GAUSS:
% Define the coefficient matrix 'Info' and the right-hand side vector 'b'
Info = [2, -1, 1; 3, 2, -1; 1, 1, 3];
b = [8; 5; 10];
% Create the augmented matrix 'A' by concatenating 'Info' and 'b'
A = [Info, b];
% Perform Gaussian elimination with partial pivoting
n = size(A, 1); % Number of rows/columns in 'A'
x = zeros(n, 1);
for i = 1:n
% Partial Pivoting: Find the row with the maximum value in the current column
[~, pivot_row] = max(abs(A(i:n, i)));
pivot_row = pivot_row + i - 1;
% Swap rows to bring the pivot element to the current row
A([i, pivot_row], :) = A([pivot_row, i], :);
for j = i+1:n
key1 = A(j, i) / A(i, i);
A(j, :) = A(j, :) - key1 * A(i, :);
end
end
% Back-substitution
x(n) = A(n, n+1) / A(n, n);
for i = n-1:-1:1
x(i) = (A(i, n+1) - sum(A(i, i+1:n) * x(i+1:n))) / A(i, i);
end
% Display the solution
fprintf('Solution is x = \n');
disp(x);
Doolitle:
clc
clear
A = [2 7 1;3 -2 0; 1 5 3];
B = [8 6 -2 ];
[L, U] = MyDecompositionLU_Fun(A);
Y = MyCalculyFun(L, B);
X = MyCalculxFun(U, Y);
function [L, U] = MyDecompositionLU_Fun(A)
[N, ~] = size(A);
L = eye(N); % Initialise L comme une matrice identité
U = zeros(N, N);
for i = 1:N
for j = i:N
U(i, j) = A(i, j);
for k = 1:i-1
U(i, j) = U(i, j) - L(i, k) * U(k, j);
end
end
for j = i+1:N
L(j, i) = A(j, i);
for k = 1:i-1
L(j, i) = L(j, i) - L(j, k) * U(k, i);
end
L(j, i) = L(j, i) / U(i, i);
end
end
end
function Y = MyCalculyFun(L, B)
N = length(L);
Y = zeros(1, N);
for i = 1:N
Y(i) = B(i);
for j = 1:i-1
Y(i) = Y(i) - L(i, j) * Y(j);
end
end
end
function X = MyCalculxFun(U, Y)
N = length(U);
X = zeros(1, N);
for i = N:-1:1
X(i) = Y(i);
for j = i+1:N
X(i) = X(i) - U(i, j) * X(j);
end
X(i) = X(i) / U(i, i);
end
end
CROOT:
clear all
clc
A = [1 3 4 8
2 1 2 3
4 3 5 8
9 2 7 4];
B = [1
1
1
1];
matrixSize=length(A);
Lower=zeros(size(A));
Upper=zeros(size(A));
Lower(:,1)= A(:,1); %Set the first column of L to the frist column of A
Upper(1,:)=A(1,:)/Lower(1,1); % Create the first row of upper, divide by L(1,1)
Upper(1,1)=1; % Start the identity matrix
for k=2:matrixSize
for j=2:matrixSize
for i=j:matrixSize
Lower(i,j)=A(i,j) - dot(Lower(i,1:j-1),Upper(1:j-1,j));
end
Upper(k,j)=(A(k,j)-dot(Lower(k,1:k-1),Upper(1:k-1,j)))/Lower(k,k);
end
end
Upper
Lower
% L * Y = B
Y = zeros(matrixSize, 1);
% BASE CASE, SOLVE THE FIRST ONE
Y(1) = B(1);
for row = 2 : matrixSize %2 - number or rows
Y(row) = B(row);
for col = 1 : row - 1 %1 - row number
Y(row) = Y(row) - Lower(row, col) * Y(col);
end
Y(row) = Y(row) / Lower(row, row)
end
Y
% U * X = Y
X = zeros(matrixSize, 1);
X(matrixSize) = Y(matrixSize) / Upper(matrixSize,matrixSize);
for row = matrixSize - 1 : -1 : 1 %second to last row - 1
temp = 0;
for col = matrixSize : -1 : 1 %number of rows to row
temp = temp + Upper(row,col) * X(col);
end
X(row) = (Y(row) - temp) / Upper(row,row);
end
Rayleigh:
format short
clc
A= [2 -7 0; -1 2 -1; 0 -1 2];
x = [1; 1; 1];
choice1=menu('Pick the choice' , 'Larqest','Smallest','Near to 3');
if choice1==1
B=A;
elseif choice1==2
B=inv(A);
else
lam0 = input('Enter the value of LAMBDA which is near to \n ');
D= A-lam0.*eye(size(A));
B= inv(D) ;
end
iter=1;
maxerr= 2;
err = 10000;
lam1 = Inf;
fprintf('\t Iter \t Eigen Value \t EigenVector \n');
fprintf('\t ======================== \n');
while all (err>maxerr)
xold = x; Y = B*x;
eigval = max(abs (Y));
eigvec = Y./eigval;
x= eigvec;
err = abs (sum(xold-x));
laml= eigval;
disp([iter laml x']);
iter = iter+1;
end
fprintf('Method converge in %d iteration \n',iter-1);
disp('==============')
if choice1==1
fprintf('The greatest eigenvalue is %5.5f\n',laml);
elseif choice1==2
fprintf('The smallest eigenvalue is %5.5f\n',1/lam1);
elseif choice1==3
fprintf('Eigenvalue near to %5.3f is %5.5f\n', lam0,lam0+1/laml);
end
disp('The corresponding eigenvector is:');
fprintf('\t %5.5f \n',x);
cholsky:
A = [4 5 9; -2 6 0; 1/2 6 10];
L = Cholesky(A);
disp(L);
function L = Cholesky(A)
n = length(A);
L = zeros(n, n);
for i = 1:n
L(i, i) = sqrt(A(i, i) - sum(L(i, 1:i-1).^2));
for j = (i + 1):n
L(j, i) = (A(j, i) - sum(L(j, 1:i-1) .* L(i, 1:i-1))) / L(i, i);
end
end
end
Jacobi:
format short
clc
A=[82 6 -1;6 74 -2;1 1 54] ;
b=[35;72;110];
maxerr = 1e-5;
x=zeros(1,size(A,1));
n = size(A,1);
err = inf;
itr=0;
while all(err>maxerr)
xold = x;
for i=1:n
sum = 0;
for j=1:n
if j~=i
sum = sum+A(i,j)*xold(j);
end
end
x(i)=(1/A(i,i))*(b(i)-sum);
end
itr = itr+1;
y(itr,:)=x;
err=abs(xold-x);
end
fprintf('method converge in %d iteration\n',itr);
disp(x)
seidel:
format short
clc
A=[82 6 -1;6 74 -2;1 1 54]
b=[35;72;110];
maxerr= 1e-5;
x=zeros(1,size(A,1));
err1=Inf;
itr=0;
while all(err1>maxerr)
x_old=x;
for i=1:size(A,1)
sum=0;
for j=1:i-1
sum=sum+A(i,j)*x(j);
end
for j=i+1:size(A,1)
sum=sum+A(i,j)*x_old(j);
end
x(i)=(1/A(i,i))*(b(i)-sum);
end
itr=itr+1;
y(itr,:)=x;
err1=abs(x_old-x);
end
fprintf('Method Converge in %d iteration \n',itr);
disp(x)
