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Summary: In this paper, we analyze the convergence of a projected fixed‐point iteration on a Riemannian manifold of matrices with fixed rank. As a retraction method, we use the projector splitting scheme. We prove that the convergence rate of the projector splitting scheme is bounded by the convergence rate of standard fixed‐point iteration without rank constraints multiplied by the function of initial approximation. We also provide counterexample to the case when conditions of the theorem do not hold. Finally, we support our theoretical results with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a shifted symmetric higher-order power method for computing the H-eigenpairs of a real symmetric even-order tensor. The local convergence of the method is proved. In addition, by utilizing the fixed-point analysis, we can characterize exactly which H-eigenpairs can be found and which cannot be found by the method. Numerical examples are presented to illustrate the performance of the method. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of weakly nonlinear complementarity problems arising from the discretization of free boundary problems. By reformulating the complementarity problems as implicit fixed-point equations based on splitting of the system matrices, we propose a class of modulus-based matrix splitting algorithms. We show their convergence by assuming that the system matrix is positive definite. Moreover, we give several kinds of typical practical choices of the modulus-based matrix splitting iteration methods based on the different splitting of the system matrix. Numerical experiments on two model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of our modulus-based matrix splitting algorithms. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We analyze inexact fixed-point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed-point iteration. Important applications are the Picard iteration and partitioned fluid-structure interaction. For the analysis, the iteration is modeled as a perturbed fixed-point iteration, and existing analysis is extended to the nested case x = F( S( x)). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY By an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations, we construct modulus-based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high-speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss-Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus-based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H + -matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with the tight enclosure of matrix multiplication AB for two floating-point matrices A and B. The aim of this paper is to compute component-wise upper and lower bounds of the exact result C of the matrix multiplication AB by floating-point arithmetic. Namely, an interval matrix enclosing C is obtained. In this paper, new algorithms for enclosing C are proposed. The proposed algorithms are designed to mainly exploit the level 3 operations in BLAS. Although the proposed algorithms take around twice as much costs as a standard algorithm promoted by Oishi and Rump, the accuracy of the result by the proposed algorithms is better than that of the standard algorithm. At the end of this paper, we present numerical examples showing the efficiency of the proposed algorithms. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]