Информационная система "Конференции"

Abstracts

Computational Algebra

We consider the abstract linear nonsingular saddle point problem begin{equation} label{sysd} L_{0} z equiv left( matrix{ A & B cr B^T & 0 cr} right) left( matrix{ u cr p cr} right) = left( matrix{ f cr g cr} right) equiv F,, end{equation} where $ A $ is a symmetric, positive definite $ N_u times N_u $ matrix, and $ B $ is an $ N_u times N_p $ matrix (in the general case $ N_u geq N_p $), $f,, g , (g perp ker(B^T))$ are given and $u, , p $ are the unknowns. The problem~(ref{sysd}) can be reformulated in the following way ([1]): $$ M_{0} z equiv left( matrix{ Q^{-1}( A + beta ,B_0) & Q^{-1}B cr B^TQ^{-1}(nu,A + nu , beta ,B_0 - Q) & nu,B^TQ^{-1}B cr} right) left( matrix{ u cr p cr} right) = left( matrix{ tilde f cr tilde g cr} right) equiv tilde F,. $$ Here $ C = C^T > 0,, Q = Q^T > 0,, B_0 = B C^{-1} B^T$ are matrices, and $nu > 0, beta $ are parameters. We introduce the preconditioner $K_{0}$ with parameter $ alpha > 0$ for the operator $M_{0}$ in the form: $ K_{0} = {rm diag} { I , alpha, C} $, where $I$ is the identity matrix. The following theorem holds. noindent {bf Theorem}. {it The condition number: %in the algorithm $$ mbox{rm cond}_2 left( {K_0}^{-1} M_0 right) = suplimits_{x neq 0} dfrac{(M_0x,x)}{(K_0x,x)} $$ has the minimum value for $ beta + nu / alpha = 1,.$ }

Note. Abstracts are published in author's edition

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