All algorithms from 1 to 5 work just fine!
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@ -14,9 +14,11 @@ if det(U) == 0
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error('Matrix is not nonsingular!')
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end
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b(n) = b(n)/U(n, n);
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% b(n, :) so that matrices are also accepted
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b(n, :) = b(n, :)/U(n, n);
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for i = n-1:-1:1
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b(i) = (b(i) - U(i, i+1 : n)*b(i+1 : n))/U(i, i);
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b(i, :) = (b(i, :) - U(i, i+1 : n)*b(i+1 : n, :))/U(i, i);
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end
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end
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@ -10,8 +10,8 @@ if length(b) ~= n
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error('Vector b has wrong length!')
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end
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b(1) = b(1)/L(1,1);
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b(1, :) = b(1, :)/L(1,1);
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for i = 2:n
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b(i) = (b(i) - L(i, 1:i-1)*b(1:i-1))/L(i, i);
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b(i, :) = (b(i, :) - L(i, 1:i-1)*b(1:i-1, :))/L(i, i);
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end
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@ -1,31 +1,25 @@
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% Algorithm 5: Gauss-Jordan Elimination
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% Input A is an augmented matrix
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function A = gauss_jordan_elimination(A)
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[n, m] = size(A);
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if n ~= m
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if n + 1 ~= m
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error('Matrix is not squared!')
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end
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if det(A) == 0
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error('Matrix is not nonsingular!')
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end
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% if det(A) == 0
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% error('Matrix is not nonsingular!')
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% end
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A
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for k = 1 : n-1
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i = k:n;
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j = k:n;
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A(i, j);
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maximum = max(abs(A(i, j)), [], 'all');
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max_idx = find(abs(A==maximum));
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[mi, lm] = ind2sub(size(A), max_idx(1));
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A([k mi], 1:n) = deal(A([mi k], 1:n));
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A(1:n, [k lm]) = deal(A(1:n, [lm k]));
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p(k) = mi;
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q(k) = lm;
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% Perform Gaussian elimination with the greatest pivot
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if A(k, k) ~= 0
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rows = k+1 : n;
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A(rows, k) = A(rows, k)/A(k, k);
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A(rows, rows) = A(rows, rows) - A(rows, k) * A(k, rows);
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for k = 1 : m-1
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row = A(k, :);
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row = row/row(k);
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A(k, :) = row;
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for l = 1 : m-1
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if l ~= k
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A(l, :) = A(l, :)-(A(l, k))*row;
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end
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end
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end
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@ -1,4 +1,4 @@
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function [U, L] = gaussian_elimination_with_complete_pivoting(A)
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function [P, Q, L, U] = gaussian_elimination_with_complete_pivoting(A)
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[n, m] = size(A);
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if n ~= m
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@ -9,24 +9,25 @@ if det(A) == 0
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error('Matrix is not nonsingular!')
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end
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A
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p = 1:n;
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q = 1:n;
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% for k = 1 : n-1
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for k = 1 : n-1
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i = k:n;
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j = k:n;
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A(i, j)
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A(i, j);
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[max_val, rows_of_max_in_col] = max(abs(A(i, j)));
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[max_val, max_col] = max(max_val);
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max_row = rows_of_max_in_col(max_col);
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% Assing value of mi and lambda in respect to the main A matrix
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[mi, lm] = deal(max_row+k-1, max_col+k-1)
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A([k mi], 1:n) = deal(A([mi k], 1:n))
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A(1:n, [k lm]) = deal(A(1:n, [lm k]))
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p(k) = mi
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q(k) = lm
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% Perform Gaussian elimination with the greatest pivot
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[mi, lm] = deal(max_row+k-1, max_col+k-1);
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A([k mi], 1:n) = deal(A([mi k], 1:n));
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A(1:n, [k lm]) = deal(A(1:n, [lm k]));
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p([k, mi]) = p([mi, k]);
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q([k, lm]) = q([lm, k]);
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% Perform Gaussian elimination with the greatest pivot
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if A(k, k) ~= 0
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rows = k+1 : n;
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A(rows, k) = A(rows, k)/A(k, k);
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@ -36,8 +37,6 @@ end
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U = triu(A);
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L = tril(A, -1) + eye(n);
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p
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I = eye(n);
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P = I(p, :)
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q
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Q = I(:, q)
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P = I(p, :);
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Q = I(:, q);
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@ -1,15 +1,79 @@
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clear all;
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B = [2, -1, 0, 0;
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A = [2, -1, 0, 0;
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-1, 2, -1, 0;
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3, -1, 2, -1;
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0, 4, -1, 2];
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0, -1, 2, -1;
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0, 0, -1, 2];
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b = [0;0;0;5];
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[U, L] = outer_product_gaussian_elimination(B);
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back_substitution(U, b);
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B = outer_product_gaussian_elimination(A);
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U = triu(B);
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L = tril(B, -1);
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x = back_substitution(U, b);
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[U, L] = gaussian_elimination_with_complete_pivoting(B)
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L*U
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% A = gauss_jordan_elimination(B)
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%% Problem 1
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clear all;
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A = [2, -1, 0, 0;
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-1, 2, -1, 0;
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0, -1, 2, -1;
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0, 0, -1, 2];
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b = [0;0;0;5];
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B = gauss_jordan_elimination([A b])
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[P, Q, L, U] = gaussian_elimination_with_complete_pivoting(A);
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b = P*b;
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% Ly = b and Ux = y
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y = forward_substitution(L, b);
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x = Q*back_substitution(U, y);
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% L*U
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%% Problem 2
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A = [1, 1, 1;
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1, 1, 2;
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1, 2, 2];
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b = [1;2;1];
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[P, Q, L, U] = gaussian_elimination_with_complete_pivoting(A);
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b = P*b;
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% Ly = b and Ux = y
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y = forward_substitution(L, b);
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x = Q*back_substitution(U, y)
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% L*U
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%% Problem 4
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A = [0.835, 0.667;
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0.333, 0.266];
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b = [0.168; 0.067];
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bp = [0.168; 0.066];
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kappa = cond(A)
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B = gauss_jordan_elimination([A b])
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Bp = gauss_jordan_elimination([A bp])
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%% Problem 5
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% AX = I3
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A = [2, 1, 2;
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1, 2, 3;
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4, 1, 2];
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[P, Q, L, U] = gaussian_elimination_with_complete_pivoting(A);
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I = P*eye(3);
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% Ly = b and Ux = y
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y = forward_substitution(L, I);
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X = Q*back_substitution(U, y)
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inv(A)
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@ -1,5 +1,5 @@
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% Algorithm 1: Outer Product Gaussian Elimination (Alg. 3.2.1)
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function [U, L] = outer_product_gaussian_elimination(A)
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function A = outer_product_gaussian_elimination(A)
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[n, m] = size(A);
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if n ~= m
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@ -14,7 +14,4 @@ end
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rows = k + 1 : n;
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A(rows, k) = A(rows, k)/A(k, k);
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A(rows, rows) = A(rows, rows) - A(rows, k) * A(k, rows);
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end
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U = triu(A);
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L = tril(A, -1) + eye(n);
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end
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