Cholesky

This method belongs to the LU factoring system. From matrix A, there are two matrices L and U, in this method the matrix L is lower triangular and the matrix U is upper triangular, but the two matrices L and U have the same values in the diagonal. However, if A is symmetric and positive definite, the factors can be chosen such that U is the transpose of L.

The factorization can be calculated directly through the following formulas

Pseudocode.

Input A, b

n= # de filas de A

L=eye(n);

U=eye(n);

for k=1:n

suma=0;

for p=1:k-1

suma=suma+L(k,p)*U(p,k);

end

L(k,k)= sqrt(A(k,k)-suma);

U(k,k) = L(k,k);

for i=k+1:n

suma=0;

for p=1:k-1

suma=suma+L(i,p)*U(p,k);

end

L(i,k)=(A(i,k)-suma)/L(k,k);

end

for j= k+1:n

suma=0;

for p=1:k-1

suma=suma+L(k,p)*U(p,j);

end

U(k,j)=(A(k,j)-suma)/L(k,k);

end

end

for i=1:n

suma=0;

for p=1:i-1

suma=suma+L(i,p)*d(p,1);

end

d(i,1)=(b(i,1)-suma)/L(i,i);

end

for i=n:-1:1

suma=0;

for p=i+1:n

suma=suma+U(i,p)*X(p,1);

end

X(i,1)=(d(i,1)-suma)/U(i,i);

end

Code.

clc

A=input('ingrese la matriz: ');

b=input('ingrese la matriz de términos independientes: ');

n=size(A,1);

L=eye(n);

U=eye(n);

for k=1:n

suma=0;

for p=1:k-1

suma=suma+L(k,p)*U(p,k);

end

L(k,k)= sqrt(A(k,k)-suma);

U(k,k) = L(k,k);

for i=k+1:n

suma=0;

for p=1:k-1

suma=suma+L(i,p)*U(p,k);

end

L(i,k)=(A(i,k)-suma)/L(k,k);

end

for j= k+1:n

suma=0;

for p=1:k-1

suma=suma+L(k,p)*U(p,j);

end

U(k,j)=(A(k,j)-suma)/L(k,k);

end

end

disp('U=')

disp(U)

disp('L=')

disp(L)

disp('L*U=')

disp(L*U)

for i=1:n

suma=0;

for p=1:i-1

suma=suma+L(i,p)*d(p,1);

end

d(i,1)=(b(i,1)-suma)/L(i,i);

end

disp('d=');

disp(d);

for i=n:-1:1

suma=0;

for p=i+1:n

suma=suma+U(i,p)*X(p,1);

end

X(i,1)=(d(i,1)-suma)/U(i,i);

end

disp('La solucion es: ')

disp(X)

B=A*X