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Incomplete LU factorization - ILU

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== Algorithm ILU ==
== Algorithm ILU ==
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Algorithm for computing LU for a n by n matrix A is given by <br>
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Algorithm for computing ILU for a '''n''' by '''n''' matrix '''A''' is given by <br>
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----
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  for r:= 1 step 1 until n-1 do
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:  for r:= 1 step 1 until n-1 do
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      d := 1/ a<sub>rr</sub> <br>
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::      d := 1/ a<sub>rr</sub> <br>
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      for i := (r+1) step 1 until n do <br>
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::      for i := (r+1) step 1 until n do <br>
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          if (i,r)<math>\in</math>S then <br>
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::          if (i,r)<math>\in</math>S then <br>
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            e := da<sub>i,r</sub>; <br>
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:::            e := da<sub>i,r</sub>; <br>
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            a<sub>i,r</sub> := e ; <br>
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:::            a<sub>i,r</sub> := e ; <br>
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            for j := (r+1) step 1 until n do <br>
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:::            for j := (r+1) step 1 until n do <br>
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              if ( (i,j)<math>\in</math>S ) and ( (r,j)<math>\in</math>S ) then <br>
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::::              if ( (i,j)<math>\in</math>S ) and ( (r,j)<math>\in</math>S ) then <br>
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                  a<sub>i,j</sub> := a<sub>i,j</sub> - e a<sub>r,j</sub> <br>
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::::                  a<sub>i,j</sub> := a<sub>i,j</sub> - e a<sub>r,j</sub> <br>
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              end if <br>
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::::              end if <br>
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            end (j-loop) <br>
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:::            end (j-loop) <br>
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          end if <br>
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::          end if <br>
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      end (i-loop) <br>
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::      end (i-loop) <br>
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  end (r-loop) <br>
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:  end (r-loop) <br>
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----
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Here S represents the set of elements of matrix A. The same algorithm could be applied to full matrix A.
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Here S represents the set of elements of matrix A.
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== Reference ==
== Reference ==
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''Tony F. Chan and Hank A. Van Der Vorst'' , Approaximate and Incomplete Factorizations
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'''Tony F. Chan and Hank A. Van Der Vorst''' , "Approaximate and Incomplete Factorizations"
 +
 
 +
 
 +
----
 +
<i> Return to [[Numerical methods | Numerical Methods]] </i>

Latest revision as of 12:38, 19 December 2008

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)\inS then
e := dai,r;
ai,r := e ;
for j := (r+1) step 1 until n do
if ( (i,j)\inS ) and ( (r,j)\inS ) 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"


Return to Numerical Methods

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