# Incomplete LU factorization - ILU

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== Algorithm ILU == | == Algorithm ILU == | ||

- | Algorithm for computing ILU for a n by n matrix A is given by <br> | + | Algorithm for computing ILU for a '''n''' by '''n''' matrix '''A''' is given by <br> |

+ | ---- | ||

- | + | : for r:= 1 step 1 until n-1 do | |

- | + | :: d := 1/ a<sub>rr</sub> <br> | |

- | + | :: for i := (r+1) step 1 until n do <br> | |

- | + | :: if (i,r)<math>\in</math>S then <br> | |

- | + | ::: e := da<sub>i,r</sub>; <br> | |

- | + | ::: a<sub>i,r</sub> := e ; <br> | |

- | + | ::: for j := (r+1) step 1 until n do <br> | |

- | + | :::: if ( (i,j)<math>\in</math>S ) and ( (r,j)<math>\in</math>S ) then <br> | |

- | + | :::: a<sub>i,j</sub> := a<sub>i,j</sub> - e a<sub>r,j</sub> <br> | |

- | + | :::: end if <br> | |

- | + | ::: end (j-loop) <br> | |

- | + | :: end if <br> | |

- | + | :: end (i-loop) <br> | |

- | + | : end (r-loop) <br> | |

+ | ---- | ||

- | + | 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. | + | |

== Reference == | == Reference == | ||

- | ''Tony F. Chan and Hank A. Van Der Vorst'' , Approaximate and Incomplete Factorizations | + | '''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/ a
_{rr}

- for i := (r+1) step 1 until n do

- if (i,r)S then

- e := da
_{i,r};

- a
_{i,r}:= e ;

- for j := (r+1) step 1 until n do

- if ( (i,j)S ) and ( (r,j)S ) then

- a
_{i,j}:= a_{i,j}- e a_{r,j}

- end if

- if ( (i,j)S ) and ( (r,j)S ) then
- end (j-loop)

- e := da
- end if

- end (i-loop)

- d := 1/ a
- 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 *