So tired

Direct Methods for Solving Linear Systems/gaussian_elimination_with_complete_pivoting.m

@@ -1,6 +1,4 @@
-function A = gaussian_elimination_with_complete_pivoting(A)
-
-% det(A(mi, lm))
+function [U, L] = gaussian_elimination_with_complete_pivoting(A)
 
 [n, m] = size(A);
 if n ~= m
@@ -11,21 +9,35 @@ if det(A) == 0
     error('Matrix is not nonsingular!')
 end
 
+A
+
+% for k = 1 : n-1
 for k = 1 : n-1
     i = k:n;
     j = k:n;
-    A(i, j);
-    maximum = max(abs(A(i, j)), [], 'all');
-    max_idx = find(abs(A==maximum));
-    [mi, lm] = ind2sub(size(A), max_idx(1));
-    A([k mi], 1:n) = deal(A([mi k], 1:n));
-    A(1:n, [k lm]) = deal(A(1:n, [lm k]));
-    p(k) = mi;
-    q(k) = lm;
+    A(i, j)
+    [max_val, rows_of_max_in_col] = max(abs(A(i, j)));
+    [max_val, max_col] = max(max_val);
+    max_row = rows_of_max_in_col(max_col);
+    % Assing value of mi and lambda in respect to the main A matrix
+    [mi, lm] = deal(max_row+k-1, max_col+k-1)
+    A([k mi], 1:n) = deal(A([mi k], 1:n))
+    A(1:n, [k lm]) = deal(A(1:n, [lm k]))
+    p(k) = mi
+    q(k) = lm
     % Perform Gaussian elimination with the greatest pivot
+    
     if A(k, k) ~= 0
         rows = k+1 : n;
         A(rows, k) = A(rows, k)/A(k, k);
         A(rows, rows) = A(rows, rows) - A(rows, k) * A(k, rows);
     end
 end
+
+U = triu(A);
+L = tril(A, -1) + eye(n);
+p
+I = eye(n);
+P = I(p, :)
+q
+Q = I(:, q)

Direct Methods for Solving Linear Systems/main.m

@@ -1,10 +1,15 @@
 clear all;
 
-B = [2, -1, 0, 0; -1, 2, -1, 0;0, -1, 2, -1; 0, 0, -1, 2];
+B = [2, -1, 0, 0;
+    -1, 2, -1, 0;
+    3, -1, 2, -1;
+    0, 4, -1, 2];
 
 b = [0;0;0;5];
 
-[U, Mk] = outer_product_gaussian_elimination(B);
+[U, L] = outer_product_gaussian_elimination(B);
 back_substitution(U, b);
 
-A = gaussian_elimination_with_complete_pivoting(B);
+[U, L] = gaussian_elimination_with_complete_pivoting(B)
+L*U
+% A = gauss_jordan_elimination(B)

Direct Methods for Solving Linear Systems/outer_product_gaussian_elimination.m

@@ -1,5 +1,5 @@
 % Algorithm 1: Outer Product Gaussian Elimination (Alg. 3.2.1)
-function [U, Mk] = outer_product_gaussian_elimination(A)
+function [U, L] = outer_product_gaussian_elimination(A)
 
 [n, m] = size(A);
 if n ~= m
@@ -17,4 +17,4 @@ end
  end
  
 U = triu(A);
-Mk = diag(A,-1); % Gauss vector
+L = tril(A, -1) + eye(n);