Drawing graphs by numerical solution of a system of second order ordinary differential equations
Mohammad ali Ebrahimi and Michael Monagan
In this poster we present the spring algorithm for drawing directed and undirected graphs of vertices and edges in 2D or 3D. Starting from random initial position for the vertices, the algorithm finds the minimal total energy of the graph. While graph drawing is a complicated problem, this algorithm requires no special knowledge about graph theory. The Maple implementation of this algorithm will be used by the Graph Theory Package. We will demo a Maple implementation of the spring algorithm.
Visualizing system of Differential Equations in Maple
Mohammad ali Ebrahimi and Michael Monagan
We present new options for Maple's DEplot command for enhancing the visualization of direction fields and for animating direction fields and solution curves. We also present a new Maplet, a database of different systems of differential equations with parameters and initial values which can be selected for display and modified by the instructor.
Univariate Polynomial Factorization in Maple via Combinatorial Trial Division
Al Erickson, Michael Monagan and Ha Le
We present Maple implementation for polynomial factorization in Z [x].
The implementation uses "recden", a recursive dense polynomial representation. The algorithm uses the Cantor Zanssenhaus/Distinct Degree or Belekamp's big prime algorithm followed by a quadratic parallel Hensel lift with trial division of combinations of 1 and 2 modular factors at each step. The algorithm finishes with a combinatorial algorithm, checking the necessary combinations of lifted factors to find real factors. We will also show a bivariate version of the Hensel lift.
Multivariate Interpolation in Maple
We present a new Maple package named MultiInterp. This package includes efficient routines for multivariate polynomial interpolation, multivariate rational function interpolation and multivariate Pade approximation. We illustrate these commands with examples (including a hands-on computer demo) and briefly explain the theory behind them. In several cases our multivariate implementations outperform the existing univariate commands in Maple.
A Generalized Apollonius Problem
The Apollonius problem is: Find all circles that touch 3 given circles, in the Euclidean plane.
We generalize this problem to: Find all circles that touch 3 given ellipses in the Euclidean plane.
Optimization Methods for Binary Sequences — The Merit Factor Problem
Optimization procedures for binary sequences, applied in particular to the merit factor problem, have been a subject of much interest in combinatorial optimization, communications engineering, and analytic number theory. Exact methods have confirmed optimal sequences up to length 60, but this is not far enough to adequately make projections. We combine both stochastic and non-stochastic search methods to obtain what we consider comprehensive results to length 90 in general and to length 180 for restriction to skew-symmetric sequences and develop statistical arguments to support this claim.
A Graph Theory Package for Maple
We present a new graph theory package for Maple. The package is presently intended for teaching and research usage, and expected to treat graphs of up to 1000 vertices in a reasonable time. One design criterion for the new GraphTheory package is a simple, yet flexible, data structure designed primarily for solving problems related to graphs rather than networks. All of the operations present in the networks package and most of the standard operations for graphs are available in the GraphTheory package. The package also includes a drawing component.
Numerical Integration in the Student Subpackage "Numerical Analysis"
Howard Wen Hao Liu and Ha Le
In this poster we present the numerical integration module (ApproximateInt) in the new student subpackage "numerical analysis" for Maple. This module is an extension to the ApproximateInt module in the subpackage Student[Calculus1], with new features including Adaptive Newton-Cotes methods, Open Newton-Cotes methods, the Romberg Integration method, and the Gaussian Quadrature method.
Computing Characteristic Polynomials over Z
We compare Maple implementations of a modular algorithm for computing the characteristic polynomial of an integer marix with the Berkowitz algorithm. The computation modulo a prime is implemented in. The best results come from simulating mod p arithmetic using floats. Two improvements have been made to further improve the timings.
An Inversion Algorithm for Multiple-Source Dispersion and Deposition of Particulate Matter
The project investigates the dispersion and deposition of pollutant particles under the influence of diffusion, advection by the wind and settling effects. In this poster we briefly explain a new approach that we have been following in solving the (inverse) problem of estimating the emitted quantity of particulate matter from the four area sources in Trail, B.C. Our solution incorporates an exact solution to an idealized steady-state (usually known as a "Gaussian plume solution") into an optimization solver to find the emission rates from the amount of deposited matter in the receivers, as well as allows to input site-specific constraints derived from the chemical processes.
Computations of Water Wave Interfaces
The poster is be a quick introduction on the water waves interface problem, one of the free-boundary flow problems calculated via Boundary Integral Methods, and the modern computational methods usually used to study them and their properties. Two different implementations are considered and discussed, as well as the numerical difficulties encountered and how they are dealt with. Some numerical examples of wave evolutions, with and without surface tension, and also breaking waves for infinite depth flow, are presented.
Rational Expression Simplification with Algebraic Side Relations
I will present two algorithms for simplifying rational expressions modulo a set of polynomial side relations. The methods use Groebner bases.
Variable Step-size Implicit-Explicit linear Multistep Methods (VSIMEX) forTime-Dependent PDEs
Time-dependent PDEs can conveniently be transformed into a large system ofODEs by doing spatial discretizations. For large systems of ODEs with both stiffand nonstiff parts, Implicit-Explicit (IMEX) schemes treat the stiff parts implicitlyand the nonstiff parts explicitly. This has proven to be a powerful technique.
For solutions with different time scales, variable step-size schemes are oftenessential in computation. However, standard linear multistep methods aredesigned for constant step-sizes. Changing the step-size for standard methodsinvolves computing the corresponding starting values for each time step. This issufficiently complicated that it is often avoided in practice.
One of the objectives of this research is to provide the end-users of IMEX methodswith easily implemented variable step-size IMEX schemes.
In this research, we derive the family of two-step second order VSIMEX schemeswith 2 free parameters. A zero stability analysis of VSIMEX schemes is providedwhich leads to analytical results on the restriction of the step-size ratio for generalsecond order VSIMEX schemes. The family of three-step, third order VSIMEXschemes with 3 free parameters is also derived. The zero stability analysis ofthese VSIMEX schemes gives numerical values for the step-size restrictions.Fourth order VSIMEX schemes and their stability properties are also studied.
Numerically, we apply our new VSIMEX schemes to the advection-diffusionequation and Burgers equation. In our tests the expected orders of convergenceare achieved and correct approximate solutions are obtained. Our resultsdemonstrate the superiority of VSIMEX schemes over classical IMEX schemes.