Some numerical results on best uniform rational approximation ofx α on [0,1]

Varga, Richard S.; Carpenter, Amos J.

doi:10.1007/bf02145384

Cited by 31 publications

(45 citation statements)

References 9 publications

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“…In Table , in parentheses, we show the computed values from the asymptotic formula . These and other computations (see, e.g., tables 2.1–2.7 in the work of Varga et al,) show that asymptotic formula is quite accurate and that the relation could be used for fairly low k .…”

Section: Solution Strategysupporting

confidence: 56%

“…Among various classes of best rational approximations, the diagonal sequences

r \in scriptR false(k, k false)

of the Walsh table of t α , 0< α <1 are studied in the greatest detail; see, for example, other works . The existence of the BURA; the distribution of the poles, zeros, and extreme points; and the asymptotic behavior of E α ( k , k ; β ) when k → ∞ are well known.…”

Section: Solution Strategymentioning

confidence: 99%

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Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Harizanov

Lazarov

Margenov

et al. 2018

Numerical Linear Algebra App

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Summary In this paper, we consider efficient algorithms for solving the algebraic equation Aαboldu=boldf, 0<α<1, where scriptA is a properly scaled symmetric and positive definite matrix obtained from finite difference or finite element approximations of second‐order elliptic problems in Rd, d=1,2,3. This solution is then written as boldu=Aβ−αboldF with boldF=A−βboldf with β positive integer. The approximate solution method we propose and study is based on the best uniform rational approximation of the function tβ−α for 0

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Section: Solution Strategysupporting

confidence: 56%

“…Among various classes of best rational approximations, the diagonal sequences

r \in scriptR false(k, k false)

Section: Solution Strategymentioning

confidence: 99%

Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Harizanov

Lazarov

Margenov

et al. 2018

Numerical Linear Algebra App

View full text Add to dashboard Cite

show abstract

“…Varga, A. Ruttan and R.S. Carpenter in [26], [29] and [27]. Starting with numerical investigations of the Bernstein conjecture, R. S. Varga has developed numerical tools that are based on the Remez algorithm, Richardson extrapolation and the use of high numbers of significant digits, which allow mathematical conjectures to be checked by numerical means (for a survey of different applications, see [25]).…”

Section: En~(x~[o 1])~no(c2(ol)ca(ct))mentioning

confidence: 99%

Best uniform rational approximation of xα on [0, 1]

Stahl

2003

Acta Math.

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“…For the case (1.8) the optimal grid was computed in the manner of [24], where optimal rational approximants of a slightly different type were computed to very high precision. For the latter case the optimal error asymptotically decays as…”

Section: −Iω √mentioning

confidence: 99%

“…This algorithm converges quadratically, provided a good initial guess is given, but otherwise it may diverge. Therefore, we implemented an extrapolation procedure in the spirit of [24] to obtain good initial iterants. The Fortran 90 multiple precision package [5] was incorporated into our Fortran program realizing the Remez method.…”

Section: Approximation On [0 1]mentioning

confidence: 99%

On Optimal Finite-Difference Approximation of PML

Asvadurov¹,

Druskin²,

Guddati³

et al. 2003

SIAM J. Numer. Anal.

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Abstract.A technique derived from two related methods suggested earlier by some of the authors for optimization of finite-difference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0, 1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finitedifference scheme with optimal grids. Results of numerical calculations are presented.Key words. absorbing boundary conditions, exponential convergence, finite differences, hyperbolic problems, perfectly matched layers, wave propagation, optimal rational approximations AMS subject classifications. 65N06, 73C02PII. S00361429013914511. Introduction. This paper is a sequel to a number of papers on so-called optimal finite-difference grids or finite-difference Gaussian rules [11,12,3,4,18,13], where exponential superconvergence of standard second order finite-difference schemes at a priori given points was obtained due to a special grid optimization procedure. This approach was successfully applied to the approximation of many nontrivial practically important problems, including elliptic PDEs for both bounded and unbounded domains. For the latter, in [18] the optimal finite-difference grid was obtained, which can be considered as the boundary condition requiring minimal arithmetic work for a given spectral interval. For hyperbolic problems, however, optimal grids were introduced only for the approximation in the interior part of the domain. Here we consider exterior hyperbolic problems. For this kind of problem a closely related method of continued fraction boundary conditions was suggested in [16], where absorbing boundary conditions were reduced to a three-term equations resembling finite-difference relations. Combining the approaches of [18] and [16] we obtain frequency independent finite-difference discretization of Berenger's perfectly matched layer (PML) absorbing boundary conditions (ABCs) [7], which produces the minimal possible impedance error for a given number of discrete layers. Similarly to the optimal grid for the Laplace equation with a solution from a Sobolev space considered in [18], the obtained discretizations show the exponential of the square root rates of convergence, though they use only the three-point stencil for second derivatives. Our solution exhibits much smaller reflection coefficients compared to examples of optimized PMLs (for the same

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Some numerical results on best uniform rational approximation ofx α on [0,1]

Cited by 31 publications

References 9 publications

Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Optimal solvers for linear systems with fractional powers of sparse SPD matrices

Best uniform rational approximation of xα on [0, 1]

On Optimal Finite-Difference Approximation of PML

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