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1897 results in Engineering mathematics and programming

Page 56 of 95

  • 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.

  • Preface to the First Edition

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

  • Summary

    This book is targeted primarily toward engineers and engineering students of advanced standing (juniors, seniors, and graduate students). Familiarity with a computer language is required; knowledge of engineering mechanics (statics, dynamics, and mechanics of materials) is useful, but not essential.

    The text attempts to place emphasis on numerical methods, not programming. Most engineers are not programmers, but problem solvers. They want to know what methods can be applied to a given problem, what are their strengths and pitfalls, and how to implement them. Engineers are not expected to write computer code for basic tasks from scratch; they are more likely to utilize functions and subroutines that have been already written and tested. Thus, programming by engineers is largely confined to assembling existing bits of code into a coherent package that solves the problem at hand.

    The “bit” of code is usually a function that implements a specific task. For the user the details of the code are unimportant. What matters is the interface (what goes in and what comes out) and an understanding of the method on which the algorithm is based. Since no numerical algorithm is infallible, the importance of understanding the underlying method cannot be overemphasized; it is, in fact, the rationale behind learning numerical methods.

    This book attempts to conform to the views outlined above. 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.

  • 10 - Introduction to Optimization

  • Jaan Kiusalaas, Pennsylvania State University
  • Book:
    Numerical Methods in Engineering with MATLAB®
    Published online:
    05 June 2012
    Print publication:
    30 September 2009, pp 380-414

  • Summary

    Find x that minimizes F(x) subject to g(x) = 0, h(x) ≤ 0

    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.

    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.

    More often than not, the design is also subjected to restrictions, or constraints, which may have the form of equalities or inequalities. As an example, take the minimum weight design of a roof truss that has to carry a certain loading. Assume that the layout of the members is given, so that the design variables are the cross-sectional areas of the members.

Page 56 of 95