ISSN:
1572-9125
Keywords:
65F10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The behaviour of PCG methods for solving a finite difference or finite element positive definite linear systemAx=b with a (pre)conditioning matrixB=U TP−1 U (whereU is upper triangular andP=diag(U)) obtained from a modified incomplete factorization, isunpredictable in the present status of knowledge whenever the upper triangular factor is not strictly diagonally dominant and 2P −D, whereD=diag(A), is not symmetric positive definite. The origin of this rather surprising shortcoming of the theory is that all upper bounds on the associated spectral condition number κ(B −1 A) obtained so far require either the strict diagonal dominance of the upper triangular factor or the strict positive definiteness of 2P −D. It is our purpose here to improve the theory in this respect by showing that, when the triangular factors are “S/P consistently ordered”M-matrices, nonstrict diagonal dominance is generally a sufficient requirement, without additional condition on 2P −D. As a consequence, the new analysis does not require diagonal perturbations (otherwise needed to keep control of the diagonal dominance ofU or of the positive definiteness of 2P −D). Further, the bounds obtained here on κ(B −1 A) are independent of the lower spectral bound ofD −1 A meaning that quasi-singular problems can be solved at the same speed as regular ones, an unexpected result.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01932739
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