The Experts below are selected from a list of 408 Experts worldwide ranked by ideXlab platform
Linas Vepštas - One of the best experts on this subject based on the ideXlab platform.
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AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS
2007Co-Authors: Linas VepštasAbstract:ABSTRACT. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function”[4, 5], to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general values of complex s and the region of complex z values given by ∣ ∣ z 2 /(z − 1) ∣ ∣ < 4. This region includes the the Hurwitz zeta ζ(s,q) for general complex s and real 1/4 ≤ q ≤ 3/4. By using the Duplication Formula and the inversion Formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Similarly, the Hurwitz zeta can be extended to the whole real interval 0 < q < 1. The speed of the algorithm is significantly superior to a Taylor’s series for the evaluation of the Hurwitz zeta; quadratically so when a large number of digits of precision is required. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included. 1
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AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS
2007Co-Authors: Linas VepštasAbstract:ABSTRACT. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function”[4, 5], to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general values of complex s and the region of complex z values given by ∣ ∣ z 2 /(z − 1) ∣ ∣ < 4. This region includes the the Hurwitz zeta ζ(s,q) for general complex s and real 1/4 ≤ q ≤ 3/4. By using the Duplication Formula and the inversion Formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Similarly, the Hurwitz zeta can be extended to the whole real interval 0 < q < 1. The speed of the algorithm is significantly superior to a Taylor’s series for the evaluation of the Hurwitz zeta; quadratically so when a large number of digits of precision is required. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group is included. 1
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AN EFFICIENT ALGORITHM FOR ACCELERATING THE CONVERGENCE OF OSCILLATORY SERIES, USEFUL FOR COMPUTING THE POLYLOGARITHM AND HURWITZ ZETA FUNCTIONS
2007Co-Authors: Linas VepštasAbstract:ABSTRACT. This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein’s “An efficient algorithm for computing the Riemann zeta function”[4, 5], to more general series. The algorithm provides a rapid means of evaluating Lis(z) for general values of complex s and a kidney-shaped region of complex z values given by ∣ ∣ z 2 /(z − 1) ∣ ∣ < 4. By using the Duplication Formula and the inversion Formula, the range of convergence for the polylogarithm may be extended to the entire complex z-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex s and z values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor’s series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included. 1
Robert C. Griffiths - One of the best experts on this subject based on the ideXlab platform.
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Reproducing kernel orthogonal polynomials on the multinomial distribution
Journal of Approximation Theory, 2019Co-Authors: Persi Diaconis, Robert C. GriffithsAbstract:Abstract Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new Duplication Formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The Duplication Formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
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Reproducing kernel orthogonal polynomials on the multinomial distribution
arXiv: Probability, 2018Co-Authors: Persi Diaconis, Robert C. GriffithsAbstract:Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials $Q_n(\bm{x},\bm{y};N,\bm{p})$ on the multinomial distribution which are sums of products of orthonormal polynomials in $\bm{x}$ and $\bm{y}$ of fixed total degree $n=0,1,\ldots , N$. The important Poisson Kernel for these polynomials $ \sum_{n=0}^N\rho^nQ_n(\bm{x},\bm{y};N,\bm{p}) $ arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new Duplication Formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The Duplication Formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The $\chi^2$ cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
Persi Diaconis - One of the best experts on this subject based on the ideXlab platform.
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Reproducing kernel orthogonal polynomials on the multinomial distribution
Journal of Approximation Theory, 2019Co-Authors: Persi Diaconis, Robert C. GriffithsAbstract:Abstract Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new Duplication Formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The Duplication Formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
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Reproducing kernel orthogonal polynomials on the multinomial distribution
arXiv: Probability, 2018Co-Authors: Persi Diaconis, Robert C. GriffithsAbstract:Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials $Q_n(\bm{x},\bm{y};N,\bm{p})$ on the multinomial distribution which are sums of products of orthonormal polynomials in $\bm{x}$ and $\bm{y}$ of fixed total degree $n=0,1,\ldots , N$. The important Poisson Kernel for these polynomials $ \sum_{n=0}^N\rho^nQ_n(\bm{x},\bm{y};N,\bm{p}) $ arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new Duplication Formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The Duplication Formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The $\chi^2$ cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
Griffiths Robert - One of the best experts on this subject based on the ideXlab platform.
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Reproducing kernel orthogonal polynomials on the multinomial distribution
2019Co-Authors: Diaconis Persi, Griffiths RobertAbstract:Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q_n(x,y};N,p) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n=0,1,.., N. sum_{n=0}^N rho^nQ_n(x,y);N,p) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new Duplication Formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The Duplication Formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The \chi^2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains
Tae-young Seo - One of the best experts on this subject based on the ideXlab platform.
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A PROOF OF THE LEGENDRE Duplication Formula FOR THE GAMMA FUNCTION
East Asian mathematical journal, 1998Co-Authors: In-hyok Park, Tae-young SeoAbstract:There have been various proofs of the Legendre Duplication Formula for the Gamma function. Another proof of the Formula is given here and a brief history of the Gamma function is also provided.
