An object-oriented hybrid symbolic/numerical approach for the development of finite element codes

The association of object-oriented programming and symbolic computation techniques introduces certain changes in finite element code organization. The purpose of this approach is to speed up the design of new formulations. Previous papers have described the basic concepts of the method. In this paper, the focus is placed on functional aspects of symbolic tools for the development of finite element formulations. Two practical examples are used to illustrate this point. The first is a space-time formulation for an incompressible flow driven by the Navier–Stokes equations, and the second is a finite element derivation of the total potential energy for linear elasticity.     

An effective hybrid algorithm for computing symbolic determinants

A hybrid algorithm for computing the determinant of a matrix whose entries are polynomials is presented. It is based on the dimension-decreasing algorithm [22] and the parallel algorithm for computing a symbolic determinant of [19]. First, through the dimension-decreasing algorithm, a given multivariate matrix can be converted to a bivariate matrix. Then, the parallel algorithm can be applied to effectively compute the determinant of the bivariate matrix. Experimental results show that the new algorithm can not only reduce enormously the intermediate expression swell in the process of symbolic computation, but also achieve higher degree of parallelism, compared with the single parallel algorithm given in [19].     

A hybrid branch-and-bound approach for exact rational mixed-integer programming

We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP  solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from the Miplib and Mittelmann libraries and for a new collection of numerically difficult instances.     

A function space model approach to the rational evaluation subgroups

Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G _*(U, X; f), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba (Trans Am Math Soc 349, 4931–4951, 1997). In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith (The evaluation subgroup of a fibre inclusion, 2006) is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S ¹-bundle over the n-dimensional torus is determined explicitly in the non-rational case.      

A Method for the Computer Calculation of Edgeworth Expansions for Smooth Function Models

Abstract

In this article I describe, in detail, a method for the computer calculation of Edgeworth expansions for a smooth function model accurate in the O(n ^–1) term. For such models, these expansions are an important tool for the analysis of normalizing transformations, the correction of an approximately normally distributed quantity for skewness, and the comparison of bootstrap inference procedures. The method is straightforward and is efficient in a sense described in the article. The implementation of the method in general is clear from its implementation in the Mathematica program (available through StatLib) for the particular case of the studentized mean.     

Space Saving Calculation of Symbolic Resultants

We describe an approach to the computation of symbolic resultants in which factors are removed during the course of the calculation, so reducing the stack size required for intermediate expressions and the storage space needed. We apply the technique to three well-established methods for calculating resultants. We demonstrate the advantages of our approach when the resultants are large and show that some otherwise intractable problems can be resolved. In certain cases a significant reduction in the cpu time required to calculate the resultant is also evident.      

A fast numerical algorithm for the integration of rational functions

A new iterative method for high-precision numerical integration of rational functions on the real line is presented. The algorithm transforms the rational integrand into a new rational function preserving the integral on the line. The coefficients of the new function are explicit polynomials in the original ones. These transformations depend on the degree of the input and the desired order of the method. Both parameters are arbitrary. The formulas can be precomputed. Iteration yields an approximation of the desired integral with mth order convergence. Examples illustrating the automatic generation of these formulas and the numerical behaviour of this method are given.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

一种自适应的四阶Newton-Cotes求积方法

本文给出了一种基于四阶Newton-Cotes公式的自适应求积算法，该算法能根据给定的容许误差，由计算机自动选取积分步长，克服了由于被积函数的性态不好而导致积分较复杂的缺陷．     

An estimate for the modulus of a rational function

The paper presents sharp inequalities for the moduli of rational functions under certain constraints on the modulus of the independent variable. These inequalities supplement some results of Govil, Mohapatra, and Dubinin. Bibliography: 6 titles.     

Experiments with a symbolic programming system for complex analysis

The paper reports the results of work on the construction of LISP programs for various symbolic operations in complex analysis, including the evaluation of integrals around closed contours by the use of Cauchy's Theorem. It is concluded that the only difficulty in the way of the preparation of programs for all of the important and useful textbook calculations in complex analysis is that large amounts of fast storage in the computer are needed.     

Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation

We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.     

A stochastic approach to global error estimation in ODE multistep numerical integration

In order to assess the quality of approximate solutions obtained in the numerical integration of ordinary differential equations related to initial-value problems, there are available procedures which lead to deterministic estimates of global errors. The aim of this paper is to propose a stochastic approach to estimate the global errors, especially in the situations of integration which are often met in flight mechanics and control problems. Treating the global errors in terms of their orders of magnitude, the proposed procedure models the errors through the distribution of zero-mean random variables belonging to stochastic sequences, which take into account the influence of both local truncation and round-off errors. The dispersions of these random variables, in terms of their variances, are assumed to give an estimation of the errors. The error estimation procedure is developed for Adams-Bashforth-Moulton type of multistep methods. The computational effort in integrating the variational equations to propagate the error covariance matrix associated with error magnitudes and correlations is minimized by employing a low-order (first or second) Euler method. The diagonal variances of the covariance matrix, derived using the stochastic approach developed in this paper, are found to furnish reasonably precise measures of the orders of magnitude of accumulated global errors in short-term as well as long-term orbit propagations.     

Reduction theory for a rational function field

LetG be a split reductive group over a finite field F_q. LetF = F_q(t) and let A denote the adèles ofF. We show that every double coset inG(F)/G(A)/K has a representative in a maximal split torus ofG. HereK is the set of integral adèlic points ofG. WhenG ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.     

A rational Arnoldi approach for ill-conditioned linear systems

For the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a suitable function of matrix. In this sense, the method can be referred to as an iterative refinement process. Numerical experiments arising from integral equations and interpolation theory are presented. Finally, the method is extended to work in connection with the standard Tikhonov regularization with the right-hand side contaminated by noise.     

Lax pair, Bäcklund transformation and N-soliton-like solution for a variable-coefficient Gardner equation from nonlinear lattice, plasma physics and ocean dynamics with symbolic computation

In this paper, a generalized variable-coefficient Gardner equation arising in nonlinear lattice, plasma physics and ocean dynamics is investigated. With symbolic computation, the Lax pair and Bäcklund transformation are explicitly obtained when the coefficient functions obey the Painlevé-integrable conditions. Meanwhile, under the constraint conditions, two transformations from such an equation either to the constant-coefficient Gardner or modified Korteweg-de Vries (mKdV) equation are proposed. Via the two transformations, the investigations on the variable-coefficient Gardner equation can be based on the constant-coefficient ones. The N-soliton-like solution is presented and discussed through the figures for some sample solutions. It is shown in the discussions that the variable-coefficient Gardner equation possesses the right- and left-travelling soliton-like waves, which involve abundant temporally-inhomogeneous features.     

Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations

With the aid of symbolic computation Maple, an extended Jacobi elliptic function expansion method is presented and successfully applied to variant Boussinesq equations. As a result, abundant periodic wave solutions in terms of the Jacobi elliptic functions are obtained. When the modulus m → 1 or m → 0, exact solitary wave solutions and trigonometric function solutions are also derived. The properties of four new solutions are graphically studied.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.