Generating a CRS-based <IMG ALIGN=BOTTOM SRC="_22900_tex2html_wrap6389.gif">-<IMG ALIGN=BOTTOM SRC="_22900_tex2html_wrap7381.gif"> Incomplete Factorization

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### Generating a CRS-based - Incomplete Factorization

In this subsection we will consider a matrix split as in diagonal, lower and upper triangular part, and an incomplete factorization preconditioner of the form . In this way, we only need to store a diagonal matrix containing the pivots of the factorization.

Hence,it suffices to allocate for the preconditioner only a pivot array of length (pivots(1:n)). In fact, we will store the inverses of the pivots rather than the pivots themselves. This implies that during the system solution no divisions have to be performed.

Additionally, we assume that an extra integer array diag_ptr(1:n) has been allocated that contains the column (or row) indices of the diagonal elements in each row, that is, .

The factorization begins by copying the matrix diagonal

```for i = 1, n
pivots(i) = val(diag_ptr(i))
end;```
Each elimination step starts by inverting the pivot
```for i = 1, n
pivots(i) = 1 / pivots(i)```
For all nonzero elements with , we next check whether is a nonzero matrix element, since this is the only element that can cause fill with .
```    for j = diag_ptr(i)+1, row_ptr(i+1)-1
found = FALSE
for k = row_ptr(col_ind(j)), diag_ptr(col_ind(j))-1
if(col_ind(k) = i) then
found = TRUE
element = val(k)
endif
end;```
If so, we update .
```        if (found = TRUE)
pivots(col_ind(j)) = pivots(col_ind(j))
- element * pivots(i) * val(j)
end;
end;```

Jack Dongarra
Mon Nov 20 08:52:54 EST 1995