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Franck Wielonsky - One of the best experts on this subject based on the ideXlab platform.
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type ii hermite pade approximation to the Exponential Function
Journal of Computational and Applied Mathematics, 2007Co-Authors: Arno B J Kuijlaars, Herbert Stahl, W Van Assche, Franck WielonskyAbstract:We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials a(3nz), b(3nz), and c(3nz) where a, b, and c are the type II Hermite-Pade approximants to the Exponential Function of respective degrees 2n+2, 2n and 2n, defined by a(z)e^-^z-b(z)=O(z^3^n^+^2) and a(z)e^z-c(z)=O(z^3^n^+^2) as z->0. Our analysis relies on a characterization of these polynomials in terms of a 3x3 matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Pade approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.
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quadratic hermite pade approximation to the Exponential Function a riemann hilbert approach
Constructive Approximation, 2005Co-Authors: Arno B J Kuijlaars, W Van Assche, Franck WielonskyAbstract:We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite–Pade approximation to the Exponential Function, defined by p(z)e-z + q(z) + r(z) ez = O(z3n+2) as z → 0. These polynomials are characterized by a Riemann–Hilbert problem for a 3 × 3 matrix valued Function. We use the Deift–Zhou steepest descent method for Riemann–Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and Functions derived from it. Our work complements the recent results of Herbert Stahl.
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quadratic hermite pade approximation to the Exponential Function a riemann hilbert approach
arXiv: Classical Analysis and ODEs, 2003Co-Authors: Arno B J Kuijlaars, W Van Assche, Franck WielonskyAbstract:We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the Exponential Function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued Function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and Functions derived from it. Our work complements recent results of Herbert Stahl.
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rational approximation to the Exponential Function with complex conjugate interpolation points
Journal of Approximation Theory, 2001Co-Authors: Franck WielonskyAbstract:In this paper, we study asymptotic properties of rational Functions that interpolate the Exponential Function. The interpolation is performed with respect to a triangular scheme of complex conjugate points lying in bounded rectangular domains included in the horizontal strip |Imz|<[email protected] Moreover, the height of these domains cannot exceed some upper bound which depends on the type of rational Functions. We obtain different convergence results and precise estimates for the error Function in compact sets of C that generalize the classical properties of Pade approximants to the Exponential Function. The proofs rely on, among others, Walsh's theorem on the location of the zeros of linear combinations of derivatives of a polynomial and on Rolle's theorem for real Exponential polynomials in the complex domain.
Alexandru Zaharescu - One of the best experts on this subject based on the ideXlab platform.
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diophantine approximation of the Exponential Function and sondowʼs conjecture
Advances in Mathematics, 2013Co-Authors: Bruce C Berndt, Sun Kim, Alexandru ZaharescuAbstract:Abstract We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of e 2 / a published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the convergents of e and prove a conjecture of J. Sondow that there are only two instances when the convergents of the continued fraction of e coalesce with partial sums of e. We conclude with open questions about the zeros of certain p-adic Functions naturally occurring in our proofs.
Gavin C Cawley - One of the best experts on this subject based on the ideXlab platform.
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on a fast compact approximation of the Exponential Function
Neural Computation, 2000Co-Authors: Gavin C CawleyAbstract:Recently Schraudolph (1999) described an ingenious, fast, and compact approximation of the Exponential Function through manipulation of the components of a standard (IEEE-754 (IEEE, 1985)) floating-point representation. This brief note communicates a recoding of this procedure that overcomes some of the limitations of the original macro at little or no additional computational expense.
Nicol N Schraudolph - One of the best experts on this subject based on the ideXlab platform.
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a fast compact approximation of the Exponential Function
Neural Computation, 1999Co-Authors: Nicol N SchraudolphAbstract:Neural network simulations often spend a large proportion of their time computing Exponential Functions. Since the exponentiation routines of typical math libraries are rather slow, their replacement with a fast approximation can greatly reduce the overall computation time. This paper describes how exponentiation can be approximated by manipulating the components of a standard (IEEE-754) floating-point representation. This models the Exponential Function as well as a lookup table with linear interpolation, but is significantly faster and more compact.
Stefan M Stefanov - One of the best experts on this subject based on the ideXlab platform.
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minimization of a convex separable Exponential Function subject to linear equality constraint and box constraints
2013Co-Authors: Stefan M StefanovAbstract:In this paper, we consider the problem of minimizing a convex separable Exponential Function over a feasible region defined by a linear equality constraint and box constraints (bounds on the variables). Problems of this form are interesting from both theoretical and practical point of view because they arise in some mathematical optimization problems as well as in various practical applications. Algorithms of polynomial computational complexity are proposed for solving problems of this form and their convergence is proved. Some examples and results of numerical experiments are also presented.
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minimizing a convex separable Exponential Function subject to linear equality constraint and bounded variables
Journal of Interdisciplinary Mathematics, 2006Co-Authors: Stefan M StefanovAbstract:Abstract In this paper, we consider the problem of minimizing a convex separable Exponential Function over a region defined by a linear equality constraint and bounds on the variables. Such problems are interesting from both theoretical and practical point of view because they arise in some mathematical programming problems as well as in various practical problems. Polynomial algorithms are proposed for solving problems of this form and their convergence is proved. Some examples and results of numerical experiments are also presented.
