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%% Labwork 3: Solving Linear Systems
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%% V. Gauss Elimination
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%% Exercise 1: Solve the system of linear equations:
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%%
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%
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% 3x1 + 2x2 = 18
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% -x1 + 2x2 = 2
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%
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% a) Graphical Method
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% (As the graph is provided in the lab, this part focuses on creating the plot in MATLAB)
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% Define the equations as functions to plot
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eq1 = @(x1) (18 - 3*x1) / 2;
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eq2 = @(x1) (2 + x1) / 2;
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% Create the x1 vector for plotting
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x1_vals = 0:0.1:6; % Range of x1 values
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% Calculate the corresponding x2 values
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x2_eq1 = eq1(x1_vals);
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x2_eq2 = eq2(x1_vals);
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% Plot the equations
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figure; % Create a new figure
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plot(x1_vals, x2_eq1, 'b-', 'LineWidth', 2); % Plot the first equation (blue line)
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hold on; % Hold the current plot to add the second equation
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plot(x1_vals, x2_eq2, 'r-', 'LineWidth', 2); % Plot the second equation (red line)
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% Add labels and title
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xlabel('x1');
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ylabel('x2');
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title('Graphical Solution of Linear Equations');
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legend('3x1 + 2x2 = 18', '-x1 + 2x2 = 2');
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axis([0 6 0 9]); % Set the axis limits to match the lab's graph
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grid on; % Add grid lines
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hold off; % Release the plot hold
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% Mark the solution point (x1=4, x2=3) as given in the lab
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plot(4, 3, 'ko', 'MarkerSize', 8, 'MarkerFaceColor', 'k'); % Black circle for the solution
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text(4.2, 3.2, 'Solution: (4, 3)'); % Label the solution
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% b) Naive Gauss Elimination Method
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% (Code provided in the lab)
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function x = GaussNaive(A, b)
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% GaussNaive: naive Gauss elimination
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% x = GaussNaive(A, b): Gauss elimination without pivoting.
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% input:
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% A = coefficient matrix
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% b = right hand side vector
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% output:
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% x = solution vector
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[m, n] = size(A);
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if m ~= n, error('Matrix A must be square'); end
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nb = n + 1;
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Aug = [A b]; % Augmented matrix
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% forward elimination
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for k = 1:n-1
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for i = k+1:n
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factor = Aug(i, k) / Aug(k, k);
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Aug(i, k:nb) = Aug(i, k:nb) - factor * Aug(k, k:nb);
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end
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end
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% back substitution
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x = zeros(n, 1);
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x(n) = Aug(n, nb) / Aug(n, n);
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for i = n-1:-1:1
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x(i) = (Aug(i, nb) - Aug(i, i+1:n) * x(i+1:n)) / Aug(i, i);
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end
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end
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% Define the coefficient matrix A and the right-hand side vector b
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A = [3 2; -1 2];
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b = [18; 2];
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% Solve using the GaussNaive function
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x_naive = GaussNaive(A, b)
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%% Exercise 2: Solve the system of linear equations:
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%%
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%
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% 2x + y - z = 8
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% -3x - y + 2z = -11
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% -2x + y + 2z = -3
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%
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% a) Using inv or mldivide in Matlab
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% Define the coefficient matrix A and the right-hand side vector b
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A = [2 1 -1; -3 -1 2; -2 1 2];
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b = [8; -11; -3];
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% Solve using inv
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x_inv = inv(A) * b
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% Solve using mldivide (backslash operator \) - RECOMMENDED
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x_mldivide = A \ b
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% b) Use the function GaussPivot.m
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% (Code provided in the lab)
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function x = GaussPivot(A, b)
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% GaussPivot: Gauss elimination pivoting
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% x = GaussPivot(A, b): Gauss elimination with pivoting.
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% input:
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% A = coefficient matrix
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% b = right hand side vector
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% output:
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% x = solution vector
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[m, n] = size(A);
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if m ~= n, error('Matrix A must be square'); end
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nb = n + 1;
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Aug = [A b];
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% forward elimination
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for k = 1:n-1
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% partial pivoting
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[big, i] = max(abs(Aug(k:n, k)));
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ipr = i + k - 1;
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if ipr ~= k
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Aug([k, ipr], :) = Aug([ipr, k], :);
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end
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for i = k+1:n
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factor = Aug(i, k) / Aug(k, k);
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Aug(i, k:nb) = Aug(i, k:nb) - factor * Aug(k, k:nb);
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end
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end
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% back substitution
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x = zeros(n, 1);
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x(n) = Aug(n, nb) / Aug(n, n);
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for i = n-1:-1:1
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x(i) = (Aug(i, nb) - Aug(i, i+1:n) * x(i+1:n)) / Aug(i, i);
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end
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end
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% Solve using GaussPivot
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x_pivot = GaussPivot(A, b)
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% c) Modify the code above to Gauss Elimination without Pivoting
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function x = GaussNaive_modified(A, b)
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% GaussNaive_modified: Gauss elimination without pivoting (modified from GaussPivot).
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% input:
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% A = coefficient matrix
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% b = right hand side vector
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% output:
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% x = solution vector
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[m, n] = size(A);
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if m ~= n, error('Matrix A must be square'); end
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nb = n + 1;
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Aug = [A b];
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% forward elimination (WITHOUT PIVOTING)
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for k = 1:n-1
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for i = k+1:n
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factor = Aug(i, k) / Aug(k, k);
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Aug(i, k:nb) = Aug(i, k:nb) - factor * Aug(k, k:nb);
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end
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end
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% back substitution
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x = zeros(n, 1);
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x(n) = Aug(n, nb) / Aug(n, n);
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for i = n-1:-1:1
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x(i) = (Aug(i, nb) - Aug(i, i+1:n) * x(i+1:n)) / Aug(i, i);
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end
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end
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% Solve using GaussNaive_modified
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x_naive_mod = GaussNaive_modified(A, b)
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% d) Explain the code
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% (Explanation embedded in the comments within the code above)
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%% VI. LU Decomposition
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%% Exercise 3: LU decomposition
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%%
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%
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% 2x + y - z = 8
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% -3x - y + 2z = -11
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% -2x + y + 2z = -3
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%
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% a) Using function lu in Matlab
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% Define A and b (same as before)
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A = [2 1 -1; -3 -1 2; -2 1 2];
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b = [8; -11; -3];
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% Perform LU decomposition
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[L, U, P] = lu(A);
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% Solve Ly = Pb
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y = L \ (P * b);
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% Solve Ux = y
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x_lu_matlab = U \ y
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% b) Use the function LU_pivot.m code
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% (Code provided in the lab)
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function [L, U, P] = LU_pivot(A)
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% LU_pivot: LU decomposition with pivoting
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[n, ~] = size(A);
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L = eye(n);
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P = eye(n);
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U = A;
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for j = 1:n
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[pivot, m] = max(abs(U(j:n, j)));
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m = m + j - 1;
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if m ~= j
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U([m, j], :) = U([j, m], :); % interchange rows m and j in U
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P([m, j], :) = P([j, m], :); % interchange rows m and j in P
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if j >= 2
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L([m, j], 1:j-1) = L([j, m], 1:j-1); % interchange rows m and j in columns 1:j-1 of L
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end
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end
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for i = j+1:n
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L(i, j) = U(i, j) / U(j, j);
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U(i, :) = U(i, :) - L(i, j) * U(j, :);
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end
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end
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end
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% Perform LU decomposition using LU_pivot
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[L_pivot, U_pivot, P_pivot] = LU_pivot(A);
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% Solve Ly = Pb
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y_pivot = L_pivot \ (P_pivot * b);
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% Solve Ux = y
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x_lu_pivot = U_pivot \ y_pivot
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% c) Change the code of LU with pivoting in LU_pivot.m to LU non-pivoting code
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function [L, U] = LU_nopivot(A)
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% LU_nopivot: LU decomposition without pivoting
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[n, ~] = size(A);
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L = eye(n);
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U = A;
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for j = 1:n
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for i = j+1:n
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L(i, j) = U(i, j) / U(j, j);
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U(i, :) = U(i, :) - L(i, j) * U(j, :);
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end
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end
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end
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% Perform LU decomposition without pivoting
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[L_nopivot, U_nopivot] = LU_nopivot(A);
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% Solve Ly = b
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y_nopivot = L_nopivot \ b; % No pivoting, so use original b
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% Solve Ux = y
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x_lu_nopivot = U_nopivot \ y_nopivot
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% d) Comparison and explain the difference between LU_pivot.m and mylup.m code:
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% (Explanation provided as comments below)
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function [L, U, P] = mylup(A)
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% MYLUP LU DECOMPOSITION WITH PIVOTING
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[n, m] = size(A);
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C = zeros(n, n+1);
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% initialize
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C(1:n, 1:n) = A;
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C(1:n, n+1) = (1:n)';
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% main algorithm
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for j = 1:n-1
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if (sum(C(1:n, j)) == 0)
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error('mylup:input', 'The solution is not determined.');
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end
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[~, i] = max(abs(C(1:n, j)));
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if (i ~= j)
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tt = C(i, 1:n+1);
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C(i, 1:n+1) = C(j, 1:n+1);
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C(j, 1:n+1) = tt;
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end
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for i = j+1:n
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t = C(i, j) / C(j, j);
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C(i, j) = t;
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C(i, j+1:n) = C(i, j+1:n) - t * C(j, j+1:n);
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end
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end
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% refine the results
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PP = C(1:n, n+1);
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Z = C(1:n, 1:n);
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L = tril(C, -1);
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U = triu(C);
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P = zeros(n, n);
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for i = 1:n
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L(i, i) = 1;
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P(i, PP(i)) = 1;
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end
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end
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% Differences between LU_pivot.m and mylup.m:
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% - LU_pivot.m is more straightforward, directly manipulating U, L, and P.
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% - mylup.m uses an augmented matrix C to keep track of row permutations,
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% which can be less intuitive but might be optimized for certain implementations.
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% - Both achieve the same goal: LU decomposition with partial pivoting.
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% - mylup.m includes a check for a column of zeros, raising an error if found.
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%% VII. Common
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%% Exercise 1: Create a vector x that:
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%%
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%
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% - is a column vector
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% - has all integer value elements from 1 to 20 respectively from the top.
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%
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% Create the column vector x
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x = (1:20)'; % Or x = [1:20]';
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% a) Create a vector y that has the same length and size as x's but all entries is 1.
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% Create the vector y
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y = ones(size(x));
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% b) Calculate vector z that is the sum of x and y
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% Calculate the vector z
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z = x + y;
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% c) Calculate the dot product between x and y
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% Calculate the dot product
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dot_product_xy = dot(x, y); % Or sum(x .* y);
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% d) Increment the first, the 5th, the 16th and the last element of z by 3
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% Increment the specified elements
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z([1, 5, 16, end]) = z([1, 5, 16, end]) + 3;
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% e) After step d, multiply those elements with 6.
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% Multiply the incremented elements by 6
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z([1, 5, 16, end]) = z([1, 5, 16, end]) * 6;
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%% Exercise 2: Create a matrix A that:
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%%
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%
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% - has size of 3*3
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% - all the entries is 1
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%
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% Create the matrix A
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A = ones(3, 3);
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% a) Create matrix b that is transpose of A
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% Create the matrix B
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B = A'; % Or B = transpose(A); Since A is all ones, A' == A, but this is the general way
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% b) Compute sum of A and B
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% Compute the sum of A and B
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sum_AB = A + B;
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% c) Compute A*B and B*A
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% Compute A * B and B * A
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AB = A * B;
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BA = B * A;
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% d) Increment all the elements of A by 2.
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% Increment all elements of A
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A = A + 2;
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%% Exercise 3: Give a MATLAB expression that multiplies two vectors to obtain:
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% (a) the matrix
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% 1 2 3 4 5
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% 1 2 3 4 5
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% 1 2 3 4 5
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% Create the vectors to produce the matrix
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vec1a = ones(3, 1);
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vec2a = 1:5;
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% Matrix (a)
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matrix_a = vec1a * vec2a;
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% (b) the matrix
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% 0 0 0
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% 1 1 1
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% 2 2 2
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% 3 3 3
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% 4 4 4
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% Create the vectors to produce the matrix
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vec1b = (0:4)';
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vec2b = ones(1, 3);
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% Matrix (b)
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matrix_b = vec1b * vec2b;
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%% Exercise 4: Create filled 2-D patches
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% a) Create matrices x and y.
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% Then plot the filled region in red as the figure below using fill in Matlab.
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% Define the x and y coordinates for the patches
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x_red1 = [0 4 0];
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y_red1 = [4 2 2];
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x_red2 = [2 4 2];
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y_red2 = [0 0 1];
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% Create the plot
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figure;
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fill(x_red1, y_red1, 'c');
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hold on;
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fill(x_red2, y_red2, 'y');
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hold off;
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% Add labels and title
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xlabel('x');
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ylabel('y');
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title('Filled 2D Patches');
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Reference in new issue
Block a user