Added shitty Algorithm 6

Direct Methods for Solving Linear Systems/Alg2_gaussian_elimination_with_complete_pivoting.m

@@ -16,7 +16,6 @@ q = 1:n;
 for k = 1 : n-1
     i = k:n;
     j = k:n;
-    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);

Direct Methods for Solving Linear Systems/Alg5_gauss_jordan_elimination.m

@@ -1,18 +1,10 @@
-% Algorithm 5: Gauss-Jordan Elimination (Alg.
-% Argument A is an augmented matrix
 function A = Alg5_gauss_jordan_elimination(A)
+% Algorithm 5: Gauss-Jordan Elimination
+% Argument A is an augmented matrix
 
 % M – rows, N – columns
 
 [M, N] = size(A);
-% 
-% if M + 1 ~= N
-%     error('Matrix is not squared!')
-% end
-%     
-% if det(A) == 0
-%     error('Matrix is not nonsingular!') 
-% end
 
 for m = 1 : M
     
@@ -26,3 +18,4 @@ for m = 1 : M
         end
     end
 end
+end

Direct Methods for Solving Linear Systems/Alg6_RREF.m

@@ -0,0 +1,51 @@
+function A = Alg6_RREF(A)
+%  Algorithm 6: Reduced Row Echelon Form (RREF)
+
+% M – rows, N – columns
+
+[M, N] = size(A);
+
+n = 0;
+
+for m = 1 : M
+    n = n + 1
+    if n > N
+        break
+    end
+    A
+    % We want the left-most coefficient to be 1 (pivot)
+    row = A(m, :);
+    if row(m) == 0
+        n = n + 1
+    end
+    [m ,n]
+    row = row/row(n);
+    A(m, :) = row;
+    
+    for i = 1 : M
+        if i ~= m
+            A(i, :) = A(i, :)-(A(i, n))*row;
+        end
+    end
+    
+    A
+    
+    for i = m + 1 : M
+       A(i:end, m+1:end); % Partial matrix (in which we are looking for non-zero pivots)
+       A(i:end, m+1); % Left-most column
+       if ~any(A(i:end, m+1)) % If the left-most column has only zeros check the next one
+           m = m + 1;
+       end
+       A(i:end, m+1:end);
+       if A(i, m+1) == 0
+           non_zero_row = find(A(i:end,m+1), 1);
+           if isempty(non_zero_row)
+               continue
+           end
+           A([i, i+non_zero_row-1], :) = deal(A([i+non_zero_row-1, i], :));
+       end
+    end
+end
+
+end
+

Direct Methods for Solving Linear Systems/main.m

@@ -60,8 +60,8 @@ bp = [0.168; 0.066];
 
 kappa = cond(A)
 
-B = gauss_jordan_elimination([A b])
+B = Alg5_gauss_jordan_elimination([A b])
-Bp = gauss_jordan_elimination([A bp])
+Bp = Alg5_gauss_jordan_elimination([A bp])
 
 %% Problem 5
 % AX = I3
@@ -79,3 +79,18 @@ X = Q*back_substitution(U, y)
 inv(A)
 
 %% Problem 6
+
+%% Problem 10
+
+% A = [1 2 2 3 1;
+%      2 4 4 6 2;
+%      3 6 6 9 6;
+%      1 2 4 5 3]
+
+A = [0.835, 0.667;
+     0.333, 0.266];
+b = [0.168; 0.067];
+ 
+Alg6_RREF([A b])
+
+