For simplicity, we adopt the following rules: i, j, m, n, k denote natural numbers, x denotes a set, K denotes a field, a, a1, a2 denote elements of K, D denotes a non empty set, d, d1, d2 denote elements of D, M , M1, M2 denote matrices over D, A, A1, A2, B1, B2 denote matrices over K, and f , g denote finite sequences of elements of N. One can prove the following propositions: (1) Let K be a ...

In this whitepaper, we describe the problem of permuting sparse square matrices into block diagonal form with overlap (BDO) and propose a graph partitioning algorithm for solving this problem. A block diagonal matrix with overlap is a block diagonal matrix whose consecutive diagonal blocks may overlap. The objective in this permutation problem is to minimize the total overlap size, whereas the ...

The solution of large-scale chemical process simulation and optimization problems using parallel computation requires algorithms that can take advantage of multiprocessing when solving the large, sparse matrices that arise. Parallel algorithms require that the matrices be partitioned in order to distribute computational work across processors. One way to accomplish this is to reorder the matrix...

In this work, the notion of extended eigenvalues a 2 ? lower triangular operator matrix has been researched. More precisely, relations between spectrum with spectrum, point and its diagonal entries have investigated. The obtained results supplemented by examples. addition, some properties block matrices displayed.

This paper considers a decentralized H2 control problem for multichannel linear time-invariant (LTI) descriptor systems. Our interest is to design a low order dynamic output feedback controller. The control problem is reduced to a feasibility problem of a bilinear matrix inequality (BMI) with respect to variables of a coefficient matrix defining the controller, a Lyapunov matrix and a matrix re...

We present a class of parallel preconditioning strategies built on a multilevel block incomplete LU (ILU) factorization technique to solve large sparse linear systems on distributed memory parallel computers. The preconditioners are constructed by using the concept of block independent sets. Two algorithms for constructing block independent sets of a distributed sparse matrix are proposed. We c...

We use i and i for the falling and rising factorials respectively and we adopt the convention a binomial coeÆcient is zero if either of its parameters is negative. Let Fn;r denote the n-by-n matrix (i 1) j i j 1 r 1 1 i;j n . Note that Fn;r is a block matrix 0 0 Gn;r 0 with Gn;r lower triangular of size n r. Let Hn;r denote the (n r)-by-(n r) lower triangular banded matrix ( 1) j) i 1 j 1 r j ...

A new parallel preconditioning algorithm for 3D nonconforming FEM elasticity systems is presented. The preconditioner is constructed in two steps. First displacement decomposition of the stiffness matrix is used. Than MIC(0) factorization is applied to a proper auxiliary M-matrix to get an approximate factorization of the obtained blockdiagonal matrix. The auxiliary matrix has a special block s...