Numerical Methods in C

10 Key excerpts on "Numerical Methods in C"

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  eBook - PDF

  Introduction to Computer Programming and Numerical Methods

  • Jocelyn O. Padallan(Author)
  • 2020(Publication Date)
  • Arcler Press

    (Publisher)

  With the significant increase in demand of modern-day computer for carrying out the numerical procedures in finding solution to mathematical models of the world, an area widely recognized as computational science or scientific computing and initiated during the 1980s and 1990s. This area is majorly interested in taking a look at numerical analysis from the perspective of computer science. It is primarily concerned with using the most advanced and powerful tools of computer graphics, numerical analysis, graphical user interfaces, and symbolic mathematical computations for making it easily for their user to set up, understand, and solve complex mathematical models of the real-world situation. 2.7.1. Historical Background It is worth noticing that the numerical algorithms are almost as old as human civilization. The Rhind Papyrus (˜1650 BC) of ancient Egypt throw light on a root finding method that help in solving a simple equation. Archimedes of Syracuse (287–212 BC) is responsible for creation of a new mathematics, includes the “method of exhaustion” for computation of lengths, areas, and volumes of geometric figures When this is used as a method for finding approximations, it carries higher importance in modern numerical integration; and was considered as an important executor in the development of the calculus by Gottfried Leibnitz and Isaac Newton. An Overview To Numerical Methods 47 A much-needed fuel for the development of numerical procedures was the invention of the calculus by Leibnitz and Newton, as this resulted in precise mathematical models for physical reality, initially in the physical sciences and eventually in the other sciences, medicine, engineering, and business. It is worth noticing that these mathematical models cannot be explicitly solved, and there is need of numerical methods for attaining the approximate solutions.

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  Matrix, Numerical, and Optimization Methods in Science and Engineering

  • Kevin W. Cassel(Author)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  6.1 Introduction to Numerical Methods 319 division). In algebra and trigonometry, the emphasis is on functions and how they are manipulated and represented. In calculus, the attention is on differentiation and integration of those functions. Finally, differential equations provide our primary con- nection between calculus and the physical world. All of these topics are primarily taught from the point of view of obtaining exact closed-form solutions. In calculus, for example, we perform differentiation and integration on smooth and continuous func- tions; we seek exact solutions of algebraic, trigonometric, and differential equations. As we quickly learn as budding engineers and scientists, however, these exact analytical methods only take us so far when considering practical applications in our chosen fields. It is not long before it is necessary to employ numerical techniques in order to obtain useful information about a set of data or a system. Numerical, or com- putational, methods is the general term given to approaches for obtaining approximate solutions to algebraic, differential, and integral equations. Because modern numerical methods lead to algorithms that are typically designed to be performed by digital com- puters, there is also an intimate connection between numerical methods and computer architectures. A prominent theme in numerical methods is development of techniques for turn- ing complex problems, such as nonlinear and/or very large problems, into ones that are amenable to solution using standard matrix methods. The trade-off in doing so is nearly always that the resulting solution is only an approximation of the exact solution. This takes us from the exact to the approximate, and from the continu- ous to the discrete. For example, a continuous function is represented as a smooth curve for the dependent variable with respect to the independent variable, say u(x ).

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  Numerical Analysis for Engineers

  Methods and Applications, Second Edition

  • Bilal Ayyub, Richard H. McCuen, Bilal M. Ayyub(Authors)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  3

  Introduction to Numerical Methods

  3.1  Introduction

  Many operations in mathematical analysis can be identified as belonging to one of two types: analytical or numerical. For example, when we need to solve a second-order polynomial aX 2 + bX + c = 0, for the values of X that satisfy the equality, it is common to use the closed-form solution

  X =

  (

  - b

  ±

  b 2

  - 4 a c

  ) / 2 a

  . This is an analytical solution. Alternatively, we could assume some value for the solution (X ), and through a systematic method of trial and error, we could obtain essentially the same solution as the analytical approach. However, this numerical approach is much more tedious, so when an analytical solution exists, it is usually the preferred method. For many types of problems, such as solving a fifth-order polynomial, a closed-form or analytical solution does not exist. Then the iterative, or numerical, approach must be used.

  In spite of the simplicity of most numerical methods, it is somewhat difficult to give a specific definition for the term numerical method. Instead of trying to define the term, we will discuss the characteristics, advantages, and disadvantages of numerical methods and then provide an example to illustrate these characteristics.

  The solution to many engineering problems involves the use of complex functional forms. Numerical methods can be used to solve a variety of quantitative problems faced by engineers. For many engineering problems, an analytical solution is either impossible or the cost or effort of performing an analytical solution would be prohibitive. Furthermore, in an effort to achieve greater accuracy in engineering solutions, more complex analyses are being used. Very often, the complexity of the problem becomes such that an analytical solution is not possible. In cases where analytical solutions are not practical or the problem is very complex, numerical methods can often be used to find a solution.

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  Design Engineer's Reference Guide

  Mathematics, Mechanics, and Thermodynamics

  • Keith L. Richards(Author)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  29 Introduction to Numerical Methods 2.1 INTRODUCTION Before the advent of electronic digital computers, all computing was accomplished using manual methods. The aim of this introduction is to present some of the methods used in analysing practical prob-lems arising in mechanical engineering that are not generally solvable using classical methods. These methods were developed prior to the development of the digital computer and have stood the test of time. The basic mathematics used in numerical analysis is covered in Chapter 1. The analysis of a physical problem will involve four basic steps: 1. Development of a suitable mathematical model that realistically represents the physical system 2. Derivation of the systems governing equations 3. Solution of the governing equations 4. Interpretation of the results While an analytical solution using classical methods will be exact, if it exists, a numerical method will require a number of iterations to generate a solution; this is only an approximation and cannot be considered to be exact by any means. It is important to understand the errors that arise in numerical analysis, and for this reason a sepa-rate section devoted to errors is covered in Section 2.2 and covers the difference between round-off errors and truncation errors. The numerical methods will be able to solve most complex problems, and one of the advantages is easy programming on a computer using programs such as MATLAB ® , MathCAD, Maple and Mathematica. All of these programs are user-friendly. 2.2 NUMERICAL METHODS FOR INTEGRATION There may be a point where the engineer is required to calculate a definite integral that cannot be solved using analytical integration or it is preferred to integrate tabulated data. The following methods may be used to solve for a definite integral: 1. Manual method 2. Mid-ordinate rule 3. Trapezoidal rule 4. Simpson’s rule 5. Quadratic triangulation 6. Romberg integration 7. Gauss quadratic 2

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  MATLAB Programming

  Mathematical Problem Solutions

  • Dingyü Xue(Author)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  One is to implement the existing numerical algorithms with general-purpose computer languages, such as C and Fortran. The other way is to use specific computer languages with good reputation. Appropriate computer languages such as MATLAB, Mathematica [ 17 ], and Maple [ 13 ] can be adopted. In this book, the languages are referred to as the computer mathematics languages. Numerical algorithms can only be used to tackle computation problems with numbers, while for problems like finding the solutions to the quadratic equation a x 2 + b x + c = 0, where a, c, d are not given numerical values but symbolic variables, numerical algorithms, no matter how sophisticated, cannot be adopted. The professional computer mathematics languages capabilities should be adopted instead. The term “mathematical computation” will be used throughout the series, whereas the term really means both numerical and analytical computation of solutions to mathematical problems. Normally, analytical solutions are explored first, and if there are no analytical solutions, numerical solutions are obtained. The following examples are shown so that the readers may understand and appreciate the necessity of using the computer mathematics languages. Example 1.1. Consider a simple problem. What is the last digit of 2 019 2 019 ? Solutions. If computers are not used, all the mathematicians know about the result is its last digit, since tricks can be used in finding it. In fact, the solution is useless in reality, since if you spend some money to buy something, you will never care what the last digit is. It makes almost no difference to you whether the last digit is 1 or 9. What you care about is what the highest digit is and how many digits there are in total. It is always useful if you can find all the digits. However, these problems cannot be solved at all without the use of computers. Dedicated computer software tools are also essential

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  C Programming and Numerical Analysis

  An Introduction

  • Seiichi Nomura(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  PART II Numerical Analysis 91 Now that the basic syntax of the C language has been explained, you are ready to write C pro- grams to solve many problems arising in engineering and science. In Part II, numerical meth- ods for solving nonlinear equations, a set of simultaneous equations and ordinary differential equations as well as numerically differentiating and integrating given functions are discussed. Solving these equations analytically requires advanced levels of mathematics. However, solving equations numerically often only involves intuitive or visual interpretation and not necessarily requires higher mathematics. Although all the essential topics in numerical analysis are elucidated, it is not possible to cover every single aspect of numerical analysis. For a comprehensive reference, Numerical Recipes in C 33 is suggested. 33 Press et al., Numerical Recipes in C, Cambridge, Cambridge University Press, 1996. 93 C H A P T E R 3 Note on Numerical Errors In Part I, the data type, float, was used for all real numbers where 4 bytes are allocated for each floating number. A float variable can represent 10 38 to 10 38 which covers most of the practical range. However, this range is translated into 6 to 8 decimal digits of precision and many problems in science and engineering require more precision that this range. Consider the following examples: 1. #include int main() { float s=0.0;int i; for (i=0;i <10000;i++) s=s+0.1; printf("%f\n",s); return 0; } The intention of the program is to add 0:1 for 10;000 times. The result should be 1;000. However, the program outputs the following result: $ gcc prog.c $ ./a.out 999.902893 The output is not 1,000 but 999:902893 which is almost 1,000 but when precision is criti- cal, this result is not acceptable. The error in this example comes from conversion between decimal numbers and binary numbers.

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  Computer Methods for Engineering with MATLAB Applications

  • Yogesh Jaluria(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  21 2 Basic Considerations in Computer Methods 2.1 INTRODUCTION In the numerical solution of engineering problems, there are several important aspects that need to be considered in order to ensure the validity of the chosen approach for a given problem and the accuracy of the results obtained. The computational procedure involves a consideration of the methods available for solv-ing the given problem, the appropriate programming language, the computational environment and software being employed, the computer and its operating system, and so forth, before proceeding to the development of the numerical scheme, or algorithm, and the corresponding program. Since these considerations are funda-mental to most computer methods, this chapter discusses the general approach to the development of the computational scheme. Also considered are the interfacing with available computer software and the verification and validation of the numerical results by a comparison with available analytical and experimental results, as discussed in Chapter 1. The consideration of numerical errors and the accuracy of the results is important in the numerical solution of any given problem. The various types of errors that arise in the computational approach are discussed, along with methods that may be employed for reducing the error. The accuracy of the solution may often be estimated by comparing the numerical results with those from the analytical solution for sim-pler problems, since the analytical solution of the given problem is presumably not available. Frequently, satisfactory analytical results are not available for comparison. In such cases, the numerical scheme itself is first employed to check the accuracy of the numerical results by ensuring that numerical parameters, such as the chosen time step and grid size, do not significantly affect the results. This process is often known as verification of the numerical method.

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  Applied Engineering Analysis

  • Tai-Ran Hsu(Author)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Numerical techniques have greatly expanded the types of problems that engineers can solve, as illustrated in a number of publications in the open literature (Chapra, 2012; Ferziger, 1998; Hoffman, 1992). The goal of this chapter is to present to readers the basic principles of some of these techniques that are frequently used in engineering analysis. The author's own experience indicates that engineers who understand the principles of the numerical methods are usually more effective and intelligent users of these methods than most technical personnel who are trained to carry out the same computational assignments using “turn-key” software packages such as the finite-difference method and the MatLAB software package in Appendix 4. This chapter will cover the principles of commonly used numerical techniques for (1) the solution of nonlinear polynomial and transcendental equations; (2) integration involving complex forms of functions; and (3) the solution of differential equations by the basic finite-difference schemes and the Runge–Kutta methods. The chapter will also cover the overviews of two popular commercial software packages called Mathematica and MatLAB.

  The principal task in numerical methods for engineering analysis is to develop algorithms that involve arithmetic and logical operations so that such operations can be performed at incredible speed by digital computers with enormous data storage capacities. Because readers of this book are expected to be users of numerical methods, we will present only the principles that are relevant to the development of these algorithms, not the theories and the proof of these methods.

  10.2 Engineering Analysis with Numerical Solutions

  Most engineering problems require enormous computational effort when numerical methods are used. Digital computers are essential tools for obtaining numerical solutions. Digital computers have incredible power in computational speed and enormous memory capacity. Unfortunately, these machines have no intelligence of their own, and they are not capable of making independent judgment on their own. Additionally, engineers need to realize the fact that digital computers can only perform simple arithmetic operations with (+, −, ×, ÷) and handle the logical flow of data. It cannot perform higher mathematical operations even in such simple cases as evaluating exponential and trigonometric functions without proper algorithms that convert the evaluation of these functions into simple arithmetic operations; thus, all complicated mathematical operations have to be converted into simple arithmetical operations. Numerical methods that enable engineers to develop algorithms for various mathematical functions and operations using digital computers have thus become essential knowledge and skills for solving many advanced engineering problems using mathematical tools.

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  Numerical Methods Fundamentals

  • R. V. Dukkipati, R. V. Dukkipati(Authors)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  1

  NUMERICAL COMPUTATIONS

  Numerical methods are methods for solving problems on computers by numerical calculations, often giving a table of numbers and/or graphical representations or figures. Numerical methods tend to emphasize the implementation of algorithms. The aim of numerical methods is therefore to provide systematic methods for solving problems in a numerical form. The process of solving problems generally involves starting from initial data, using high-precision digital computers, following the steps in the algorithms, and finally obtaining the results. Often the numerical data and the methods used are approximate. Hence, the error in a computed result may be caused by the errors in the data, the errors in the method, or both.

  In this chapter, we will describe Taylor’s theorem, a few basic ideas and concepts regarding numerical computations, number representation, (including binary, decimal, and hexadecimal numbers), errors considerations, absolute and relative errors, inherent errors, round-off errors and truncation errors, error estimation, general error formulae (including approximation of a function), stability and condition, uncertainty in data, linear convergence, quadratic convergence, and Aitken’s acceleration formulae.

  1.1 TAYLOR’S THEOREM

  Taylor’s theorem allows us to represent, exactly, and fairly the general functions in terms of polynomials with a known, specified, and boundable error. Taylor’s theorem is stated as follows:

  Let f (x ) have n + 1 continuous derivatives on [a , b ] for some n ≥ 0, and let x , x 0 ∈ [a , b ]. Then

  f (x ) = p n (x ) + R n (x ) (1.1)

  for     (1.2)

  and     (1.3)

  Also, there exists a point ξ x , between x and x 0 such that

  (1.4)

  where R n (x ) is the remainder

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  Computational Methods in Physics and Engineering

  • Samuel S M Wong(Author)
  • 1997(Publication Date)
  • WSPC

    (Publisher)

  Chapter 1 Computational Methods Modern electronic computers owe their origin, to a large extent, to the needs in science and engineering. In the 50 years or so since their appearance, computers have out-performed their original goals of solving numerical problems and keeping tracks of information. They are now an essential tool in almost every aspect of the daily routines of engineers and scientists, from data collection to writing technical reports. The high speed of computation available to us these days opens up not only new ways of carrying out traditional tasks but also new areas of endeavor that have implications going well beyond what we can realize at the moment. Our concern here is limited to a small, albeit important, corner of the role of modern computers in science and engineering, namely some of the general techniques to solve common problems encountered in physics and engineering. 1-1 Numerical calculations and beyond When we use a computer to solve a problem in science, the general assumption is that it is done numerically. Indeed, the proper name of most computers in the market is digital computer, reminding us of the fact that numbers are being manipulated. However, in addition to mathematical operations, such as addition and multiplica-tion, the central processor of a computer is also capable of logical operations, that is, making decisions depending on whether a particular condition is true or false. Furthermore, a binary digit, or bit, of the computer memory may be regarded as a logical unit, representing the value true if it is on (= 1) and false if it is off (= 0). In this way, a computer can be programmed equally well to carry out logical decisions or, more generally, symbolic manipulations. At the same time, the computer screen is made of lines of horizontal dots or pixels. On a monochrome screen, each dot can be turned on or off. For a color monitor, there is the further capability of displaying different colors at each dot.

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