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Essentials of scientific computing: Numerical methods for science and engineering

V Zalizniak, Siberian Federal University, Russia

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

 - 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
 - an introduction to MATLAB is included, together with a brief overview of modern software widely used in scientific computations

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.

ISBN 1 904275 32 X
ISBN-13: 978 1 904275 32 9
March 2008
212 pages  234 x 156mm  paperback  
£60.00 / US$100.00 / €70.00

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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.

Titles which may also be of interest:
Practical scientific computing
Computational functional analysis
Machine learning and data mining
Circuit analysis

Contents

Errors in computer arithmetic operations

Solving equations of the form f( x)=0
 - The bisection method
 - Calculation ofroots with the use ofiterative functions
 - Concluding remarks

Solving systems of linear equations
 - Linear algebra background
 - Systems oflinear equations
 - Types ofmatrices that arise from applications and analysis
 - Error sources
 - Condition number
 - Direct methods
 - Iterative methods
 - Comparative efficacy of direct and iterative methods

Computational eigenvalue problems
 - Basic facts concerning eigenvalue problems
 - Localization of eigenvalues
 - Power method
 - Inverse iteration
 - Iteration with a shift of origin
 - The QR method
 - Concluding remarks

Solving systems of nonlinear equations
 - Fixed-point iteration
 - Newton's method
 - Method with cubic convergence
 - Modification ofNewton's method
 - Making the Newton-based techniques more reliable

Numerical integration
 - Simple quadrature formulae
 - Computation ofintegrals with prescribed accuracy
 - Integration formulae of Gaussian type
 - Dealing with improper integrals
 - Multidimensional integration

Introduction to finite difference schemes for ordinary differential equations
 - Elementary example of a finite difference scheme
 - Approximation and stability
 - Numerical solution ofinitial-value problems
 - Numerical solution ofboundary-value problems
 - Error estimation and control

Interpolation and Approximation
 - Interpolation
 - Approximation offunctions and data representation

Programming in MATLAB
 - Numbers, variables and special characters
 - Arithmetic and logical expressions
 - Conditional execution
 - Loops
 - Arrays
 - Functions
 - Input and output
 - 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

Bibliography