Fixed Algs 1 and 4 and introduced gaussian_elimination_with_complete_pivoting

.gitignore

@@ -0,0 +1 @@
+*.asv

Direct Methods for Solving Linear Systems/back_substitution.m

@@ -1,3 +1,4 @@
+% Argorithm 4: Back Substitution (Alg. 3.1.2)
 function back_substitution(U,b)
 
 [n, m] = size(U);
@@ -9,13 +10,13 @@ if length(b) ~= n
     error('Vector b has wrong length!')
 end
 
-if det(A) == 0
+if det(U) == 0
     error('Matrix is not nonsingular!')
 end
 
 b(n) = b(n)/U(n, n)
-for i = n-1:-1:n
-    
+for i = n-1:-1:1
+    b(i) = (b(i) - U(i, i+1 : n)*b(i+1 : n))/U(i, i)
 end
 
 end

Direct Methods for Solving Linear Systems/gaussian_elimination_with_complete_pivoting.m

@@ -0,0 +1,30 @@
+function gaussian_elimination_with_complete_pivoting(A)
+
+% det(A(mi, lm))
+
+[n, m] = size(A);
+if n ~= m
+    error('Matrix is not squared!')
+end
+    
+if det(A) == 0
+    error('Matrix is not nonsingular!') 
+end
+
+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, 1:n) = A(mi, 1:n)
+    A(1:n, k) = A(1:n, lm)
+    p(k) = mi
+    q(k) = lm
+    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

Direct Methods for Solving Linear Systems/outer_product_gaussian_elimination.m

@@ -1,5 +1,5 @@
-% Outer Product Gaussian Elimination (Alg. 3.2.1)
-function U, Mk = outer_product_gaussian_elimination(A)
+% Algorithm 1: Outer Product Gaussian Elimination (Alg. 3.2.1)
+function [U, Mk] = outer_product_gaussian_elimination(A)
 
 [n, m] = size(A);
 if n ~= m
@@ -10,13 +10,13 @@ end
     error('Matrix is not nonsingular!') 
  end
  
- A
+ A;
  
  for k = 1 : n-1
      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);
-     A
+     A;
  end
  
 U = triu(A)