Refactored Direct Methods

Direct Methods for Solving Linear Systems/Alg1.m

@@ -1,4 +1,4 @@
-function A = Alg1_outer_product_gaussian_elimination(A)
+function A = Alg1(A)
 % Algorithm 1: Outer Product Gaussian Elimination
 % Performs a gaussian eliminaion on a square matrix A.
 
@@ -22,4 +22,5 @@ for k = 1 : m-1
     A(rows, rows) = A(rows, rows) - A(rows, k) * A(k, rows);
 end
  
-end
+end
+

Direct Methods for Solving Linear Systems/Alg11.m

@@ -1,5 +1,5 @@
 function [Q R] = Alg11(A)
-% Algorithm 11: QR factorization via Householder algorithm.
+% Algorithm 11: QR factorization via Householder transformation.
 
 [m, n] = size(A);
 

Direct Methods for Solving Linear Systems/Alg2.m

@@ -1,4 +1,4 @@
-function [P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A)
+function [P, Q, L, U] = Alg2(A)
 % Algorithm 2: Gaussian Elimination with Complete Pivoting.
 % [P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A)
 % computes the complete pivoting factorization PAQ = LU.

Direct Methods for Solving Linear Systems/Alg3.m

@@ -1,4 +1,4 @@
-function b = Alg3_forward_substitution(L, b)
+function b = Al3(L, b)
 % Algorithm 3: Forward Substitution
 % b = Alg3_forward_substitution(L, b) overwrites b with the solution to
 % Lx = b.

Direct Methods for Solving Linear Systems/Alg4.m

@@ -1,4 +1,4 @@
-function b = Alg4_back_substitution(U,b)
+function b = Alg4(U,b)
 % Argorithm 4: Back Substitution 
 % b = Alg4_back_substitution(U,b) returns vetor b with solution to the
 % Ux = b.
@@ -17,9 +17,9 @@ if length(b) ~= m
     error('Vector b has wrong length!')
 end
 
-if det(U) < eps
+% if det(U) < eps
-    error('Matrix is not nonsingular!')
+%     error('Matrix is not nonsingular!')
-end
+% end
 
 
 % The following algorithm is based on the Algrotihm 3.1.2 from [2].

Direct Methods for Solving Linear Systems/Alg5.m

@@ -1,9 +1,9 @@
-function A = Alg5_gauss_jordan_elimination(A)
+function A = Alg5(A)
 % Algorithm 5: Gauss-Jordan Elimination
 % A = Alg5_gauss_jordan_elimination(A) performs Gauss-Jordan elimination
 % on an augmented matrix A.
 
-[m, n] = size(A);
+[m, ~] = size(A);
 
 for k = 1 : m
     
@@ -12,7 +12,7 @@ for k = 1 : m
     A(k, :) = row;
     
     for i = 1 : m
-        if i ~= k
+        if i ~= k && A(i, k) ~= 0
             A(i, :) = A(i, :)-(A(i, k))*row;
         end
     end

Direct Methods for Solving Linear Systems/main.m

@@ -1,54 +1,28 @@
-clear all;
-clc;
-
-A = [2, -1, 0, 0;
-    -1, 2, -1, 0;
-    0, -1, 2, -1;
-    0, 0, -1, 2];
-
-b = [0;0;0;5];
-
-B = Alg1_outer_product_gaussian_elimination(A);
-U = triu(B);
-x = Alg4_back_substitution(U, b);
-
 %% Problem 1
 clear all;
+clc;
 A = [2, -1, 0, 0;
     -1, 2, -1, 0;
     0, -1, 2, -1;
     0, 0, -1, 2];
-
 b = [0;0;0;5];
-
+B = Alg5([A b])
-B = gauss_jordan_elimination([A b])
-
-[P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A);
-
-b = P*b;
-% Ly = b and Ux = y
-y = forward_substitution(L, b);
-
-x = Q*back_substitution(U, y);
-
-% L*U
 
 %% Problem 2
+clear all;
+clc;
 A = [1, 1, 1;
      1, 1, 2;
      1, 2, 2];
-
 b = [1;2;1];
 
-[P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A);
+[P, Q, L, U] = Alg2(A);
 
 b = P*b;
 % Ly = b and Ux = y
-y = Alg3_forward_substitution(L, b);
+y = Alg3(L, b);     % Forward substitution
+x = Q*Alg4(U, y)	% Backward substitution
 
-x = Q*Alg4_back_substitution(U, y)
-
-% L*U
 
 
 %% Problem 4
@@ -60,25 +34,34 @@ bp = [0.168; 0.066];
 
 kappa = cond(A)
 
-B = Alg5_gauss_jordan_elimination([A b])
+B = Alg5([A b])
-Bp = Alg5_gauss_jordan_elimination([A bp])
+Bp = Alg5([A bp])
 
 %% Problem 5
-% AX = I3
+clear all;
+clc;
 A = [2, 1, 2;
      1, 2, 3;
      4, 1, 2];
  
-[P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A);
+[P, Q, L, U] = Alg2(A)
 
 I = P*eye(3);
 % Ly = b and Ux = y
-y = forward_substitution(L, I);
+y = Alg3(L, I);     % Forward substitution
-
+x = Q*Alg4(U, y)    % Backward substitution
-X = Q*back_substitution(U, y)
+inv(A) - x
-inv(A)
 
 %% Problem 6
+clc;
+
+A = [1 2 3 4;
+    -1 1 2 1;
+     0 2 1 3;
+     0 0 1 1];
+
+[L, U, P] = Alg8(A)
+det(A) - prod(diag(U))
 
 %% Problem 10