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4162 results in Numerical analysis and computational science

Page 102 of 209

  • Computing a Chief Series and the Soluble Radical of a Matrix Group Over a Finite Field

  • Derek F. Holt, Mark J. Stather
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 11 / 2008
    Published online by Cambridge University Press:
    01 February 2010, pp. 223-251
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  • We describe an algorithm for computing a chief series, the soluble radical, and two other characteristic subgroups of a matrix group over a finite field, which is intended for matrix groups that are too large for the use of base and strong generating set methods. The algorithm has been implemented in Magma by the second author.

  • On Homogenous Minimal Involutive Varieties

  • L. C. O. Almeida, S. C. Coutinho
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 8 / 2005
    Published online by Cambridge University Press:
    01 February 2010, pp. 301-315
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  • Ѕ(2n,k) be the variety of homogeneous polynomials of degree k in 2n variables. The authors of this paper give a computer-assisted proof that there is an analytic open set Ω of Ѕ(4,3) such that the surface F = 0 is a minimal homogeneous involutive variety of ℂ4 for all F ∈ Ω. As part of the proof, they give an explicit example of a polynomial with rational coefficients that belongs to Ω.

  • On a Result of Darboux

  • Javier Chavarriga, Jaume Llibre, Jean Moulin Ollagnier
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 4 / 2001
    Published online by Cambridge University Press:
    01 February 2010, pp. 197-210
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  • This paper is concerned with a relation of Darboux in enumerative geometry, which has very useful applications in the study of polynomial vector fields. The original statement of Darboux was not correct. The present paper gives two different elementary proofs of this relation. The first one follows the ideas of Darboux, and uses basic facts about the intersection index of two plane algebraic curves; the second proof is rather more sophisticated, and gives a stronger result, which should also be very useful. The power of the relation of Darboux is then illustrated by the provision of new, simple proofs of two known results. First, it is shown that an irreducible invariant algebraic curve of degree n > 1 without multiple points for a polynomial vector field of degree m satisfies n ≤ m + 1. Second, a proof is given that quadratic polynomial vector fields have no algebraic limit cycles of degree 3.

  • Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

  • Willem A. de Graaf
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 11 / 2008
    Published online by Cambridge University Press:
    01 February 2010, pp. 280-297
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  • Let G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

  • A New Geometric Algorithm to Generate Smooth Interpolating Curves on Riemannian Manifolds

  • Rui C. Rodrigues, F. Silva Leite, Janusz Jakubiak
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 8 / 2005
    Published online by Cambridge University Press:
    01 February 2010, pp. 251-266
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  • This paper presents a new geometric algorithm to construct a Ck-smooth spline curve that interpolates a given set of data (points and velocities) on a complete Riemannian manifold. Although based on a modification of the De Casteljau procedure, this new algorithm is implemented in only three steps, independently of the required degree of smoothness, and therefore introduces a significant reduction in complexity. The key role is played by the choice of an appropriate smoothing function, which is defined as soon as the degree of smoothness is fixed.

  • 8 - Two-Point Boundary Value Problems

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 290-318

  • Summary

    Introduction

    In two-point boundary value problems, the auxiliary conditions associated with the differential equation, called the boundary conditions, are specified at two different values of x. This seemingly small departure from initial value problems has a major repercussion — it makes boundary value problems considerably more difficult to solve. In an initial value problem we were able to start at the point where the initial values were given and march the solution forward as far as needed. This technique does not work for boundary value problems, because there are not enough starting conditions available at either endpoint to produce a unique solution.

    One way to overcome the lack of starting conditions is to guess the missing values. The resulting solution is very unlikely to satisfy boundary conditions at the other end, but by inspecting the discrepancy we can estimate what changes to make to the initial conditions before integrating again. This iterative procedure is known as the shooting method. The name is derived from analogy with target shooting — take a shot and observe where it hits the target, then correct the aim and shoot again.

    Another means of solving two-point boundary value problems is the finite difference method, where the differential equations are approximated by finite differences at evenly spaced mesh points. As a consequence, a differential equation is transformed into set of simultaneous algebraic equations.

    The two methods have a common problem: They give rise to nonlinear sets of equations if the differential equations are not linear. As we noted in Chapter 2, all methods of solving nonlinear equations are iterative procedures that can consume a lot of computational resources.

  • 1 - Introduction to Python

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 1-26

  • Summary

    General Information

    Quick Overview

    This chapter is not a comprehensive manual of Python. Its sole aim is to provide sufficient information to give you a good start if you are unfamiliar with Python. If you know another computer language, and we assume that you do, it is not difficult to pick up the rest as you go.

    Python is an object-oriented language that was developed in the late 1980s as a scripting language (the name is derived from the British television show Monty Python's Flying Circus). Although Python is not as well known in engineering circles as some other languages, it has a considerable following in the programming community — in fact, Python is used by more programmers than Fortran. Python may be viewed as an emerging language, because it is still being developed and refined. In the current state, it is an excellent language for developing engineering applications — Python's facilities for numerical computation are as good as those of Fortran MATLAB.®

    Python programs are not compiled into machine code, but are run by an interpreter. The great advantage of an interpreted language is that programs can be tested and debugged quickly, allowing the user to concentrate more on the principles behind the program and less on programming itself. Because there is no need to compile, link, and execute after each correction, Python programs can be developed in a much shorter time than equivalent Fortran or C programs. On the negative side, interpreted programs do not produce stand-alone applications. Thus, a Python program can be run only on computers that have the Python interpreter installed.

  • 10 - Introduction to Optimization

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 374-415

  • Summary

    Introduction

    Optimization is the term often used for minimizing or maximizing a function. It is sufficient to consider the problem of minimization only; maximization of F(x) is achieved by simply minimizing —F(x). In engineering, optimization is closely related to design. The function F(x), called the merit function or objective function, is the quantity that we wish to keep as small as possible, such as the cost or weight. The components of x, known as the design variables, are the quantities that we are free to adjust. Physical dimensions (lengths, areas, angles, etc.) are common examples of design variables.

    Optimization is a large topic with many books dedicated to it. The best we can do in limited space is to introduce a few basic methods that are good enough for problems that are reasonably well behaved and do not involve too many design variables. By omitting the more sophisticated methods,we may actually not miss all that much. All optimization algorithms are unreliable to a degree — any one of them may work on one problem and fail on another. As a rule of the thumb, by going up in sophistication we gain computational efficiency, but not necessarily reliability.

    The algorithms for minimization are iterative procedures that require starting values of the design variables x. If F(x) has several local minima, the initial choice of x determines which of these will be computed. There is no guaranteed way of finding the global optimal point. One suggested procedure is to make several computer runs using different starting points and pick the best result.

  • 5 - Numerical Differentiation

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp 177-192

  • Summary

    Introduction

    Numerical differentiation deals with the following problem: We are given the function y = f(x) and wish to obtain one of its derivatives at the point x = xk. The term “given” means that we either have an algorithm for computing the function or possess a set of discrete data points (xi, yi), i = 0, 1, …, n. In either case, we have access to a finite number of (x, y) data pairs from which to compute the derivative. If you suspect by now that numerical differentiation is related to interpolation, you are right — one means of finding the derivative is to approximate the function locally by a polynomial and then differentiate it. An equally effective tool is the Taylor series expansion of f(x) about the point xk, which has the advantage of providing us with information about the error involved in the approximation.

    Numerical differentiation is not a particularly accurate process. It suffers from a conflict between roundoff errors (due to limited machine precision) and errors inherent in interpolation. For this reason, a derivative of a function can never be computed with the same precision as the function itself.

Numerical Methods in Engineering with Python
  • 2nd edition
  • Jaan Kiusalaas
  • Published online:
    05 June 2012
    Print publication:
    29 January 2010
  • This text is for engineering students and a reference for practising engineers, especially those who wish to explore Python. This new edition features 18 additional exercises and the addition of rational function interpolation. Brent's method of root finding was replaced by Ridder's method, and the Fletcher-Reeves method of optimization was dropped in favor of the downhill simplex method. Each numerical method is explained in detail, and its shortcomings are pointed out. The examples that follow individual topics fall into two categories: hand computations that illustrate the inner workings of the method and small programs that show how the computer code is utilized in solving a problem. This second edition also includes more robust computer code with each method, which is available on the book website. This code is made simple and easy to understand by avoiding complex bookkeeping schemes, while maintaining the essential features of the method.

  • Preface to the Second Edition

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with Python
    Published online:
    05 June 2012
    Print publication:
    29 January 2010, pp x-x

  • Summary

    The major change in the second edition is the replacement of NumArray (a Python extension that implements array objects) with NumPy. As a consequence, most routines listed in the text required some code changes. The reason for the changeover is the imminent discontinuance of support for NumArray and its predecessor Numeric.

    We also took the opportunity to make a few changes in the material covered:

  • Rational function interpolation was added to Chapter 3.

  • Brent's method of root finding in Chapter 4 was replaced by Ridder's method. The full-blown algorithm of Brent is a complicated procedure involving elaborate bookkeeping (a simplified version was presented in the first edition). Ridder's method is as robust and almost as efficient as Brent's method, but much easier to understand.

  • The Fletcher—Reeves method of optimization was dropped in favor of the downhill simplex method in Chapter 10. Fletcher—Reeves is a first-order method that requires knowledge of the gradients of the merit function. Because there are few practical problems where the gradients are available, the method is of limited utility. The downhill simplex algorithm is a very robust (but slow) zero-order method that often works where faster methods fail.

Page 102 of 209