Efficient and Validated Numerical Evaluation of Abelian Integrals

In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

We consider perturbed pendulum-like equations on the cylinder of the form x¨+sin⁡(x)=ε∑s=0mQn,s(x)x˙s where Qn,s are trigonometric polynomials of degree n, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case ε=0 in terms of m and n. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.

Perturbations of non-Hamiltonian reversible quadratic systems with cubic orbits

An integrable system on the moduli space of rational functions and its variants

In analyzing conducting bodies composed of surfaces connected to wires via a method of moments discretization of the electric field integral equation, junction basis functions are defined at connection points between wires and surfaces to correctly capture the physics of the current flux in the vicinity of the junction. In evaluating the associated self-term vector potential surface integrals, two singularities are present: one is related to the Green's function, and the other is due to the 1/ρ behavior of the junction basis function that models the current near a junction. Here we present a simple and efficient numerical procedure for evaluating singular and near-singular source vector potential integrals involving junction basis functions. The approach is based on a double transformation that cancels both singularities and also overcomes some limitations of singularity subtraction methods. The proposed scheme is then optimized to obtain at least four correct digits in the evaluation of these integrals. Essentially arbitrary accuracy is achievable by suitably increasing the quadrature order.

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.

In order to solve Toeplitz linear systems An(f)x=b generated by a nonnegative integrable function f, through use of the preconditioned conjugate gradient (PCG) method, several authors have proposed An(g) as preconditioner in the case where g is a trigonometric polynomial [10, 14, 27, 12, 28]. In preceding works, we studied the distribution and the extremal properties of the spectrum of the preconditioned matrix G=An−1(g) An(f). In this paper we prove that the union of the spectra of all the Gn is dense on the essential range of f/g, i.e.,ER(f/g) and we obtain asymptotic information about the rate of convergence of the smallest eigenvalue λln of Gn to r (and of λnn to R). As a consequence of this second order result, it is possible to handle the case where f has zeros of any order θ, through the PCG methods proposed in [10, 14]. This is a noteworthy extension since the techniques developed in [10, 14, 27, 12, 28] are shown to be effective only when f has zeros of even orders. The cost of this procedure is O(n1+c(θ) log n) arithmetic operations (ops) where the quantity c(θ) belongs to interval [0,2−1] and takes the maximum value 2−1 when f has a zero of odd order. Finally, for the special case of zeros of odd orders, we propose a further algorithm which makes use of the PCG techniques proposed in [10, 14, 27, 12, 28] for theeven order case, reducing the cost to O(n long n) ops.

In this paper, we present a numerical method for rigorously finding the monodromy of linear differential equations. Beginning at a base point where certain particular solutions are explicitly given by series expansions, we first compute the value of fundamental system of solutions using interval arithmetic to rigorously control truncation and rounding errors. The solutions are then analytically continued along a prescribed contour encircling the singular points of the differential equation via a rigorous integrator. From these computations, the monodromy matrices are derived, generating the monodromy group of the differential equation. This method establishes a mathematically rigorous framework for addressing the monodromy problem in differential equations. For a notable example, we apply our computer-assisted proof method to resolve the monodromy problem for a Picard–Fuchs differential equation associated with a family of K3 toric hypersurfaces.

Szegő polynomials are orthogonal with respect to an inner product on the unit circle. Numerical methods for weighted least-squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of Szegő polynomials. This paper discusses the conditioning of several mappings involving Szegő polynomials and, thereby, sheds light on the sensitivity of some approximation problems involving trigonometric polynomials.

Supersonic flows about conical bodies

A method is described for the numerical evaluation of definite integrals over the complete real line where the integrand, as a function of a complex variable, has singularities near the real axis. The main method is concerned with the case when the singularities are poles of arbitrary order, but an extension to deal with branch singularities is briefly considered. A modification for the evaluation of Cauchy principal value integrals is also described.

Density functional theory requires precise numerical values for three-center nuclear attraction integrals, best obtained over Slater-type orbitals (STOs). Efficient evaluation of three-center nuclear attraction integrals over STOs to predetermined accuracy is made possible by applying the nonlinear D and D transformations. These methods are implemented in Fortran subroutines. This work shows how the conditions of applicability for these transformations are readily proven to be satisfied using the Axiom symbolic computation system. Axiom also provides exact values of the integrals for comparison. Their evaluation by standard numerical quadrature methods (Gauss–Laguerre), proves inadequate: Certain parameters lead to inaccurate values for the integrals, whereas the transformed integrals are highly accurate and rapidly evaluated. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 181–188, 1998

Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives

Accuracy optimal methods for evaluating hypersingular integrals

The present work concerns the development of an extremely accurate and rapid algorithm for the numerical evaluation of the three-centre nuclear attraction integrals over Slater-type functions. These integrals are numerous, they occur in many millions of terms, even for small molecules and they require rapid and accurate evaluation. The new algorithm is based on nonlinear transformation methods, on numerical quadrature and on properties of the sine and Bessel functions. The section with numerical results shows the high accuracy and the substantial gain in calculation time realized using the new algorithm. The complete expressions of the three-centre nuclear attraction integrals over B functions and over Slater-type functions are evaluated for different values of the quantum numbers to show the efficiency of the new approach. Numerical results obtained with linear and nonlinear systems and comparisons with numerical results from the literature are listed.