Essentials of Scientific Computing: Numerical Methods for Science and Engineering

by Victor Zalizniak

Overview

Modern development of science and technology is based to a large degree on computer modelling. To understand the principles and techniques of computer modelling, students should first get a strong background in classical numerical methods, which are the subject of this book. This text is intended for use in a numerical methods course for engineering and science students, but will also be useful as a handbook on numerical techniques for research students.

Essentials of Scientific Computing is as self-contained as possible and considers a variety of methods for each type of problem discussed. It covers the basic ideas of numerical techniques, including iterative process, extrapolation and matrix factorization, and practical implementation of the methods shown is explained through numerous examples. An introduction to MATLAB is included, together with a brief overview of modern software widely used in scientific computations.

• Outlines classical numerical methods, which is essential for understanding the principles and techniques of computer modelling
• Intended for use in a numerical methods course for engineering and science students, but will also be useful as a handbook on numerical techniques for research students
• Covers the basic ideas of numerical techniques, including iterative process, extrapolation and matrix factorization

Product Details

ISBN-13:	9780857099365
Publisher:	Elsevier Science
Publication date:	03/01/2008
Sold by:	Barnes & Noble
Format:	eBook
Pages:	220
File size:	6 MB

About the Author

Dr. Victor Zalizniak was awarded his Masters in Physics at Krasnoyarsk State University, Russia before becoming a Research Fellow at the Centre for Scientific Computing at the Russian Academy of Sciences (Siberian Branch). He then moved to the Department of Aerospace Engineering at the Royal Melbourne Institute of Technology where he obtained his PhD. In 2001 he returned to the Department of Computer Science at his alma mater, Krasnoyarsk State University, where he continues to lecture, research and write in his particular fields of computational physics and mathematical physics. He is the author of several books on scientific computing including Essentials of Computational Physics Parts 1 and 2.

Table of Contents

• About the Author
• Dedication
• List of Examples
• Preface
• 1: Errors in computer arithmetic operations
• 2: Solving equations of the form f(x)=0
  • 2.1 THE BISECTION METHOD
  • 2.2 CALCULATION OF ROOTS WITH THE USE OF ITERATIVE FUNCTIONS
  • 2.3 CONCLUDING REMARKS
• 3: Solving systems of linear equations
  • 3.1 LINEAR ALGEBRA BACKGROUND
  • 3.2 SYSTEMS OF LINEAR EQUATIONS
  • 3.3 TYPES OF MATRICES THAT ARISE FROM APPLICATIONS AND ANALYSIS
  • 3.4 ERROR SOURCES
  • 3.5 CONDITION NUMBER
  • 3.6 DIRECT METHODS
  • 3.7 ITERATIVE METHODS
  • 3.8 COMPARATIVE EFFICACY OF DIRECT AND ITERATIVE METHODS
• 4: Computational eigenvalue problems
  • 4.1 BASIC FACTS CONCERNING EIGENVALUE PROBLEMS
  • 4.2 LOCALIZATION OF EIGENVALUES
  • 4.3 POWER METHOD
  • 4.4 INVERSE ITERATION
  • 4.5 ITERATION WITH A SHIFT OF ORIGIN
  • 4.6 THE QR METHOD
  • 4.7 CONCLUDING REMARKS
• 5: Solving systems of nonlinear equations
  • 5.1 FIXED-POINT ITERATION
  • 5.2 NEWTON’S METHOD
  • 5.3 METHOD WITH CUBIC CONVERGENCE
  • 5.4 MODIFICATION OF NEWTON’S METHOD
  • 5.5 MAKING THE NEWTON-BASED TECHNIQUES MORE RELIABLE
• 6: Numerical integration
  • 6.1 SIMPLE QUADRATURE FORMULAE
  • 6.2 COMPUTATION OF INTEGRALS WITH PRESCRIBED ACCURACY
  • 6.3 INTEGRATION FORMULAE OF GAUSSIAN TYPE
  • 6.4 DEALING WITH IMPROPER INTEGRALS
  • 6.5 MULTIDIMENSIONAL INTEGRATION
• 7: Introduction to finite difference schemes for ordinary differential equations
  • 7.1 ELEMENTARY EXAMPLE OF A FINITE DIFFERENCE SCHEME
  • 7.2 APPROXIMATION AND STABILITY
  • 7.3 NUMERICAL SOLUTION OF INITIAL VALUE PROBLEMS
  • 7.4 NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS
  • 7.5 ERROR ESTIMATION AND CONTROL
• 8: Interpolation and Approximation
  • 8.1 INTERPOLATION
  • 8.2 APPROXIMATION OF FUNCTIONS AND DATA REPRESENTATION
• 9: Programming in MATLAB
  • 9.1 NUMBERS, VARIABLES AND SPECIAL CHARACTERS
  • 9.2 ARITHMETIC AND LOGICAL EXPRESSIONS
  • 9.3 CONDITIONAL EXECUTION
  • 9.4 LOOPS
  • 9.5 ARRAYS
  • 9.6 FUNCTIONS
  • 9.7 INPUT AND OUTPUT
  • 9.8 VISUALIZATION
• A: Integration formulae of Gaussian type
• B: Transformations of integration domains
• C: Stability regions for Runge-Kutta and Adams schemes
• D: A brief survey of available software
  • LAPACK library for problems in numerical linear algebra
  • IMSL mathematics and statistics libraries
  • Numerical Algorithms Group (NAG) numerical libraries
• Bibliography
• Index

Editorial Reviews

Zalizniak brings an unparalleled and novel treatment to this field and is to be commended for the sufficient level of detail provided in each chapter. The strength of the work lies in its conciseness, mathematical rigor and clear explanations of the underlying principles …Recommended, Choice
…invaluable to research students as a handbook on numerical techniques, Mathematical Reviews

From the Publisher