LU factorization with complete pivoting.

An implementation of algorithm 3.4.2 from Matrix Computations.

To compute the LU factorization under default settings:

[L U p q] = lucp(A)

This produces a factorization such that L*U = A(p,q). Vectors p and q permute the rows and columns, respectively.

The pivot tolerance can be controlled:

[L U p q] = lucp(A,tol)

The algorithm will terminate if the absolute value of the pivot is less than tol.

Permutation matrices can be generated:

[L U P Q] = lucp(A,tol,'matrix')
[L U P Q] = lucp(A,tol,'sparse')

The first will generate full permutation matrices P and Q such that L*U = P*A*Q. The second generates sparse P and Q.

If A is sparse, L and U will be sparse. However, no effort is taken to reduce fill in.

This function works on non-square matrices.