real

lab2/lab2.m

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+%% Labwork 2 Solutions
+
+%%% III. Polynomials
+
+%% Exercise 1: Arithmetic Operations
+
+% a) 12^2 - 13^2
+a = 12^2 - 13^2
+
+% b) 24 * 5 / 1.5
+b = 24 * 5 / 1.5
+
+% c) 7! (7 factorial)
+c = factorial(7)
+
+% d) 4^2 + 3^3 - 124
+d = 4^2 + 3^3 - 124
+
+% e) 6/5 * (1/2 * 1/4 - 1)
+e = (6/5) * ((1/2) * (1/4) - 1)
+
+%% Exercise 2: fzero
+
+% Given f(x) = x^2 - 9
+
+% a) Find the positive root using fzero
+f = @(x) x^2 - 9;  % Define the function
+positive_root = fzero(f, 2)  % Initial guess of 2 (can be any positive number)
+
+% b) Initial guess of zero
+root_zero_guess = fzero(f, 0) % It might give a warning, and the result can vary slightly due to how fzero works
+
+% c) Display all iterations
+options = optimset('Display', 'iter');
+[root_iterations, fval_iterations, exitflag_iterations, output_iterations] = fzero(f, 2, options)
+
+% d) Set tolerance: tolx = 10^-3
+options_tol = optimset('TolX', 10^-3);
+root_tolerance = fzero(f, 2, options_tol)
+
+% e) Verify with roots
+coefficients = [1 0 -9]; % Coefficients of the quadratic equation
+all_roots = roots(coefficients)
+
+
+%% Exercise 3: plot
+
+% Given v(t) = sqrt(gm/cd) * tanh(sqrt(g*cd/m) * t)
+% g = 9.81, m = 68.1, cd = 0.25
+
+g = 9.81;
+m = 68.1;
+cd = 0.25;
+
+t = 0:2:20; % Time vector from 0 to 20 seconds in steps of 2
+v = sqrt((g*m)/cd) .* tanh(sqrt((g*cd)/m) .* t);  % Calculate velocity
+
+% a) Plot (t, v)
+figure; % Create a new figure
+plot(t, v, 'b-o'); % Plot v against t ('b-o' for blue line with circles)
+
+% b) Customize the graph
+title('Plot of v versus t');
+xlabel('Values of t');
+ylabel('Values of v');
+grid on;
+
+
+%%% IV. System of Non-linear Equations
+
+%% Exercise 1: Newton-Raphson
+
+% a) Use ezplot to derive good guesses
+%   (This part is visual, so I'll show the code to generate the plot for you to estimate)
+figure;
+ezplot(@(x1, x2) x1^2 + x1*x2 - 10, [-6 6]); % Plot the first equation
+hold on;
+ezplot(@(x1, x2) x2 + 3*x1*x2^2 - 57, [-6 6]); % Plot the second equation
+hold off;
+title('Plot of equations for initial guesses');
+xlabel('x1');
+ylabel('x2');
+grid on;
+%   Inspect the plot to get approximate intersection points (guesses).
+
+% b) Explain the given code
+%   (Explanation embedded as comments in the code below)
+
+clc  % Clear command window
+clear all  % Clear variables
+close all  % Close all figures
+
+% Define the initial guess as a column vector
+x = [1; 1];  
+
+N = 50;  % Maximum number of iterations
+
+for k = 1:N  % Loop for iterations
+    % Define the system of equations
+    F = [x(1)^2 + x(1)*x(2) - 10; 
+         x(2) + 3*x(1)*x(2)^2 - 57];  
+    
+    % Calculate the Jacobian matrix
+    A = [2*x(1) + x(2), x(1); 
+         3*x(2)^2, 1 + 6*x(1)*x(2)];  
+    
+    % Solve for the change in x (dx)
+    dx = A \ F;  
+    
+    % Update x
+    x = x - dx;  
+    
+    % Increment the iteration counter (though it's redundant here, as the loop does it)
+    %k = k + 1; 
+end
+
+k  % Display the final iteration number
+x'  % Display the solution as a row vector
+F  % Display the function values at the solution (should be close to zero)
+
+
+% c) Create root2d.m and use fsolve
+%
+%   Create a new MATLAB function file named root2d.m with the following content:
+%   ------------------------------------------------------------------
+%   function F = root2d(x)
+%       F(1) = x(1)^2 + x(1)*x(2) - 10;
+%       F(2) = x(2) + 3*x(1)*x(2)^2 - 57;
+%   end
+%   ------------------------------------------------------------------
+%
+%   Then, in a separate script:
+
+% Call fsolve with the function and an initial guess
+f_system = @root2d;
+x0 = [0; 0];  % Initial guess
+[solution_fsolve, fval_fsolve] = fsolve(f_system, x0)
+
+%   Explanation:
+%   - root2d.m defines the system of equations as a function that takes a vector x 
+%     and returns a vector F representing the equations.
+%   - fsolve finds the solution to the system of nonlinear equations.  It takes the
+%     function handle (@root2d) and an initial guess (x0) as input.
+
+% d) Use newtmult.m
+%   (Assuming newtmult.m is provided as in the lab, place it in the same directory)
+
+% Define the function (same as in root2d.m)
+func = @(x) [x(1)^2 + x(1)*x(2) - 10; x(2) + 3*x(1)*x(2)^2 - 57];
+x0 = [0; 0];  % Initial guess
+es = 0.0001; % Desired relative error
+maxit = 50;  % Maximum iterations
+
+[x_newtmult, f_newtmult, ea_newtmult, iter_newtmult] = newtmult(func, x0, es, maxit)
+
+%   Explanation of newtmult.m (from the lab):
+%   - It implements the Newton-Raphson method for systems of nonlinear equations.
+%   - Inputs: function handle (func), initial guess (x0), desired relative error (es), 
+%     maximum iterations (maxit), and any additional parameters for the function.
+%   - Outputs: solution (x), function values at the solution (f), approximate relative error (ea), 
+%     and number of iterations (iter).
+%   - The code iteratively calculates the Jacobian, solves a linear system, and updates the 
+%     solution until convergence or the maximum number of iterations is reached.
+
+
+%%% V. Common
+
+%% Exercise 1: Calculate the value of x
+
+% a) log(x) = 770  (Assuming base 10 logarithm)
+x_a = 10^770
+
+% b) e^x = 15
+x_b = log(15)
+
+% c) sin(x) = 0.12 (in radians)
+x_c = asin(0.12)
+
+% d) sin(x + pi/5) = 0.25 (in radians)
+x_d = asin(0.25) - pi/5
+
+% e) tan(x + 2*pi/3) = 3 (in radians)
+x_e = atan(3) - 2*pi/3
+
+%% Exercise 2: Create a vector x
+
+% Create the row vector
+x_vector = [2 3 1 5 11]
+
+% a) Find length and size
+length_x = length(x_vector)
+size_x = size(x_vector)
+%   - length() returns the number of elements in the longest dimension.
+%   - size() returns a row vector with the number of rows and columns.
+
+% b) Find the transpose
+x_transpose = x_vector'
+
+% c) Calculate the product of x and its transpose
+product_x_xt = x_vector * x_transpose
+
+
+%% Exercise 3: Build the matrix B
+
+% Build the matrix B
+B = [1:7; 9:-2:-3; 4.^(1:7)]