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Added Algorithm 8 and refactored code

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Sergiusz Warga 2021-03-13 20:23:23 +01:00
parent d6b6b159af
commit 19cba61d44
9 changed files with 110 additions and 56 deletions
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1
.gitignore vendored
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@ -1 +1,2 @@
# MATLAB
*.asv

17
Direct Methods for Solving Linear Systems/Alg1_outer_product_gaussian_elimination.m
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@ -1,20 +1,21 @@
function A = Alg1_outer_product_gaussian_elimination(A)
% ADDME Algorithm 1: Outer Product Gaussian Elimination (Golub, Loan, 3.2.1)
% Matrix A has to be a square singular matrix.
% Algorithm 1: Outer Product Gaussian Elimination (Golub, Loan, 3.2.1)
% Performs a gaussian eliminaion on a square matrix A.
[m, n] = size(A);
if m ~= n
error('Matrix is not squared!')
end
[m, n] = size(A);
if m ~= n
error('Matrix is not squared!')
end
% if det(A) == 0
% error('Matrix is not nonsingular!')
% end
for k = 1 : m-1
for k = 1 : m-1
rows = k + 1 : m;
A(rows, k) = A(rows, k)/A(k, k);
A(rows, rows) = A(rows, rows) - A(rows, k) * A(k, rows);
end
end
end

18
Direct Methods for Solving Linear Systems/Alg2_gaussian_elimination_with_complete_pivoting.m
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@ -1,27 +1,27 @@
function [P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A)
% ADDME Algorithm 2: Gaussian Elimination with Complete Pivoting.
% Algorithm 2: Gaussian Elimination with Complete Pivoting.
% [P, Q, L, U] = Alg2_gaussian_elimination_with_complete_pivoting(A)
[n, m] = size(A);
if n ~= m
error('Matrix is not squared!')
end
if det(A) == 0
error('Matrix is not nonsingular!')
end
% if det(A) == 0
% error('Matrix is not nonsingular!')
% end
p = 1:n;
q = 1:n;
% for k = 1 : n-1
for k = 1 : n-1
i = k:n;
j = k:n;
[max_val, rows_of_max_in_col] = max(abs(A(i, j)));
[max_val, max_col] = max(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
% Assign value of mu 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]));
@ -36,8 +36,8 @@ for k = 1 : n-1
end
end
U = triu(A);
L = tril(A, -1) + eye(n);
I = eye(n);
U = triu(A);
L = tril(A, -1) + I;
P = I(p, :);
Q = I(:, q);

27
Direct Methods for Solving Linear Systems/Alg3_forward_substitution.m
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@ -1,17 +1,18 @@
% Algorithm 3: Forward Substitution (Alg. 3.1.1)
function b = Alg3_forward_substitution(L, b)
% Algorithm 3: Forward Substitution (Golub, Loan, Alg. 3.1.1)
[n, m] = size(L);
if n ~= m
error('Matrix is not squared!')
end
if length(b) ~= n
error('Vector b has wrong length!')
end
b(1, :) = b(1, :)/L(1,1);
for i = 2:n
b(i, :) = (b(i, :) - L(i, 1:i-1)*b(1:i-1, :))/L(i, i);
[m, n] = size(L);
if m ~= n
error('Matrix is not squared!')
end
if length(b) ~= m
error('Vector b has wrong length!')
end
b(1, :) = b(1, :)/L(1,1);
for i = 2:m
b(i, :) = (b(i, :) - L(i, 1:i-1)*b(1:i-1, :))/L(i, i);
end
end

2
Direct Methods for Solving Linear Systems/Alg4_back_substitution.m
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@ -1,5 +1,5 @@
function b = Alg4_back_substitution(U,b)
% ADDME Argorithm 4: Back Substitution (Golub, Loan, Alg. 3.1.2)
% Argorithm 4: Back Substitution (Golub, Loan, Alg. 3.1.2)
% Returns vetor b with solution to he Ux = b.
[m, n] = size(U);

18
Direct Methods for Solving Linear Systems/Alg5_gauss_jordan_elimination.m
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@ -2,19 +2,17 @@ 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);
[M, N] = size(A);
for m = 1 : M
for k = 1 : m
row = A(m, :);
row = row/row(m);
A(m, :) = row;
row = A(k, :);
row = row/row(k);
A(k, :) = row;
for n = 1 : M
if n ~= m
A(n, :) = A(n, :)-(A(n, m))*row;
for i = 1 : m
if i ~= k
A(i, :) = A(i, :)-(A(i, k))*row;
end
end
end

4
Direct Methods for Solving Linear Systems/Alg6_RREF.m
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@ -1,9 +1,7 @@
function A = Alg6_RREF(A)
% ADDME Algorithm 6: Reduced Row Echelon Form (RREF)
% Algorithm 6: Reduced Row Echelon Form (RREF)
% A = Alg6_RREF(A) returns RREF of matrix A.
% M – rows, N – columns
[M, N] = size(A);
n = 0;

25
Direct Methods for Solving Linear Systems/Alg7.m
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@ -1,7 +1,8 @@
function [L, U] = Alg7(A)
% ADDME LU Factorization without pivoting
% [L, U] = Alg7(A) decomposes matrix A into U – upper triangular matrix and
% L – lower unit triangular matrix such, that A = LU.
% LU Factorization without pivoting
% [L, U] = Alg7(A) decomposes matrix A using Crout's algorithm into
% U – upper triangular matrix and L – unit lower triangular matrix
% such that A = LU.
[m, n] = size(A);
@ -14,21 +15,21 @@ function [L, U] = Alg7(A)
% L = tril(A, -1) + eye(m);
% Instead, we should use the Crout's algorithm
% It returns an Unit Upper Triangular matrix and a Lower Traingular matrix
% (in oppose to the Gaussian Elimination).
A
L = zeros(m, n)
U = eye(m, n)
L = zeros(m, n);
U = eye(m, n);
for k = 1:n
for i = k:n
L(i, k) = A(i, k) - dot(L(i,1:k-1), U(1:k-1, k));
end
for j = k : n
U(k, j) = (A(k, j) - dot(L(k, 1:k-1), U(1:k-1, j))) / L(k, k);
U(k, j) = A(k, j) - dot(L(k, 1:k-1), U(1:k-1, j));
end
for i = k:n
L(i, k) = (A(i, k) - dot(L(i,1:k-1), U(1:k-1, k))) / U(k,k);
end
end
end

54
Direct Methods for Solving Linear Systems/Alg8.m Normal file
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function [L, U, P] = Alg8(A)
% LU Factorization with partial pivoting
% [L, U, P] = Alg8(A) decomposes matrix A using Crout's algorithm into
% U – upper triangular matrix and L – unit lower triangular matrix
% using partial pivoting, interchanging A's rows, such that AP = LU.
[m, n] = size(A);
L = zeros(m, n);
U = eye(m, n);
P = eye(m, n);
for k = 1:m
if A(k, k) == 0
non_zero_col = find(A(k, :), 1);
A(:, [k non_zero_col]) = deal(A(:, [non_zero_col k]));
P(:, [k non_zero_col]) = deal(P(:, [non_zero_col k]));
end
end
% TODO: Permutation A*P does not work
%
% 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
for k = 1:n
for j = k : n
U(k, j) = A(k, j) - dot(L(k, 1:k-1), U(1:k-1, j));
end
for i = k:n
L(i, k) = (A(i, k) - dot(L(i,1:k-1), U(1:k-1, k))) / U(k,k);
end
end
end