# Incomplete LU factorization - ILU

## Algorithm ILU

Algorithm for computing ILU for a n by n matrix A is given by

for r:= 1 step 1 until n-1 do
d := 1/ arr
for i := (r+1) step 1 until n do
if (i,r)$\in$S then
e := dai,r;
ai,r := e ;
for j := (r+1) step 1 until n do
if ( (i,j)$\in$S ) and ( (r,j)$\in$S ) then
ai,j := ai,j - e ar,j
end if
end (j-loop)
end if
end (i-loop)
end (r-loop)

Here S represents the set of elements of matrix A. The same algorithm could be applied to full matrix A.

## Reference

Tony F. Chan and Hank A. Van Der Vorst , "Approaximate and Incomplete Factorizations"