We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [−1, 1] with weight function We show that computing higher-order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available. * alfredo.deanho@uc3m.es † daan.huybrechs@cs.kuleuven.be ‡ peter.opsomer@cs.kuleuven.be (corresponding author) 1 arXiv:1502.07191v4 [cs.MS] 22 Oct 2015The main advantage of these expansions is that they lead to increasing accuracy for increasing degree of the polynomials, at a computational cost that is actually independent of the degree. In contrast, the typical use of the recurrence relation for orthogonal polynomials in computations leads to a cost that is at least linear in the degree. Furthermore, the expansions may be used to compute Gaussian quadrature rules in O(n) operations, rather than O(n 2 ) based on the recurrence relation.

Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval [0, ∞) with respect to a weight function of the formThe classical Laguerre polynomials correspond to Q(x) = x. The computation of higher-order terms of the asymptotic expansions of these polynomials for large degree becomes quite complicated, and a full description seems to be lacking in literature. However, this information is implicitly available in the work of Vanlessen [28], based on a non-linear steepest descent analysis of an associated so-called Riemann-Hilbert problem. We will extend this work and show how to efficiently compute an arbitrary number of higherorder terms in the asymptotic expansions of Laguerre and Laguerre-type polynomials. This effort is similar to the case of Jacobi and Jacobi-type polynomials in a previous paper. We supply an implementation with explicit expansions in four different regions of the complex plane. These expansions can also be extended to Hermite-type weights of the form exp(− m k=0 q k x 2k ) on (−∞, ∞), and to general non-polynomial functions Q(x) using contour integrals. The expansions may be used, e.g., to compute Gauss-Laguerre quadrature rules in a lower computational complexity than based on the recurrence relation, and with improved accuracy for large degree. They are also of interest in random matrix theory.The methodology of [28] is based on the non-linear steepest descent method by Deift and Zhou [9] for a 2 × 2 Riemann-Hilbert problem that is generically associated with orthogonal polynomials by Fokas, Its and Kitaev [10]. This is further detailed in § 2.1. The general strategy is to apply a sequence of transformations, such that the final matrix-valued function R(z) is asymptotically close to the identity matrix as n or z tends to ∞. The asymptotic result for Y (z), and subsequently for the polynomials, is obtained by inverting these transformations. The transformations involve a normalization of behaviour at infinity, the so-called 'opening of a lens' around the interval of orthogonality, and the introduction of local parametrices in disks around special points like endpoints, which are matched to global parametrices elsewhere in the complex plane. These transformations split C into different regions, where different formulas for the asymptotics are valid.In our case of Laguerre-type polynomials, one also first needs an n-dependent rescaling of the x axis using the so-called MRS numbers (defined further on in this paper). After this step the roots of the rescaled polynomials accumulate in a fixed and finite interval. We provide an algorithm to obtain an arbitrary number of terms in the expansions, where we set up series expansions using many convolutions that follow the chain of transformations and their inverses in this steepest descent method. While doing this, we keep computational efficiency in mind, as well as the use of the correct branch cuts in the complex plane.The strategy outlined above and in § 2.1 was also followed in our earlier article about asymptotic expansions...

Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green's function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green's function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the sparsifying modification of the Green's function with other accelerating schemes, such as the fast multipole method, appears possible in principle and is a future research topic. * daan.huybrechs@cs.kuleuven.be † peter.opsomer@cs.kuleuven.be arXiv:1606.09178v4 [math.NA]

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