Determination of the rank of an integration lattice

The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations. In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B

Notes on lattice rules

AbstractAn elementary introduction to lattices, integration lattices and lattice rules is followed by a description of the role of the dual lattice in assessing the trigonometric degree of a lattice rule. The connection with the classical lattice-packing problem is established: any s-dimensional cubature rule can be associated with an index ρ=δs/s!N, where δ is the enhanced degree of the rule and N its abscissa count. For lattice rules, this is the packing factor of the associated dual lattice with respect to the unit s-dimensional octahedron.An individual cubature rule may be represented as a point on a plot of ρ against δ. Two of these plots are presented. They convey a clear idea of the relative cost-effectiveness of various individual rules and sequences of rules

The Number of Lattice Rules of Specified Upper Class and Rank

The upper class of a lattice rule is a convenient entity for classification and other purposes. The rank of a lattice rule is a basic characteristic, also used for classification. By introducing a rank proportionality factor and obtaining certain recurrence relations, we show how many lattice rules of each rank exist in any prime upper class. The Sylow p-decomposition may be used to obtain corresponding results for any upper class

On quadrature error expansions. Part I

AbstractWe treat the theory of numerical quadrature over a square using an m2 copy Q(m)ƒ of a one-point quadrature rule. For some integrand functions the quadrature error Q(m)ƒ − Iƒ may be expressed as an asymptotic expansion in inverse powers of m or other simple functions of m. We determine in some cases the nature of this expansion and derive integral representations for both the coefficients and the remainder term. In this part we deal only with smooth functions and those having algebraic line singularities along edges. In some of these cases the form is already known but some of the integral representations are new. These results form the basis for Part II in which new expansions, for integrands having algebraic singularities along intersecting edges and point algebraic singularities at the vertices will be presented

Spin-1/2 J1-J2 model on the body-centered cubic lattice

Using exact diagonalization (ED) and linear spin wave theory (LSWT) we study the influence of frustration and quantum fluctuations on the magnetic ordering in the ground state of the spin-1/2 J1-J2 Heisenberg antiferromagnet (J1-J2 model) on the body-centered cubic (bcc) lattice. Contrary to the J1-J2 model on the square lattice, we find for the bcc lattice that frustration and quantum fluctuations do not lead to a quantum disordered phase for strong frustration. The results of both approaches (ED, LSWT) suggest a first order transition at J2/J1 $\approx$ 0.7 from the two-sublattice Neel phase at low J2 to a collinear phase at large J2.Comment: 6.1 pages 7 figure

Higher order numerical methods for solving fractional differential equations

The final publication is available at Springer via http://dx.doi.org/10.1007/s10543-013-0443-3In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results

Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals

High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat