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Unformatted text preview: oting is performed and sparsity is not included. 52 Solution Techniques for Elliptic Problems for i = 2 to N do for k = 1 to i ; 1 do aik = aik =akk for j = k + 1 to N do aij = aij ; aik akj end for end for end for Figure 9.3.9: Gaussian elimination loop. The Gaussian elimination update in Figure 9.3.9 is aij := aij ; aik akj : Let the current level of ll of aij be denoted as levij . Then the size of updated element aij should be levij ; levik levkj = levij ; (levik +levkj ) : Roughly speaking, the size of the updated element aij will be the maximum of the two sizes levij and (levik +levkj ) . Thus, it makes sense to de ne the new level of ll as levij := min(levij levik + levkj ): It is common (cf. 4], Section 10.3) to shift all levels by -1 from this de nition. Thus, the initial level of ll of aij is set as aij 6= 0 levij := 0 ioftherwise or i = j : (9.3.33a) 1 The updated level of ll is levij := min(levij levik + levkj + 1): (9.3.33b) We see that levij cannot increase during the factorization. Thus, if aij 6= 0 in the original matrix A, levij never becomes nonzero. Equations (9.3.33) suggest a natural way of discarding elements during the incomplete factorization. With an ILU(p) algorithm, we'll keep all elements whose level of ll does not exceed p. The zero pattern of A for the ILU(p) procedure may be de ned as the set Zp(A) := f(i j )jlevij > pg: (9.3.33c) 9.3. Conjugate Gradient Methods 53 This de nition is consistent with the ILU(0) preconditioning described in Example 9.3.5. De ne levij according to (9.3.33a) for i = 2 to N do for k = 1 to i ; 1 do if levik p then aik = aik =akk for j = k + 1 to N do if levij p then aij = aij ; aik akj Update levij according to (9.3.33b) end if end for for j = k + 1 to N do if levij > p then aij = 0 levij = 1 end if end for end if end for end for Figure 9.3.10: Incomplete ILU(p) factorization. A pseudocode ILU(p) algorithm appears in Figure 9.3.10. It is intended to give the general idea of the procedure and, as stated, is not ve...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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