ('----');
+bisection_over_epsilon;
+
+methods = {
+    @min_bisection;
+    @min_golden_section;
+    @min_fibonacci;
+    @min_bisection_der
+};
+
+disp (" ");
+disp (" ");
+disp ('2. Number of iterations for different lambda values');
+disp ('----');
+for i = 1:length(methods)
+    iterations_over_lambda(methods{i});
+end
+
+
+disp (" ");
+disp (" ");
+disp ('3. [a, b] interval convergence');
+disp ('----');
+for i = 1:length(methods)
+    interval_over_iterations(methods{i});
+end

Work 1/scripts/bisection/bisection_interval.m

@@ -1,48 +0,0 @@
-%
-% Keeping epsilon fixed, plot the [a,b] interval over the iterations for
-% different lambda values (min, mid, max))
-%
-
-
-% Clear workspace and load the functions and interval
-clear
-addpath('..');
-GivenEnv;
-
-% * epsilon: e = 0.001
-% * lambda:  l > 2e = 0.001
-% * dl: A small step away from 2e
-%       dl = 0.0001
-% * lambda_max: 0.1
-% * N: 3 lambda values
-
-N = 3;
-epsilon = 0.001;
-dl = 0.0001;
-lambda_max= 0.1;
-lambda = linspace(2*epsilon + dl, lambda_max, N);
-k = zeros(1, N); % preallocate k
-
-
-%
-% * Call the bisection method for each lambda value for each function
-% * Plot the [a,b] interval over iterations for each lambda for each function 
-%
-
-for i = 1:length(funs)
-    figure;
-    for j = 1:N
-        [a, b, k(j)] = bisection(funs{i}, a_0, b_0, epsilon, lambda(j));
-        subplot(length(funs), 1, j)
-        plot(1:length(a), a, 'ob')
-        hold on
-        plot(1:length(b), b, '*r')
-        if j == 1
-            title(titles(i), 'Interpreter', 'latex')
-        end
-        xlabel("Iterations @lambda=" + lambda(j))
-        ylabel('[a_k, b_k]')
-    end
-end
-
-  

Work 1/scripts/bisection/bisection_over_epsilon.m

@@ -1,41 +0,0 @@
-%
-% Keeping lambda (accuracy) fixed, test the iteration needed for different
-% epsilon values.
-%
-
-
-% Clear workspace and load the functions and interval
-clear
-addpath('..');
-GivenEnv;
-
-% * lambda = 0.01
-% * epsilon: e < lambda/2 = 0.005
-% * de: A small step away from zero and lambda/2
-%       de = 0.0001
-% * N: 50 points
-
-N = 50;
-lambda = 0.01;
-de = 0.0001;
-epsilon = linspace(de, (lambda/2)-de, N);
-k = zeros(1,N); % preallocate k
-
-
-%
-% * Call the bisection method for each epsilon value for each function and
-%   keep the number of iterations needed.
-% * Plot the iterations k(epsilon) for each function
-%
-for i = 1:length(funs)
-    for j = 1:N
-        [a, b, k(j)] = bisection(funs{i}, a_0, b_0, epsilon(j), lambda);
-    end
-    subplot(1, length(funs), i)
-    plot(epsilon, k, '-b', 'LineWidth', 1.0)
-    title(titles(i), 'Interpreter', 'latex')
-    xlabel('epsilon')
-    ylabel('Iterations')
-end
-
-    

Work 1/scripts/bisection/bisection_over_lambda.m

@@ -1,44 +0,0 @@
-%
-% Keeping epsilon fixed, test the iteration needed for different lambda
-% values.
-%
-
-
-% Clear workspace and load the functions and interval
-clear
-addpath('..');
-GivenEnv;
-
-% * epsilon: e = 0.001
-% * lambda:  l > 2e = 0.001
-% * dl: A small step away from 2e
-%       dl = 0.0001
-% * lambda_max: 0.1
-% * N: 50 points
-
-N = 50;
-epsilon = 0.001;
-dl = 0.0001;
-lambda_max= 0.1;
-lambda = linspace(2*epsilon + dl, lambda_max, N);
-k = zeros(1, N); % preallocate k
-
-
-%
-% * Call the bisection method for each lambda value for each function and
-%   keep the number of iterations needed.
-% * Plot the iterations k(lambda) for each function
-%
-
-for i = 1:length(funs)
-    for j = 1:N
-        [a, b, k(j)] = bisection(funs{i}, a_0, b_0, epsilon, lambda(j));
-    end
-    subplot(1, length(funs), i)
-    plot(lambda, k, '-b', 'LineWidth', 1.0)
-    title(titles(i), 'Interpreter', 'latex')
-    xlabel('lambda')
-    ylabel('Iterations')
-end
-
-  

Work 1/scripts/bisection_over_epsilon.m

@@ -0,0 +1,62 @@
+%
+% Keeping lambda (accuracy) fixed, test the iteration needed for different
+% epsilon values.
+%
+
+
+% Load the functions and interval
+GivenEnv;
+
+fig_dir = 'figures';
+if ~exist(fig_dir, 'dir')
+    mkdir(fig_dir);
+end
+
+% Setup
+% ========================
+
+% lambda = 0.01
+% epsilon: e < lambda/2 = 0.005
+% de: A small step away from zero and lambda/2
+%     de = 0.0001
+% N: 50 points (50 epsilon values)
+
+N = 50;
+lambda = 0.01;
+de = 0.0001;
+epsilon = linspace(de, (lambda/2)-de, N);
+k = zeros(1,N); % preallocate k
+
+
+%
+% Call the min_bisection method for each epsilon value for each
+% function and keep the number of iterations needed.
+% Then plot and save the # of iterations k(epsilon) for each function.
+%
+
+figure('Name', 'iterations_over_epsilon_min_bisection', 'NumberTitle', 'off');
+set(gcf, 'Position', [100, 100, 1280, 600]); % Set the figure size to HD
+
+for i = 1:length(funs)
+    for j = 1:N
+        [a, b, k(j)] = min_bisection(funs(i), a_0, b_0, epsilon(j), lambda);
+    end
+    fprintf('%20s(%34s ):  [a, b]= [%f, %f], iterations(min, max)= (%d, %d)\n', ...
+            "min_bisection", char(funs(i)), a(end), b(end), k(1), k(N) );
+    subplot(1, length(funs), i)
+    plot(epsilon, k, '-b', 'LineWidth', 1.0)
+    title(titles(i), 'Interpreter', 'latex')
+    xlabel('epsilon')
+    ylabel('Iterations')
+end
+
+%
+% Print and save the figures
+%
+%fig_epsc = fullfile(fig_dir, "iter_over_epsilon_min_bisection" + ".epsc");
+fig_png  = fullfile(fig_dir, "iter_over_epsilon_min_bisection" + ".png");
+
+%print(gcf, fig_epsc, '-depsc', '-r300');
+print(gcf, fig_png, '-dpng', '-r300');
+
+    

Work 1/scripts/golden_section/golden_section_interval.m

@@ -1,45 +0,0 @@
-%
-% Plot the [a,b] interval over the iterations for different lambda
-% values (min, mid, max))
-%
-
-
-% Clear workspace and load the functions and interval
-clear
-addpath('..');
-GivenEnv;
-
-% * lambda_min: 0.0001
-% * lambda_max: 0.1
-% * N: 3 lambda values
-
-N = 3;
-lambda_min = 0.0001;
-lambda_max = 0.1;
-lambda = linspace(lambda_min, lambda_max, N);
-k = zeros(1, N); % preallocate k
-
-
-%
-% * Call the golden_sector method for each lambda value for each function and
-%   keep the number of iterations needed.
-% * Plot the [a,b] interval over iterations for each lambda for each function 
-%
-
-for i = 1:length(funs)
-    figure;
-    for j = 1:N
-        [a, b, k(j)] = golden_section(funs{i}, a_0, b_0, lambda(j));
-        subplot(length(funs), 1, j)
-        plot(1:length(a), a, 'ob')
-        hold on
-        plot(1:length(b), b, '*r')
-        if j == 1
-            title(titles(i), 'Interpreter', 'latex')
-        end
-        xlabel("Iterations @lambda=" + lambda(j))
-        ylabel('[a_k, b_k]')
-    end
-end
-
-  

Work 1/scripts/golden_section/golden_section_over_lambda.m

@@ -1,39 +0,0 @@
-%
-% Test the iteration needed for different lambda values.
-%
-
-
-% Clear workspace and load the functions and interval
-clear
-addpath('..');
-GivenEnv;
-
-% * lambda_min: 0.0001
-% * lambda_max: 0.1
-% * N: 50 points
-
-N = 50;
-lambda_min = 0.0001;
-lambda_max = 0.1;
-lambda = linspace(lambda_min, lambda_max, N);
-k = zeros(1, N); % preallocate k
-
-
-%
-% * Call the golden_sector method for each lambda value for each function and
-%   keep the number of iterations needed.
-% * Plot the iterations k(lambda) for each function
-%
-
-for i = 1:length(funs)
-    for j = 1:N
-        [a, b, k(j)] = golden_section(funs{i}, a_0, b_0, lambda(j));
-    end
-    subplot(1, length(funs), i)
-    plot(lambda, k, '-b', 'LineWidth', 1.0)
-    title(titles(i), 'Interpreter', 'latex')
-    xlabel('lambda')
-    ylabel('Iterations')
-end
-
-  

Work 1/scripts/interval_over_iterations.m

@@ -0,0 +1,82 @@
+function [] = interval_over_iterations(method)
+% Plot the [a,b] interval over the iterations for different lambda
+% values (min, mid, max))
+%
+% method: the minimum calculation method
+%       * bisections
+%       * golden_section
+%       * fibonacci
+%       * bisection_der
+
+
+% Load the functions and interval
+GivenEnv;
+
+fig_dir = 'figures';
+if ~exist(fig_dir, 'dir')
+    mkdir(fig_dir);
+end
+
+% Setup
+% ========================
+%
+% We need to test against the same lambda values for all the methods in
+% order to compare them. And since epsilon (which is related to lambda)
+% was given for bisection method, we base our calculations to that.
+%
+%
+% epsilon: e = 0.001
+% lambda:  l > 2e =>
+% lambda_min: 0.0021
+% lambda_max: 0.1
+% N: 3 points (3 lambda values min-mid-max)
+
+N = 3;
+epsilon = 0.001;
+lambda_min = 0.0021;
+lambda_max = 0.1;
+lambda = linspace(lambda_min, lambda_max, N);
+k = zeros(1, N); % preallocate k
+
+
+%
+% Call the minimum calculation method for each lambda value for each
+% function and keep the number of iterations needed.
+% Then Plot the [a,b] interval over iterations for each lambda for each
+% function.
+%
+% note: In order to use the same method call for all methods, we force a 
+%       common interface for minimum method functions. Thus some arguments
+%       will not be needed for some methods (epsilon is not needed for
+%       bisection _der for example).
+%
+
+disp(" ");
+for i = 1:length(funs)
+    figure('Name', "interval_over_iterations_" + char(method) + "_fun" + i, 'NumberTitle', 'off');
+    set(gcf, 'Position', [100, 100, 1280, 720]); % Set the figure size to HD
+    for j = 1:N
+        [a, b, k(j)] = method(funs(i), a_0, b_0, epsilon, lambda(j));
+        
+        fprintf('%20s(%34s ):  [a, b]= [%f, %f], @lambda=%f, iterations= %d\n', ...
+            char(method), char(funs(i)), a(end), b(end), lambda(j), k(j) );
+        
+        subplot(length(funs), 1, j)
+        plot(1:length(a), a, 'ob')
+        hold on
+        plot(1:length(b), b, '*r')
+        if j == 1
+            title(titles(i), 'Interpreter', 'latex')
+        end
+        xlabel("Iterations @lambda=" + lambda(j))
+        ylabel('[a_k, b_k]')
+    end
+    
+    % Print and save the figure
+    %fig_epsc = fullfile(fig_dir, "interval_over_iterations_" + char(method) + "_fun" + i + ".epsc");
+    fig_png  = fullfile(fig_dir, "interval_over_iterations_" + char(method) + "_fun" + i + ".png");
+
+    %print(gcf, fig_epsc, '-depsc', '-r300');
+    print(gcf, fig_png, '-dpng', '-r300');
+end
+

Work 1/scripts/iterations_over_lambda.m

@@ -0,0 +1,83 @@
+function [] = iterations_over_lambda(method)
+% Plot iteration needed for different lambda values.
+%
+%
+% method: the minimum calculation method
+%       * bisections
+%       * golden_section
+%       * fibonacci
+%       * bisection_der
+
+
+% Load the functions and interval
+GivenEnv;
+
+fig_dir = 'figures';
+if ~exist(fig_dir, 'dir')
+    mkdir(fig_dir);
+end
+
+% Setup
+% ========================
+%
+% We need to test against the same lambda values for all the methods in
+% order to compare them. And since epsilon (which is related to lambda)
+% was given for bisection method, we base our calculations to that.
+%
+%
+% epsilon: e = 0.001
+% lambda:  l > 2e =>
+% lambda_min: 0.0021
+% lambda_max: 0.1
+% N: 50 points (50 lambda values)
+
+N = 50;
+epsilon = 0.001;
+lambda_min = 0.0021;
+lambda_max = 0.1;
+lambda = linspace(lambda_min, lambda_max, N);
+k = zeros(1, N); % preallocate k
+
+
+%
+% Call the minimum calculation method for each lambda value for each
+% function and keep the number of iterations needed.
+% Then plot and save the # of iterations k(lambda) for each function.
+%
+% note: In order to use the same method call for all methods, we force a 
+%       common interface for minimum method functions. Thus some arguments
+%       will not be needed for some methods (epsilon is not needed for
+%       bisection _der for example).
+%
+
+figure('Name', "iterations_over_lambda_" + char(method), 'NumberTitle', 'off');
+set(gcf, 'Position', [100, 100, 1280, 600]); % Set the figure size to HD
+
+disp(" ");
+for i = 1:length(funs)
+    for j = N:-1:1
+        [a, b, k(j)] = method(funs(i), a_0, b_0, epsilon, lambda(j));
+    end
+
+    fprintf('%20s(%34s ):  [a, b]= [%f, %f], iterations(min, max)= (%d, %d)\n', ...
+            char(method), char(funs(i)), a(end), b(end), k(N), k(1) );
+    subplot(1, length(funs), i)
+    plot(lambda, k, '-b', 'LineWidth', 1.0)
+    title(titles(i), 'Interpreter', 'latex')
+    xlabel('lambda')
+    ylabel('Iterations')
+end
+
+
+%
+% Print and save the figures
+%
+%fig_epsc = fullfile(fig_dir, "iter_over_lambda_" + char(method) + ".epsc");
+fig_png  = fullfile(fig_dir, "iter_over_lambda_" + char(method) + ".png");
+
+%print(gcf, fig_epsc, '-depsc', '-r300');
+print(gcf, fig_png, '-dpng', '-r300');
+
+
+
+  

Work 1/scripts/min_bisection.m

@@ -1,33 +1,34 @@
-function [a, b, k] = bisection(fun, alpha, beta, epsilon, lambda)
-%
-% Detailed explanation goes here
-%
-%
-
-% Error checking
-if 2*epsilon >= lambda || lambda <= 0
-    error ('Convergence criteria not met')
-end
-
-% Init output vectors
-a = alpha;
-b = beta;
-
-k=1;
-
-while b(k) - a(k) > lambda
-    % bisect [a,b]
-    mid = (a(k) + b(k)) / 2;
-    x_1 = mid - epsilon;
-    x_2 = mid + epsilon;
-    
-    % set new search interval
-    k = k + 1;
-    if fun(x_1) < fun(x_2)
-        a(k) = a(k-1);
-        b(k) = x_2;
-    else
-        a(k) = x_1;
-        b(k) = b(k-1);
-    end
-end
+function [a, b, k] = min_bisection(fun_expression, alpha, beta, epsilon, lambda)
+%
+% Detailed explanation goes here
+%
+%
+
+% Error checking
+if 2*epsilon >= lambda || lambda <= 0
+    error ('Convergence criteria not met')
+end
+
+% Init
+a = alpha;
+b = beta;
+fun = matlabFunction(fun_expression);
+
+k=1;
+
+while b(k) - a(k) > lambda
+    % bisect [a,b]
+    mid = (a(k) + b(k)) / 2;
+    x_1 = mid - epsilon;
+    x_2 = mid + epsilon;
+    
+    % set new search interval
+    k = k + 1;
+    if fun(x_1) < fun(x_2)
+        a(k) = a(k-1);
+        b(k) = x_2;
+    else
+        a(k) = x_1;
+        b(k) = b(k-1);
+    end
+end

Work 1/scripts/min_bisection_der.m

@@ -0,0 +1,36 @@
+function [a, b, k] = min_bisection_der(fun_expression, alpha, beta, epsilon, lambda)
+%
+% Detailed explanation goes here
+%
+%
+
+% Error checking
+if lambda <= 0
+    error ('Convergence criteria not met')
+end
+
+% Init output vectors
+a = alpha;
+b = beta;
+dfun = matlabFunction(diff(fun_expression));
+
+k=1;
+while b(k) - a(k) > lambda
+    % bisect [a,b]
+    x_mid = (a(k) + b(k)) / 2;
+    
+    % set new search interval
+    k = k + 1;
+    df = dfun(x_mid);
+    if df < 0
+        a(k) = x_mid;
+        b(k) = b(k-1);
+    elseif df > 0
+        a(k) = a(k-1);
+        b(k) = x_mid;
+    else % df == 0
+        a(k) = x_mid;
+        b(k) = x_mid;
+        break;
+    end
+end

Work 1/scripts/min_fibonacci.m

@@ -0,0 +1,54 @@
+function [a, b, N] = min_fibonacci(fun_expression, alpha, beta, epsilon, lambda)
+%
+
+% Use Binet's formula instead of matlab's recursive fibonacci
+% implementation
+fib = @(n) ( ((1 + sqrt(5))^n - (1 - sqrt(5))^n) / (2^n * sqrt(5)) );
+
+% Error checking
+if lambda <= 0 || epsilon <= 0
+    error ('Convergence criteria not met')
+end
+
+% Init variables
+a = alpha;
+b = beta;
+fun = matlabFunction(fun_expression);
+
+% calculate number of iterations
+N=0;
+while fibonacci(N) < (b(1) - a(1)) / lambda
+    N = N + 1;
+end
+
+
+% calculate x1, x2 of the first iteration, since the following iteration
+% will not require to calculate both
+x_1 = a(1) + (fib(N-2) / fib(N)) * (b(1) - a(1));
+x_2 = a(1) + (fib(N-1) / fib(N)) * (b(1) - a(1));
+
+% All but the last calculation
+for k = 1:N-2
+    % set new search interval
+    if fun(x_1) < fun(x_2)
+        a(k+1) = a(k);
+        b(k+1) = x_2;
+        x_2 = x_1;
+        x_1 = a(k+1) + (fib(N-k-2) / fib(N-k)) * (b(k+1) - a(k+1));
+    else
+        a(k+1) = x_1;
+        b(k+1) = b(k);
+        x_1 = x_2;
+        x_2 = a(k+1) + (fib(N-k-1) / fib(N-k)) * (b(k+1) - a(k+1));
+    end
+end
+
+% Last calculation
+x_2 = x_1 + epsilon;
+if fun(x_1) < fun(x_2)
+    a(N) = a(N-1);
+    b(N) = x_1;
+else
+    a(N) = x_1;
+    b(N) = b(N-1);
+end

Work 1/scripts/min_golden_section.m

@@ -1,37 +1,36 @@
-function [a, b, k] = golden_section(fun, alpha, beta, lambda)
-%
-
-
-% Error checking
-if lambda <= 0
-    error ('Convergence criteria not met')
-end
-
-% Init variables
-gamma = 0.618;
-a = alpha;
-b = beta;
-
-
-% calculate x1, x2 of the first iteration, since the following iteration
-% will not require to calculate both
-k=1;
-x_1 = a(k) + (1 - gamma)*(b(k) - a(k));
-x_2 = a(k) + gamma*(b(k) - a(k));
-
-while b(k) - a(k) > lambda    
-    % set new search interval
-    k = k + 1;
-    if fun(x_1) < fun(x_2)
-        a(k) = a(k-1);
-        b(k) = x_2;
-        x_2 = x_1;
-        x_1 = a(k) + (1 - gamma)*(b(k) - a(k));
-    else
-        a(k) = x_1;
-        b(k) = b(k-1);
-        x_1 = x_2;
-        x_2 = a(k) + gamma*(b(k) - a(k));
-    end
-end
-
+function [a, b, k] = min_golden_section(fun_expression, alpha, beta, epsilon, lambda)
+%
+
+
+% Error checking
+if lambda <= 0
+    error ('Convergence criteria not met')
+end
+
+% Init variables
+gamma = 0.618;
+a = alpha;
+b = beta;
+fun = matlabFunction(fun_expression);
+
+% calculate x1, x2 of the first iteration, since the following iteration
+% will not require to calculate both
+k=1;
+x_1 = a(k) + (1 - gamma)*(b(k) - a(k));
+x_2 = a(k) + gamma*(b(k) - a(k));
+
+while b(k) - a(k) > lambda    
+    % set new search interval
+    k = k + 1;
+    if fun(x_1) < fun(x_2)
+        a(k) = a(k-1);
+        b(k) = x_2;
+        x_2 = x_1;
+        x_1 = a(k) + (1 - gamma)*(b(k) - a(k));
+    else
+        a(k) = x_1;
+        b(k) = b(k-1);
+        x_1 = x_2;
+        x_2 = a(k) + gamma*(b(k) - a(k));
+    end
+end