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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010274946 | QA76.9.M35 Z35 2008 | Open Access Book | Book | Searching... |
Searching... | 33000000008617 | QA76.9.M35 Z35 2008 | Open Access Book | Book | Searching... |
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Summary
Summary
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.
Author Notes
Dr. Victor Zalizniak, Department of Mathematical Modelling in Mechanics Siberian Federal University, Krasnoyarsk, Russia
Reviews 1
Choice Review
Essentials of Scientific Computing is one book that can be judged by its title. Zalizniak (Siberian Federal Univ.) brings 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. For each topic, Zalizniak articulates the pros and cons, efficiency, speed, stability, and error estimation of various numerical methods. The work's eight chapters cover subjects that are essential in science and engineering, including systems of linear and nonlinear equations, eigenvalue problems, numerical integration, interpolation and approximation, and finite difference for ordinary differential equations. The concepts and numerical schemes are essential for novices, and well worth revisiting by practitioners in the field. Chapters start with an elementary consideration of the numerical scheme and progress unambiguously to advanced topics. It would have been desirable to include problem sets in each chapter that reinforce the concepts and schemes through pencil and paper practice and applied programming. Summing Up: Recommended. Graduate students and above. R. N. Laoulache University of Massachusetts Dartmouth
Table of Contents
| Chapter 1 Errors in computer arithmetic operations | p. 1 |
| Chapter 2 Solving equations of the form f(x)=0 | p. 5 |
| 2.1 The bisection method | p. 7 |
| 2.2 Calculation of roots with the use of iterative functions | p. 8 |
| 2.2.1 One-point iterative processes | p. 11 |
| 2.2.2 Multi-point iterative processes | p. 16 |
| 2.2.3 One-point iterative processes with memory | p. 18 |
| 2.3 Concluding remarks | p. 19 |
| Chapter 3 Solving systems of linear equations | p. 21 |
| 3.1 Linear algebra background | p. 22 |
| 3.2 Systems of linear equations | p. 24 |
| 3.3 Types of matrices that arise from applications and analysis | p. 25 |
| 3.3.1 Sparse matrices | p. 25 |
| 3.3.2 Band matrices | p. 25 |
| 3.3.3 Symmetric positive definite matrices | p. 26 |
| 3.3.4 Triangular matrices | p. 26 |
| 3.3.5 Orthogonal matrices | p. 27 |
| 3.3.6 Reducible matrices | p. 27 |
| 3.3.7 Matrices with diagonal dominance | p. 27 |
| 3.4 Error sources | p. 27 |
| 3.5 Condition number | p. 28 |
| 3.6 Direct methods | p. 30 |
| 3.6.1 Basic principles of direct methods | p. 30 |
| 3.6.2 Error estimation for linear systems | p. 33 |
| 3.6.3 Concluding remarks | p. 33 |
| 3.7 Iterative methods | p. 34 |
| 3.7.1 Basic principles of iterative methods | p. 34 |
| 3.7.2 Jacobi method | p. 36 |
| 3.7.3 Gauss-Seidel method | p. 37 |
| 3.7.4 Method of relaxation | p. 39 |
| 3.7.5 Variational iterative methods | p. 42 |
| 3.8 Comparative efficacy of direct and iterative methods | p. 45 |
| Chapter 4 Computational eigenvalue problems | p. 47 |
| 4.1 Basic facts concerning eigenvalue problems | p. 48 |
| 4.2 Localization of eigenvalues | p. 49 |
| 4.3 Power method | p. 49 |
| 4.4 Inverse iteration | p. 51 |
| 4.5 Iteration with a shift of origin | p. 53 |
| 4.6 The QR method | p. 55 |
| 4.7 Concluding remarks | p. 58 |
| Chapter 5 Solving systems of nonlinear equations | p. 59 |
| 5.1 Fixed-point iteration | p. 60 |
| 5.2 Newton's method | p. 65 |
| 5.3 Method with cubic convergence | p. 68 |
| 5.4 Modification of Newton's method | p. 69 |
| 5.5 Making the Newton-based techniques more reliable | p. 71 |
| Chapter 6 Numerical integration | p. 74 |
| 6.1 Simple quadrature formulae | p. 75 |
| 6.2 Computation of integrals with prescribed accuracy | p. 80 |
| 6.3 Integration formulae of Gaussian type | p. 84 |
| 6.4 Dealing with improper integrals | p. 88 |
| 6.4.1 Integration over an infinite interval | p. 88 |
| 6.4.2 Singular integrands | p. 89 |
| 6.5 Multidimensional integration | p. 90 |
| Chapter 7 Introduction to finite difference schemes for ordinary differential equations | p. 97 |
| 7.1 Elementary example of a finite difference scheme | p. 98 |
| 7.2 Approximation and stability | p. 100 |
| 7.2.1 Approximation of differential equations by the difference scheme | p. 101 |
| 7.2.2 Replacement of derivatives by difference expressions | p. 105 |
| 7.2.3 Definition of stability of difference schemes | p. 107 |
| 7.2.4 Convergence as a consequence of approximation and stability | p. 107 |
| 7.3 Numerical solution of initial-value problems | p. 108 |
| 7.3.1 Stability of difference schemes for linear problems | p. 109 |
| 7.3.2 Runge-Kutta methods | p. 112 |
| 7.3.3 Adams type methods | p. 115 |
| 7.3.4 Method for studying stability of difference schemes for nonlinear problems | p. 119 |
| 7.3.5 Systems of differential equations | p. 121 |
| 7.3.6 Methods for stiff differential equations | p. 124 |
| 7.4 Numerical solution of boundary-value problems | p. 128 |
| 7.4.1 The shooting method | p. 128 |
| 7.4.2 Conversion of difference schemes to systems of equations | p. 130 |
| 7.4.3 Method of successive approximations | p. 132 |
| 7.4.4 Method of time development | p. 134 |
| 7.4.5 Accurate approximation of boundary conditions when derivatives are specified at boundary points | p. 138 |
| 7.5 Error estimation and control | p. 140 |
| Chapter 8 Interpolation and Approximation | p. 144 |
| 8.1 Interpolation | p. 144 |
| 8.1.1 Polynomial interpolation | p. 145 |
| 8.1.2 Trigonometric interpolation | p. 148 |
| 8.1.3 Interpolation by splines | p. 151 |
| 8.1.4 Two-dimensional interpolation | p. 154 |
| 8.2 Approximation of functions and data representation | p. 156 |
| 8.2.1 Least-squares approximation | p. 156 |
| 8.2.2 Approximation by orthogonal functions | p. 161 |
| 8.2.3 Approximation by interpolating polynomials | p. 166 |
| Chapter 9 Programming in MATLAB | p. 169 |
| 9.1 Numbers, variables and special characters | p. 170 |
| 9.2 Arithmetic and logical expressions | p. 171 |
| 9.3 Conditional execution | p. 171 |
| 9.4 Loops | p. 173 |
| 9.5 Arrays | p. 174 |
| 9.6 Functions | p. 176 |
| 9.7 Input and output | p. 178 |
| 9.8 Visualization | p. 180 |
| Appendix A Integration formulae of Gaussian type | p. 183 |
| Appendix B Transformations of integration domains | p. 192 |
| Appendix C Stability regions for Runge-Kutta and Adams schemes | p. 194 |
| Appendix D A brief survey of available software | p. 195 |
| LAPACK library for problems in numerical linear algebra | p. 195 |
| IMSL mathematics and statistics libraries | p. 198 |
| Numerical Algorithms Group (NAG) numerical libraries | p. 203 |
| MATLAB numerical functions | p. 210 |
| Bibliography | p. 212 |
| Index | p. 215 |
| List of Examples | |
| Example 1.1 Binary representation of a real number | |
| Example 2.1 Bisection method | |
| Example 2.2 Fixed-point iteration | |
| Example 2.3 Aitken's [delta superscript 2]-process | |
| Example 2.4 Newton's method | |
| Example 2.5 Iterative scheme (2.22) | |
| Example 2.6 Multi-point iterative scheme (2.25) | |
| Example 2.7 Secant method | |
| Example 3.1 Ill-conditioned system | |
| Example 3.2 Jacobi method | |
| Example 3.3 Gauss-Seidel method | |
| Example 3.4 Calculation of the optimal relaxation parameter for iterative scheme (3.24) | |
| Example 3.5 Method SOR | |
| Example 3.6 Rate of convergence of the method of minimal residual | |
| Example 3.7 Application of the Jacobi preconditioner |
