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+Uspeye(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*eL*U*e) ans = 1.4251e014
This shows that A
and L*U
,
where L
and U
are given by the
modified Crout ILU
, have the same rowsum.
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 fillin, has 7840
nonzeros, the same amount
as A
.
These incomplete factorizations may be useful as preconditioners for a system of linear equations being solved by iterative methods such as BICG (BiConjugate Gradients), GMRES (Generalized Minimum Residual Method).
[1] Saad, Yousef, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, 1996, Chapter 10  Preconditioning Techniques.