lu factorization in matlab

LU factorization is a useful technique in linear algebra for solving systems of linear equations. The following code snippet shows how to perform LU factorization in MATLAB:

main.m
A = [1 2 3; 4 5 6; 7 8 9]; % the matrix to be factorized
[L, U, P] = lu(A); % performing LU factorization with pivoting
120 chars
3 lines

In this code, A is the matrix to be factorized. The lu function returns three matrices: L, U, and P. L and U are the lower and upper triangular matrices, respectively, that result from the factorization. P is the permutation matrix that represents the row interchanges performed during pivoting.

Once we have obtained the L, U, and P matrices, we can use them to solve systems of linear equations. For example:

main.m
b = [1; 2; 3]; % the right-hand side vector
y = L \ (P * b); % solving Ly = Pb
x = U \ y; % solving Ux = y
107 chars
4 lines

In this code, b is the right-hand side vector of the system of linear equations. We first solve Ly = Pb for y using MATLAB's backslash operator \, which automatically takes into account the permutation matrix P. We then solve Ux = y for the solution vector x.