Affine arithmetic is a well-known tool to reduce the wrapping effect of ordinary interval arithmetic. We discuss several improvements both in theory and in terms of practical implementation. In particular details of INTLAB's affine arithmetic toolbox are presented. Computational examples demonstrate advantages and weaknesses of the approach.
Standard error estimates in numerical linear algebra are often of the form γk|R||S| where R,S are known matrices and γk:=ku/(1-u) with u denoting the relative rounding error unit. Recently we showed that for a number of standard problems γk can be replaced by ku for any order of computation and without restriction on the dimension. Such problems include LU- and Cholesky decomposition, triangular system solving by substitution, matrix multiplication and more. The theoretical bound implies a practically computable bound by estimating the error in the floating-point computation of ku|R||S|. Standard techniques, however, imply again a restriction on the dimension. In this note we derive simple computable bounds being valid without restriction on the dimension. As the bounds are mathematically rigorous, they may serve in computer assisted proofs.
This paper is concerned with real interval arithmetic. We focus on interval matrix multiplication. Well-known algorithms for this purpose require the evaluation of several point matrix products to compute one interval matrix product. In order to save computing time we propose a method that modifies such known algorithm by partially using low-precision floating-point arithmetic. The modified algorithms work without significant loss of tightness of the computed interval matrix product but are about 30% faster than their corresponding original versions. The negligible loss of accuracy is rigorously estimated.
Improved componentwise error bounds for approximate solutions of linear systems are derived in the case where the coefficient of a given linear system is an H-matrix. One of the error bounds presented in this paper proves to be tighter than the existing error bound, which is effective especially for ill-conditioned cases. Numerical experiments are performed to illustrate the effect of the improvements.
Matrix factorizations such as LU, Cholesky and others are widely used for solving linear systems. In particular, the diagonal pivoting method can be applied to symmetric and indefinite matrices. Floating-point arithmetic is extensively used for this purpose. Since finite precision numbers are treated, rounding errors are involved in computed results. In this paper rigorous backward error bounds for 2×2 linear systems which arise in the factorization process of the diagonal pivoting method are given. These bounds are much better than previously known ones.
We consider convergence and a posteriori error estimates of the classical Jacobi method for solving symmetric eigenvalue problems. The famous convergence proof of the classical Jacobi method consists of two phases. First, it is shown that all the off-diagonal elements converge to zero. Then, from a perturbation theorem, Parlett or Wilkinson shows convergence of the diagonal elements in the textbooks. Ciarlet also gives another convergence proof based on a discussion about a bounded sequence corresponding to a diagonal element. In this paper, we simplify the Ciarlet's convergence proof. Our proof does not use any perturbation theory. Moreover, employing this approach, we obtain a posteriori error estimates for eigenvectors.
We consider numerical verification methods for existence of a closed orbit in a dynamical system which is described by ODEs. Besides Zgliczynski's method using Poincaré map, the authors proposed a method of verification for closed orbits and their time period. In this paper, we derive a relationship between our method and one of bordering methods which gives some explanation of superiority of this bordering.
We present a three term recurrence relation of orthogonal polynomials for H01 and H02 spaces. If there were explicit formulas for such orthogonal polynomials, they would be very convenient for various kinds of practical computations. As long as such formulas are not available recurrence relations may effictively substitute them. As far as the authors know, these recurrence relations of orthogonal polynomials for H01 and H02 were not known up to now. In this paper, the three term recurrence relations of these orthogonal polynomials were derived by using properties of Legendre polynomials. Our results will enable us to simplify computational procedures in the spectral approximation methods.
This paper proposes analytical expressions of end-to-end throughput for IEEE 802.11e Enhanced Distributed Channel Access (EDCA) wireless string-topology multi-hop networks. For obtaining the IEEE 802.11e EDCA performance, internal collisions between Access Categories (ACs) in a node, frame collisions with external nodes, and frame-existence probabilities of buffers at each AC are expressed as functions of EDCA access parameters. Therefore, it is possible to obtain the effects of the EDCA access parameters to Quality of Service (QoS) support in the EDCA. It is possible to obtain the end-to-end throughput at any offered load with respect to each AC because the buffer states can be expressed according to ACs. The obtained analytical expressions are verified by showing the quantitative agreements with simulation results.
This study investigates an invariant three-torus (IT3) and related quasi-periodic bifurcations generated in a three-coupled delayed logistic map. The IT3 generated in this map corresponds to a four-dimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasi-periodic oscillations, we demonstrate that it is necessary that the number of iterations deleted as a transient state is large and similar to the number of iterations to be averaged as a stationary state. For example, it is necessary to delete 10,000,000 transient iterations to appropriately evaluate the Lyapunov exponents if they are averaged for 10,000,000 stationary state iterations in this higher-dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by analyzing a graph of the Lyapunov exponents, we show that the local bifurcation transition from an invariant two-torus(IT2) to an IT3 is caused by a Neimark-Sacker bifurcation, which can be a one-dimension-higher quasi-periodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT2 to an IT3 can be generated by a saddle-node bifurcation, which can be denoted as a one-dimension-higher quasi-periodic saddle-node bifurcation. Furthermore, we find a codimension-two bifurcation point at which the curves of a QH bifurcation and a quasi-periodic saddle-node bifurcation intersect.
In this paper, we give a characterization of permutation polynomials modulo 2k: the period of sequences generated by iterating such polynomials is a power of two. Chebyshev polynomials are examples of such polynomials. Based on these findings, we also evaluate the factual key space of a recently proposed key exchange protocol based on Chebyshev polynomials modulo 2k.