ilu
Incomplete LU factorization
Description
[___] = ilu(
performs the incomplete LU factorization of A
,options
)A
with options specified
by the structure options
.
For example, you can perform an incomplete LU factorization with pivoting by setting
the type
field of options
to
"ilutp"
. You can then specify a rowsum or columnsum preserving
modified incomplete LU factorization by setting the milu
field to
"row"
or "col"
. This combination of options
returns the permutation matrix P
such that L
and
U
are incomplete factors of A*P
for the
"row"
option (where U
is column permuted), or
L
and U
are incomplete factors of
P*A
for the "col"
option (where
L
is row permuted).
Examples
Different Types of Incomplete LU Factorization
The ilu
function provides three types of incomplete LU factorizations: the zerofill factorization (ILU(0)
), the Crout version (ILUC
), and the factorization with threshold dropping and pivoting (ILUTP
).
By default, ilu
performs the zerofill incomplete LU factorization of a sparse matrix input. For example, find the complete and incomplete factorization of a sparse matrix with 7840 nonzeros. Its complete LU factors have 126,478 nonzeros, and its incomplete LU factors with zerofill have 7840 nonzeros, the same number as A
.
A = gallery("neumann",1600) + speye(1600);
n = nnz(A)
n = 7840
n = nnz(lu(A))
n = 126478
n = nnz(ilu(A))
n = 7840
Because the zerofill factorization preserves the sparsity pattern of the input matrix in its LU factors, the relative error of the factorization is essentially zero on the pattern of nonzero elements of A
.
[L,U] = ilu(A); e = norm(A(L*U).*spones(A),"fro")/norm(A,"fro")
e = 4.8874e17
However, the product of these zerofill factors is not a good approximation of the original matrix.
e = norm(AL*U,"fro")/norm(A,"fro")
e = 0.0601
To improve the accuracy, you can use other types of incomplete LU factorization with fillin. For example, use the Crout version with a 1e6
drop tolerance.
options.droptol = 1e6;
options.type = "crout";
[L,U] = ilu(A,options);
The Crout version of the incomplete factorization has 51,482 nonzeros in its LU factors (less than the complete factorization). With fillin, the product of the incomplete LU factors is a better approximation of the original matrix.
n = nnz(ilu(A,options))
n = 51482
e = norm(AL*U,"fro")./norm(A,"fro")
e = 9.3040e07
For comparison, for the same input matrix A
, the incomplete factorization with threshold dropping and pivoting gives results similar to the Crout version.
options.type = "ilutp";
[L,U,P] = ilu(A,options);
n = nnz(ilu(A,options))
n = 51541
norm(P*AL*U,"fro")./norm(A,"fro")
ans = 9.4960e07
Drop Tolerance of Incomplete LU Factorization
Change the drop tolerance of incomplete LU factorization to factor a sparse matrix.
Load the west0479
matrix, which is a realvalued 479by479 nonsymmetric sparse matrix. Estimate the condition number of the matrix by using condest
.
load west0479
A = west0479;
c1 = condest(A)
c1 = 1.4244e+12
Improve the condition number of the matrix by using equilibrate
. Permute and rescale the original matrix A
to create a new matrix B = R*P*A*C
, which has a better condition number and diagonal entries of only 1 and –1.
[P,R,C] = equilibrate(A); B = R*P*A*C; c2 = condest(B)
c2 = 5.1036e+04
Specify the options to perform the incomplete LU factorization of B
with threshold dropping and pivoting that preserves the row sum. For comparison purposes, first specify a zero drop tolerance of fillin, which produces the complete LU factorization.
options.type = "ilutp"; options.milu = "row"; options.droptol = 0; [L,U,P] = ilu(B,options);
This factorization is very accurate in approximating the input matrix B
, but the factors are significantly more dense than B
.
e = norm(B*PL*U,"fro")/norm(B,"fro")
e = 1.0345e16
nLU = nnz(L)+nnz(U)size(B,1)
nLU = 19118
nB = nnz(B)
nB = 1887
You can change the drop tolerance to obtain incomplete LU factors that are more sparse but less accurate in approximating B
. For example, the following plots show the ratio of the density of the incomplete factors to the density of the input matrix plotted against the drop tolerance, and the relative error of the incomplete factorization.
ntols = 20; tau = logspace(6,2,ntols); e = zeros(1,ntols); nLU = zeros(1,ntols); for k = 1:ntols options.droptol = tau(k); [L,U,P] = ilu(B,options); nLU(k) = nnz(L)+nnz(U)size(B,1); e(k) = norm(B*PL*U,"fro")/norm(B,"fro"); end figure semilogx(tau,nLU./nB,LineWidth=2) title("Ratio of nonzeros of LU factors with respect to B") xlabel("drop tolerance") ylabel("nnz(L)+nnz(U)size(B,1)/nnz(B)",FontName="FixedWidth")
figure loglog(tau,e,LineWidth=2) title("Relative error of the incomplete LU factorization") xlabel("drop tolerance") ylabel("$BPLU_F\,/\,B_F$",Interpreter="latex")
In this example, the relative error of the incomplete LU factorization with threshold dropping is on the same order of the drop tolerance (which does not always occur).
Using ilu
as Preconditioner to Solve Linear System
Use an incomplete LU factorization as a preconditioner for bicgstab
to solve a linear system.
Load west0479
, a real 479by479 nonsymmetric sparse matrix.
load west0479
A = west0479;
Define b
so that the true solution to $\mathit{Ax}=\mathit{b}$ is a vector of all ones.
b = full(sum(A,2));
Set the tolerance and maximum number of iterations.
tol = 1e12; maxit = 20;
Use bicgstab
to find a solution at the requested tolerance and number of iterations. Specify five outputs to return information about the solution process:
x
is the computed solution toA*x = b
.fl0
is a flag indicating whether the algorithm converged.rr0
is the relative residual of the computed answerx
.it0
is the iteration number whenx
was computed.rv0
is a vector of the residual history at each half iteration for $\Vert \mathit{b}\mathit{Ax}\Vert $.
[x,fl0,rr0,it0,rv0] = bicgstab(A,b,tol,maxit); fl0
fl0 = 1
rr0
rr0 = 1
it0
it0 = 0
fl0
is 1
because bicgstab
does not converge to the requested tolerance 1e12
within the requested 20 iterations. In fact, the behavior of bicgstab
is so poor that the initial guess x0 = zeros(size(A,2),1)
is the best solution and is returned, as indicated by it0 = 0
.
To aid with the slow convergence, you can specify a preconditioner matrix. Because A
is nonsymmetric, use ilu
to generate the preconditioner $\mathit{M}=\mathit{L}\text{\hspace{0.17em}}\mathit{U}$. Specify a drop tolerance to ignore nondiagonal entries with values smaller than 1e6
. Solve the preconditioned system ${\mathit{A}\text{\hspace{0.17em}}\mathit{M}}^{1}\text{\hspace{0.17em}}\mathit{M}\text{\hspace{0.17em}}\mathit{x}=\mathit{b}$ by specifying L
and U
as inputs to bicgstab
.
options = struct("type","ilutp","droptol",1e6); [L,U] = ilu(A,options); [x_precond,fl1,rr1,it1,rv1] = bicgstab(A,b,tol,maxit,L,U); fl1
fl1 = 0
rr1
rr1 = 5.9892e14
it1
it1 = 3
The use of an ilu
preconditioner produces a relative residual rr1
less than the requested tolerance of 1e12
at the third iteration. The output rv1(1)
is norm(b)
, and the output rv1(end)
is norm(bA*x1)
.
You can follow the progress of bicgstab
by plotting the relative residuals at each half iteration. Plot the residual history of each solution with a line for the specified tolerance.
semilogy((0:numel(rv0)1)/2,rv0/norm(b),"o") hold on semilogy((0:numel(rv1)1)/2,rv1/norm(b),"o") yline(tol,"r"); legend("No preconditioner","ILU preconditioner","Tolerance",Location="East") xlabel("Iteration number") ylabel("Relative residual")
Input Arguments
A
— Input matrix
sparse square matrix
Input matrix, specified as a sparse square matrix.
options
— LU factorization options
structure
LU factorization options, specified as a structure with up to five fields:
Field Name  Summary  Description 

 Type of incomplete LU factorization  Valid values for
The default value is

 Drop tolerance of the incomplete LU factorization  Valid value for The default value is

 Type of modified incomplete LU factorization  Valid values for
The default value is 
 Whether to replace zero diagonal entries of U  Valid values for The default
value is 
 Pivot threshold  Valid values are between The default value is

Output Arguments
L
— Lower triangular factor
sparse square matrix
Lower triangular factor, returned as a sparse square matrix.
When
options.type
is specified as"nofill"
(by default) or"crout"
,L
is returned as a unit lower triangular matrix (that is, a lower triangular matrix with 1s on the main diagonal).When
options.type
is specified as"ilutp"
andoptions.milu
is specified as"col"
,L
is returned as a rowpermuted unit lower triangular matrix.
U
— Upper triangular factor
sparse square matrix
Upper triangular factor, returned as a sparse square matrix.
When
options.type
is specified as"nofill"
(by default) or"crout"
,U
is returned as an upper triangular matrix.When
options.type
is specified as"ilutp"
andoptions.milu
is specified as"row"
,U
is returned as a columnpermuted upper triangular matrix.
P
— Permutation matrix
sparse square matrix
Permutation matrix, returned as a sparse square matrix.
When options.type
is specified as "nofill"
(by
default) or "crout"
, P
is returned as an identity
matrix of the same size as A
because neither of these types performs
pivoting.
W
— Nonzeros of the LU factors
sparse square matrix
Nonzeros of the LU factors, returned as a sparse square matrix, where W = L
+ U  speye(size(A))
.
Tips
The incomplete factorizations returned by this function may be useful as preconditioners for a system of linear equations being solved by iterative methods, such as
bicg
,bicgstab
, orgmres
.
References
[1] Saad, Y. "Preconditioning Techniques", chap. 10 in Iterative Methods for Sparse Linear Systems. The PWS Series in Computer Science. Boston: PWS Pub. Co, 1996.
Extended Capabilities
ThreadBased Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports threadbased environments. For more information, see Run MATLAB Functions in ThreadBased Environment.
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
If you include the field
type
in the structure arrayoptions
, then you must set it to'nofill'
.
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced in R2007a
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