Gauss–Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss–Legendre weights is derived. Together, these two expansions provide a practical ...

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the n-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral meth...

A very general beam solution of the paraxial wave equation in circular cylindrical coordinates is presented. We call such a field a circular beam (CiB). The complex amplitude of the CiB is described by either the Whittaker functions or the confluent hypergeometric functions and is characterized by three parameters that are complex in the most general situation. The propagation through complex A...

We present an efficient quantum algorithm for estimating Gauss sums over finite fields and finite rings. This is a natural problem as the description of a Gauss sum can be done without reference to a black box function. With a reduction from the discrete logarithm problem to Gauss sum estimation we also give evidence that this problem is hard for classical algorithms. The workings of the quantu...

A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. We describe Wirtinger presentations realizable by virtual link groups.

We investigate the use of partial Gauss elimination as a preprocessing step to parity game solving. Gauss elimination is a well-known technique for solving Boolean equation systems. We present an Θ(V ) time algorithm for exhaustive partial Gauss elimination on parity games. Experimental validation shows that this approach is probably not feasible for speeding up the solving of real-world parity...

Gauss-Seidel method is often the smoother of choice within multigrid applications. In the context of unstructured meshes, however, maintaining good parallel efficiency is difficult with multiplicative iterative methods such as Gauss-Seidel. This leads us to consider alternative smoothers. We discuss the computational advantages of polynomial smoothers within parallel multigrid algorithms for po...

We present an algorithm for computing Gauss sums over Fq for large prime powers q. This allows us to find the exact values of Gauss sums in many previously intractable cases. The efficient computation of such Gauss sums up to multiplication with a root of unity is achieved by using Stickelberger’s factorization of Gauss sums, an application of the Fincke-Pohst algorithm, and fast arithmetic for...

We give an estimate of the Gauss curvature for minimal surfaces in Rm whose Gauss map omits more than m(m + 1)/2 hyperplanes in P(C).

Introduction In this paper we present a modification of Gauss-Huard's method for solving dense linear systems that allows an efficient implementation on machines with a hierarchical memory structure. GaussHuard's method resembles Gauss-Jordan's method in the fact that it reduces the given system by elementary transformations to a diagonal system and it resembles regular Gaussian elimination in ...