Incomplete LU factorization
ilu(A,setup)
[L,U] = ilu(A,setup)
[L,U,P] = ilu(A,setup)
ilu produces a unit lower triangular matrix,
an upper triangular matrix, and a permutation matrix.
ilu(A,setup) computes
the incomplete LU factorization of A. setup is
an input structure with up to five setup options. The fields must
be named exactly as shown in the table below. You can include any
number of these fields in the structure and define them in any order.
Any additional fields are ignored.
Field Name | Description |
|---|---|
| Type of factorization. Values for
If |
| Drop tolerance of the incomplete LU factorization. The nonzero
entries of abs(U(i,j)) >= droptol*norm(A(:,j)), with
the exception of the diagonal entries, which are retained regardless
of satisfying the criterion. The entries of abs(L(i,j)) >= droptol*norm(A(:,j))/U(j,j). |
| Modified incomplete LU factorization. Values for
|
| If |
| Pivot threshold between |
ilu(A,setup) returns L+U-speye(size(A)),
where L is a unit lower triangular matrix and U is
an upper triangular matrix.
[L,U] = ilu(A,setup) returns
a unit lower triangular matrix in L and an upper
triangular matrix in U.
[L,U,P] = ilu(A,setup) returns
a unit lower triangular matrix in L, an upper triangular
matrix in U, and a permutation matrix in P.
ilu works on sparse square matrices only.
Start with a sparse matrix and compute the LU factorization.
A = gallery('neumann', 1600) + speye(1600);
setup.type = 'crout';
setup.milu = 'row';
setup.droptol = 0.1;
[L,U] = ilu(A,setup);
e = ones(size(A,2),1);
norm(A*e-L*U*e)
ans =
1.4251e-014
This shows that A and L*U,
where L and U are given by the
modified Crout ILU, have the same row-sum.
Start with a sparse matrix and compute the LU factorization.
A = gallery('neumann', 1600) + speye(1600);
setup.type = 'nofill';
nnz(A)
ans =
7840
nnz(lu(A))
ans =
126478
nnz(ilu(A,setup))
ans =
7840
This shows that A has 7840 nonzeros,
the complete LU factorization has 126478 nonzeros,
and the incomplete LU factorization, with 0 level
of fill-in, has 7840 nonzeros, the same amount
as A.
[1] Saad, Yousef, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996, Chapter 10 - Preconditioning Techniques.