error bounds
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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 220
Author(s):  
Ezgi Erdoğan ◽  
Enrique A. Sánchez Pérez

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.


2022 ◽  
Author(s):  
Yingxiang Mo ◽  
Hong Chen ◽  
Yuxiang Han ◽  
Hao Deng

2021 ◽  
Vol 5 (1) ◽  
pp. 380-386
Author(s):  
Richard P. Brent ◽  

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.


Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 787-816
Author(s):  
Hongbin Lu ◽  
Weiyuan Qiu ◽  
Fei Yang

Abstract For McMullen maps f λ (z) = z p + λ/z p , where λ ∈ C \ { 0 } , it is known that if p ⩾ 3 and λ is small enough, then the Julia set J(f λ ) of f λ is a Cantor set of circles. In this paper we show that the Hausdorff dimension of J(f λ ) has the following asymptotic behavior dim H J ( f λ ) = 1 + log 2 log p + O ( | λ | 2 − 4 / p ) , as λ → 0 . An explicit error estimation of the remainder is also obtained. We also observe a ‘dimension paradox’ for the Julia set of Cantor set of circles.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Monirul Islam ◽  
Syed Shakaib Irfan

AbstractThis is the first paper dealing with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also provide some characterizations of these two sufficiency properties. We conclude the paper with a discussion of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds.


Automatica ◽  
2021 ◽  
Vol 134 ◽  
pp. 109896
Author(s):  
Emilio Tanowe Maddalena ◽  
Paul Scharnhorst ◽  
Colin N. Jones

Author(s):  
Prakash Chakraborty ◽  
Harsha Honnappa

In this paper, we establish strong embedding theorems, in the sense of the Komlós-Major-Tusnády framework, for the performance metrics of a general class of transitory queueing models of nonstationary queueing systems. The nonstationary and non-Markovian nature of these models makes the computation of performance metrics hard. The strong embeddings yield error bounds on sample path approximations by diffusion processes in the form of functional strong approximation theorems.


Author(s):  
Tobias Jawecki

AbstractPrior recent work, devoted to the study of polynomial Krylov techniques for the approximation of the action of the matrix exponential etAv, is extended to the case of associated φ-functions (which occur within the class of exponential integrators). In particular, a posteriori error bounds and estimates, based on the notion of the defect (residual) of the Krylov approximation are considered. Computable error bounds and estimates are discussed and analyzed. This includes a new error bound which favorably compares to existing error bounds in specific cases. The accuracy of various error bounds is characterized in relation to corresponding Ritz values of A. Ritz values yield properties of the spectrum of A (specific properties are known a priori, e.g., for Hermitian or skew-Hermitian matrices) in relation to the actual starting vector v and can be computed. This gives theoretical results together with criteria to quantify the achieved accuracy on the fly. For other existing error estimates, the reliability and performance are studied by similar techniques. Effects of finite precision (floating point arithmetic) are also taken into account.


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