64 research outputs found
Systems of Markov type functions: normality and convergence of Hermite-Padé approximants
This thesis deals with Hermite-Padé approximation of analytic and merophorphic
functions. As such it is embeded in the theory of vector rational approximation of
analytic functions which in turn is intimately connectd with the theory of multiple
orthogonal polynomials. All the basic concepts and results used in this thesis involving
complex analysis and measure theory may found in classical textbooks...........Programa Oficial de Doctorado en IngenierÃa MatemáticaPresidente: Francisco José Marcellán Español; Vocal: Alexander Ivanovich Aptekarev; Secretario: Andrei MartÃnez Finkelshtei
On the convergence of type I Hermite-Pade approximants
Fade approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite-Pade approximants. The convergence properties of type II Hermite-Pade approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite-Pade approximants for Nikishin systems of functions.Both authors were partially supported by research grant MTM2012-36372-C03-01 of Ministerio de EconomÃa y Competitividad, Spain
Convergence and Asymptotic of Multi-Level Hermite-Padé Polynomials
Mención Internacional en el tÃtulo de doctorPrograma de Doctorado en IngenierÃa Matemática por la Universidad Carlos III de MadridPresidente: Francisco José Marcellán Español.- Secretario: Bernardo de la Calle Ysern.- Vocal: Arnoldus Bernardus Jacobus Kuijla
Electrostatic Partners and Zeros of Orthogonal and Multiple Orthogonal Polynomials
For a given polynomial P with simple zeros, and a given semiclassical weight w, we present a construction that yields a linear second-order differential equation (ODE), and in consequence, an electrostatic model for zeros of P. The coefficients of this ODE are written in terms of a dual polynomial that we call the electrostatic partner of P. This construction is absolutely general and can be carried out for any polynomial with simple zeros and any semiclassical weight on the complex plane. An additional assumption of quasi-orthogonality of P with respect towallows us to give more precise bounds on the degree of the electrostatic partner. In the case of orthogonal and quasiorthogonal polynomials, we recover some of the known results and generalize others.
Additionally, for the Hermite–Padé or multiple orthogonal polynomials of type II, this approach yields a system of linear second-order differential equations, from which we derive an electrostatic interpretation of their zeros in terms of a vector equilibrium. More detailed results are obtained in the special cases of Angelesco, Nikishin, and generalized Nikishin systems.We also discuss the discrete-to-continuous transition of thesemodels in the asymptotic regime, as the number of zeros tends to infinity, into the known vector equilibrium problems. Finally, we discuss how the system of obtained second-order ODEs yields a third-order differential equation for these polynomials, well described in the literature. We finish the paper by presenting several illustrative examples.The first author was partially supported by Simons Foundation Collaboration Grants
for Mathematicians (grant 710499). He also acknowledges the support of the Spanish Government and the
European RegionalDevelopment Fund (ERDF) through grant PID2021-124472NB-I00, Junta deAndalucÃa
(research group FQM-229 and Instituto Interuniversitario Carlos I de FÃsica Teórica y Computacional),
and by the University of AlmerÃa (Campus de Excelencia Internacional del Mar CEIMAR) in the early
stages of this project. The second and third authors were partially supported by Spanish Ministerio de
Ciencia, Innovación y Universidades, under grant MTM2015-71352-P. The third author was additionally
supported by Junta de AndalucÃa (research group FQM-384), the University of Granada (Research
Project ERDF-UGR A-FQM-246-UGR20), and by the IMAG-Maria de Maeztu grant CEX2020-001105-
M/AEI/10.13039/501100011033.
Funding for open access publishing: Universidad de Granada/CBU
Approximating Mills ratio
Consider the Mills ratio f(x) =1 − Φ(x)/φ(x), x ≥ 0, where φ is the density function of the standard Gaussian law and Φ its cumulative distribution. We introduce a general procedure to approximate f on the whole [0, ∞) which allows to prove interesting properties where f is involved. As applications we present a new proof that 1/f is strictly convex, and we give new sharp bounds of f involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian Q-function are studied
Series Representations and Approximation of some Quantile Functions appearing in Finance
It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort
Arbitrarily Accurate Analytical Approximations for the Error Function
In this paper a spline based integral approximation is utilized to propose a
sequence of approximations to the error function that converge at a
significantly faster manner than the default Taylor series. The approximations
can be improved by utilizing the approximation erf(x) approximately equal to
one for x>>1. Two generalizations are possible, the first is based on
demarcating the integration interval into m equally spaced sub-intervals. The
second, it based on utilizing a larger fixed sub-interval, with a known
integral, and a smaller sub-interval whose integral is to be approximated. Both
generalizations lead to significantly improved accuracy. Further, the initial
approximations, and the approximations arising from the first generalization,
can be utilized as the inputs to a custom dynamical system to establish
approximations with better convergence properties. Indicative results include
those of a fourth order approximation, based on four sub-intervals, which leads
to a relative error bound of 1.43 x 10-7 over the positive real line. Various
approximations, that achieve the set relative error bounds of 10-4, 10-6, 10-10
and 10-16, over the positive real, are specified. Applications include, first,
the definition of functions that are upper and lower bounds, of arbitrary
accuracy, for the error function. Second, new series for the error function.
Third, new sequences of approximations for exp(-x2) which have significantly
higher convergence properties that a Taylor series approximation. Fourth, the
definition of a complementary demarcation function eC(x) which satisfies the
constraint eC(x)^2 + erf(x)^2 = 1. Fifth, arbitrarily accurate approximations
for the power and harmonic distortion for a sinusoidal signal subject to a
error function nonlinearity. Sixth, approximate expressions for the linear
filtering of a step signal that is modelled by the error function
Exponential integrators: tensor structured problems and applications
The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed
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