A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Abstract

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive unper and lower estimates of individual eigenvalues and of __ condition number. This includes a discussion that the condition number of preconditioned second matrices is O(h-'). Some of the methods are applied to compute certain parameters involved in preconditioner.

Maximum‐weight‐basis preconditioners

“…• Algebraic eigenvalue bounds that depend on matrices arising in the incomplete-factorization process, such as References [12][13][14]. These results are not directly applicable to MWB precoditioners, which are not constructed by such a process.…”

“…• Algebraic eigenvalue-bounds for block-incomplete-factorization preconditioners that depend on the number of blocks in the matrix partitioning, such as References [13,15]. These, too, are not directly applicable to MWB preconditioners, in which there is no natural partitioning into blocks.…”

“…Early results, such as References [16][17][18], depended on these perturbation, but later the unperturbed variants were shown to be equally e ective [19][20][21]. For further details regarding spectral analysis of preconditioners, see the monograph [11], the survey [13], the papers cited above, and the papers [13,15,18,[22][23][24][25][26][27][28].…”

Optimal Preconditioners Based on Rate of Convergence Estimates for the Conjugate Gradient Method

“…The second method is based on sparse approximate inverses on factored LL T form, which type of method has received much attention in recent years (see, [18], [19], [20], [1], [2], for instance). It is reported here only for completeness and in its basic form as described originally in [1], [2] and [18].…”

“…Here it holds c ¼ cot 1 As remarked previously, any matrix ½a ij can be identified with a transformation of the triangles.…”