Ludwig Elsner

Models of parallel chaotic iteration methods

Abstract We consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b. In the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed. In the second model any processor can update the global approximation which resides in the central processor at any time. This model is a generalization of a sequential iterative scheme due to Ostrowski called the free steering group Jacobi iterative scheme and a chaotic relaxation point iterative scheme due to Chazan and Miranker. We show that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.

The generalized spectral-radius theorem: An analytic-geometric proof

Abstract Let ∑ be a bounded set of complex matrices, ∑ m = {A 1 ... A m : A i ∈ ∑} . The generalized spectral-radius theorem states that ϱ(∑) =ρ^(∑) , where ϱ(∑) and ρ^(σ) are defined as follows: ϱ{∑) =lim sup ϱ m (∑){1/m} , where ϱ m (∑) =sup{ϱ(A): A ∈ ∑ m } with ϱ ( A ) the spectral radius; ρ^(∑) =lim sup ρ^ m (∑){1/m} , where ρ^ m (∑) =sup{‖A‖: A ∈ ∑ m } with ‖ ‖ any matrix norm. We give an elementary proof, based on analytic and geometric tools, which is in some ways simpler than the first proof by Berger and Wang.

Comparisons of weak regular splittings and multisplitting methods

SummaryComparison results for weak regular splittings of monotone matrices are derived. As an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings. This yields a very simple proof of results of Neumann-Plemmons on upper bounds, and establishes lower bounds, which has in special cases been conjectured by these authors.

Convergence of algorithms of decomposition type for the eigenvalue problem

We develop the theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR,LR,SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm performs nested subspace iteration with a change of coordinate system at each step. Thus the convergence of the GR algorithm depends on the convergence of certain sequences of subspaces. It also depends on the quality of the coordinate transformation matrices, as measured by their condition numbers. We show that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate. For matrices possessing certain special types of structure, cubic convergence can be achieved.

On the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data

Abstract Convergence of iterative processes in C k of the form x i+r i =α j i x 1+r i -1 +(1-α)P j i x i , where j i ϵ{1,2,…, n }, i = 1,2,…, is analyzed. It is shown that if the matrices P 1 ,…, P n are paracontracting in the same smooth, strictly convex norm and if the sequence { j i } ∞ i = 1 has certain regularity properties, then the above iterates converge. This result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique (ART) algorithm often used in tomographic reconstruction from incomplete data.

Convergence of sequential and asynchronous nonlinear paracontractions

SummaryWe establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.

Schur parameter pencils for the solution of the unitary eigenproblem

Abstract Let U − λV be an n × n pencil with unitary matrices U and V . An algorithm is presented which reduces U and V simultaneously to unitary block diagonal matrices G o = Q H UP and G e = Q H VP with block size at most two. It is an O ( n 3 ) process using Householder eliminations, and it is backward stable. In the special case V = I the block diagonal matrices G o , G H e can be normalized so that their entries are just the Schur parameters of the Hessenberg condensed form of U . We call G o − λG e a Schur parameter pencil. It can also be derived from U,V by a Lanczos-like process. For the solution of the eigenvalue problem for G o − λG e a QR -type algorithm can be developed based on this unitary reduction of a pencil U − λV to a Schur parameter pencil. The condensed form is preserved throughout the process. Each iteration step needs only O ( n ) operations. This method of solving the unitary eigenvalue problem seems to be the closest possible analogy to the QR method for the Hermitian eigenvalue problem.

NORMAL MATRICES : AN UPDATE

A list of seventy conditions on an n x n complex matrix A, equivalent to its being normal, published nearly ten years ago by Grone, Johnson, Sa, and Wolkowicz has proved to be very useful. Hoping that, in an extended form, it will be even more helpful, we compile here another list of about twenty conditions. They either have been overlooked by the authors. of the original list or have appeared during the last decade

On the power method in max algebra

Abstract The eigenvalue problem for an irreducible nonnegative matrix

Max-algebra and pairwise comparison matrices

A = [a_{ij}]