# Incomplete LU factorization - ILU

(Difference between revisions)
 Revision as of 04:39, 14 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 08:12, 14 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 1: Line 1: == Algorithm ILU == == Algorithm ILU == Algorithm for computing ILU for a n by n matrix A is given by
Algorithm for computing ILU for a n by n matrix A is given by
+ ---- - for r:= 1 step 1 until n-1 do + :  for r:= 1 step 1 until n-1 do - d := 1/ arr
+ ::      d := 1/ arr
- for i := (r+1) step 1 until n do
+ ::      for i := (r+1) step 1 until n do
- if (i,r)$\in$S then
+ ::          if (i,r)$\in$S then
- e := dai,r;
+ :::            e := dai,r;
- ai,r := e ;
+ :::            ai,r := e ;
- for j := (r+1) step 1 until n do
+ :::            for j := (r+1) step 1 until n do
- if ( (i,j)$\in$S ) and ( (r,j)$\in$S ) then
+ ::::              if ( (i,j)$\in$S ) and ( (r,j)$\in$S ) then
- ai,j := ai,j - e ar,j
+ ::::                  ai,j := ai,j - e ar,j
- end if
+ ::::              end if
- end (j-loop)
+ :::            end (j-loop)
- end if
+ ::          end if
- end (i-loop)
+ ::      end (i-loop)
- end (r-loop)
+ :  end (r-loop)
- + ---- Here S represents the set of elements of matrix A. The same algorithm could be applied to full matrix A. Here S represents the set of elements of matrix A. The same algorithm could be applied to full matrix A.

## 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