Efficient accelero-summation of holonomic functions

We present a SageMath implementation of the symbolic-numeric algorithm introduced by van der Hoeven in 2007 for factoring linear differential operators whose coefficients are rational functions.

In the paper we deal with the Heun functions --- solutions of the Heun\nequation, which is the most general Fuchsian equation of second order with four\nregular singular points. Despite the increasing interest to the equation and\nnumerous applications of the functions in a wide variety of physical problems,\nit is only Maple amidst known software packages which is able to evaluate the\nHeun functions numerically. But the Maple routine is known to be imperfect:\neven at regular points it may return infinities or end up with no result.\nImproving the situation is difficult because the code is not publicly\navailable. The purpose of the work is to suggest and develop alternative\nalgorithms for numerical evaluation of the Heun functions. A procedure based on\npower series expansions and analytic continuation is suggested which allows us\nto avoid numerical integration of the differential equation and to achieve\nreasonable efficiency and accuracy. Results of numerical tests are given.\n

A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\textrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disk. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions through its associated Clifford algebra and study its twisted module with respect to irregular $\mathfrak{s}\mathfrak{l}_{2}$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara-type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, in particular work by Witten [51], and more generally in higher-dimensional context. Our original motivation in [24] comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections (similar to the affine Lie algebra at critical level) and as such gives a family of deformations of the Feigin–Tipunin algebra. Our present article now describes, in the smallest case, the fibres and their representation categories over irregular connections.

With an explicit expression for the Green's function, Eq. (19), we have a common method for obtaining the radial parts of the Green's functions for a broad range of physical problems which lead to the Whittaker equation (listed in part in the introduction). This analysis emphasizes the special role played by the Whittaker equation in physical applications. Since the symmetry of the Green's function — an important property — is a consequence of the self-adjoint nature of the corresponding differential operator, the Whittaker equation is apparently the most general second-order equation with one regular singularity and one irregular singularity which retains its value for inhomogeneous physical problems, i.e., for problems leading to inhomogeneous differential equations.

A large part of the modern theory of differential equations in the complex domain is concerned with regular singularities and holonomic systems. However the theory of differential equations with irregular singularities has a long history and has become very active in recent years. Substantial links of this theory to the theory of algebraic groups, commutative algebra, resurgent functions, and Galois differential methods have been discovered. This survey attempts a general introduction to some of these aspects, with emphasis on reduction theory, asymptotic analysis, Stokes phenomena, and certain moduli problems.

Computing GCDs of Polynomials over Algebraic Number Fields

The geometric phase requires the multivaluedness of solutions to Fuchsian second-order equations. The angle, or its complement, is given by half the area of a spherical triangle in the case of three singular points, or half the area of a lune in the case of two singular points. Both are fundamental regions where the automorphic function takes a value only once, and a linear-fractional transformation tessellates the plane in replicas of the fundamental region. The condition that the homologues of the poles, representing vertices, be angles places restrictions on quantum numbers which are no longer integers, for, otherwise, the phase factors would become unity. Restriction must be made to regular singular points for only then will solutions to the differential equation be rational functions so that the covering group will be cyclic and the covering space be a ‘spiral staircase’. Many of the equations of mathematical physics, with essential singularities, become Fuchsian differential equations, with regular singularities, at zero kinetic energy. Examples of geometric phase include the phasor, the Pancharatnam phase of beams of polarized light in different states, the Aharanov-Bohm phase, and angular momentum with centripetal ‘attraction’. In the latter example, the phase is one-half the area of the lune, which disappears when the pole at infinity becomes an essential singularity thereby recovering the Schrodinger equation. The behavior of an automorphic function at a limit point on the boundary is analogous to the confluence of two regular singularities in a linear second-order differential equation to produce an essential singularity at infinity. ∗ info@bernardhlavenda.com; www.bernardhlavenda.com 1 ar X iv :1 41 1. 56 03 v1 [ ph ys ic s. ge nph ] 1 5 M ay 2 01 3

On Computations in Kummer Extensions

The compatible expansion in series of solutions of both the equations of P–Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P–Q pair are derived.

<p style='text-indent:20px;'>We study second order linear differential equations with analytic coefficients. One important case is when the equation admits a so called regular singular point. In this case we address some untouched and some new aspects of Frobenius methods. For instance, we address the problem of finding formal solutions and studying their convergence. A characterization of regular singularities is given in terms of the space of solutions. An analytic-geometric classification of such linear polynomial homogeneous ODEs is obtained by the use of techniques from geometric theory of foliations means. This is done by associating to such an ODE a rational Riccati differential equation and therefore a <i>global holonomy group</i>. This group is a computable group of Moebius maps. These techniques apply to classical equations as Bessel and Legendre equations. We also address the problem of deciding which such polynomial equations admit a Liouvillian solution. A normal form for such a solution is then obtained. Our results are concrete and (computationally) constructive and are aimed to shed a new light in this important subject.</p>

In this section, we study the behavior of solutions of a linear Pfaffian system in a neighborhood of a singular point of the coefficients, where we restrict singular points to ones corresponding to regular singular points in ordinary differential equations. Note that a singular point of analytic functions in several variables is not a point but a set of codimension one (a hypersurface).

We present a novel representation for generalized hypergeometric functions of type p+1Fp which is in fact defined by an infinite series in nonnegative integer powers of its argument. We first construct a first order vector differential equation such that the unknown vector's coefficient is the sum of a constant matrix and a matrix premultiplied by the reciprocal of the independent variable whereas its first order derivative has unit matrix coefficient. An infinite process of factor extractions and power annihilations is employed yielding finally a vector differential equation that can be easily and analytically solved. Truncation of this scheme can be used to get approximations to hypergeometric functions of type p+1Fp. These functions have regular singularities at 0 and 1 values of the independent variable together with another regular singularity at infinity. Hence the factors are chosen to reflect the expected behavior of the functions at the singular point in a descending contribution order. Factorization is realized also for regular points. A simple, yet meaningful, implementation seems to give quite promising results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

In this work, we prove that the sparse matrix resulting from a finite-element-based analysis of electrodynamic problems can be represented by an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> matrix without any approximation, and the inverse of this sparse matrix has a data-sparse <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -matrix approximation with error well controlled. Two proofs are developed. One is based on the general eigenvalue-based solution to the ordinary differential equations, and the other is based on the relationship between a partial differential operator and an integral operator. Both proofs have reached the same conclusion. Based on the proof, we develop an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -matrix-based direct finite-element solver of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">kN</i> log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) memory complexity and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> ) time complexity for solving electromagnetic problems, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> is a small parameter that is adaptively determined based on accuracy requirements, and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is the number of unknowns. Both inverse-based and LU-based direct solutions are developed. The LU-based solution is further accelerated by nested dissection. A comparison with the state-of-the-art direct finite element solver that employs the most advanced sparse matrix solution has shown clear advantages of the proposed direct solver. In addition, the proposed solver is applicable to arbitrarily-shaped three-dimensional structures and arbitrary inhomogeneity.

We present a survey of our work about applications of symbol calculus to quantum mechanics. Symbol calculus embodies the rules of algebra for the linear partial differential operators with variable coefficients, when such operators are expressed as functions of the non-commuting quantities xj and ~/~xj (j=l ..... n). In quantum mechanics the symbol calculus generates S~-C~sica£ expansions, i.e. perturbative series in powers of Planck's constant ~, that are both numerically useful and mathematically interesting. As the complexity of such expansions increases very rapidly with the order, algebraic computer systems have proven most useful in their study [l] . We present here a few of our results, obtained with the extensive use of the formula manipulation language REDUCE (except when specified) : in particular we have found new algorithms to solve the stationary SchrBdinger equation formally in powers of ~. Basic facts about quantum mechanics and semi-classical methods are found in [3] any textbook such as . More precise references, about symbol calculus for instance, are listed in our earlier articles [4j.~ ~

We consider dynamic evaluation of algebraic functions (matrix multiplication, determinant, convolution, Fourier transform, etc.) in the model of Reif and Tate; i.e., if f(x1, ..., xn) = (y1, ..., ym) is an algebraic problem, we consider serving on-line requests of the form change input xi to value v or what is the value of output yi?. We present techniques for showing lower bounds on the worst case time complexity per operation for such problems. The first gives lower bounds in a wide range of rather powerful models (for instance history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil). Using this technique, we show optimal Ω(n) bounds for dynamic matrix-vector product, dynamic matrix multiplication and dynamic discriminant and an Ω(√n) lower bound for dynamic polynomial multiplication (convolution), providing a good match with Reif and Tate's O(√n log n) upper bound. We also show linear lower bounds for dynamic determinant, matrix adjoint and matrix inverse and an Ω(√n) lower bound for the elementary symmetric functions. The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model. The third technique gives lower bounds in the weaker straight line program model. Using this technique, we show an Ω((log n)2/log log n) lower bound for dynamic discrete Fourier transform. Technical ingredients of our techniques are the incompressibility technique of Ben-Amram and Galil and the lower bound for depth-two superconcentrators of Radhakrishnan and Ta-Shma. The incompressibility technique is extended to arithmetic computation in arbitrary fields. Due to the space constraints imposed by these proceedings, in this version of the paper we only present the third technique, proving the lower bound for dynamic discrete Fourier transform and refer to the full version of the paper which is currently available as a BRICS technical report, for the rest of the proofs.