Numerical Methods for Engineers and Scientists : An Introduction with Applications Using MATLAB
by Amos Gilat (The Ohio State Univ. ); Vish SubramaniamEdition: 1
Format: Hardcover
Pub. Date: 2007-04-01
Publisher(s): WILEY
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Summary
Following a unique approach, this innovative book integrates the learning of numerical methods with practicing computer programming and using software tools in applications. It covers the fundamentals while emphasizing the most essential methods throughout the pages. Readers are also given the opportunity to enhance their programming skills using MATLAB to implement algorithms. They'll discover how to use this tool to solve problems in science and engineering.
Author Biography
Amos Gilat, Ph.D., is Professor of Mechanical Engineering at The Ohio State University. Dr. Gilat’s main research interests are in plasticity, specifically, in developing experimental techniques for testing materials over a wide range of strain rates and temperatures and in investigating constitutive relations for viscoplasticity. Dr. Gilat’s research has been supported by the National Science Foundation, NASA, Department of Energy, Department of Defense, and various industries.
Vish Subramaniam, Ph.D., is Professor of Mechanical Engineering & Chemical Physics at The Ohio State University. Dr. Subramaniam’s main research interests are in plasma and laser physics and processes, particularly those that involve non-equilibrium phenomena. Dr. Subramaniam’s research is both experimental and computational, and has been supported by the Department of Defense, National Science Foundation, and numerous industries.
Table of Contents
| Preface | p. vii |
| Introduction | p. 1 |
| Background | p. 1 |
| Representation of Numbers on a Computer | p. 4 |
| Errors in Numerical Solutions | p. 10 |
| Round-Off Errors | p. 10 |
| Truncation Errors | p. 13 |
| Total Error | p. 14 |
| Computers and Programming | p. 15 |
| Problems | p. 18 |
| Mathematical Background | p. 21 |
| Background | p. 21 |
| Concepts from Pre-Calculus and Calculus | p. 22 |
| Vectors | p. 26 |
| Operations with Vectors | p. 28 |
| Matrices and Linear Algebra | p. 30 |
| Operations with Matrices | p. 31 |
| Special Matrices | p. 33 |
| Inverse of a Matrix | p. 34 |
| Properties of Matrices | p. 35 |
| Determinant of a Matrix | p. 35 |
| Cramer's Rule and Solution of a System of Simultaneous Linear Equations | p. 36 |
| Norms | p. 38 |
| Ordinary Differential Equations (ODE) | p. 39 |
| Functions of Two or More Independent Variables | p. 42 |
| Definition of the Partial Derivative | p. 42 |
| Chain Rules | p. 43 |
| The Jacobian | p. 44 |
| Taylor Series Expansion of Functions | p. 45 |
| Taylor Series for a Function of One Variable | p. 45 |
| Taylor Series for a Function of Two Variables | p. 47 |
| Problems | p. 48 |
| Solving Nonlinear Equations | p. 53 |
| Background | p. 53 |
| Estimation of Errors in Numerical Solutions | p. 55 |
| Bisection Method | p. 57 |
| Regula Falsi Method | p. 60 |
| Newton's Method | p. 62 |
| Secant Method | p. 67 |
| Fixed-Point Iteration Method | p. 70 |
| Use of MATLAB Built-In Functions for Solving Nonlinear Equations | p. 73 |
| The fzero Command | p. 74 |
| The roots Command | p. 75 |
| Equations with Multiple Solutions | p. 75 |
| Systems of Nonlinear Equations | p. 77 |
| Newton's Method for Solving a System of Nonlinear Equations | p. 78 |
| Fixed-Point Iteration Method for Solving a System of Nonlinear Equations | p. 82 |
| Problems | p. 84 |
| Solving a System of Linear Equations | p. 93 |
| Background | p. 93 |
| Overview of Numerical Methods for Solving a System of Linear Algebraic Equations | p. 94 |
| Gauss Elimination Method | p. 96 |
| Potential Difficulties When Applying the Gauss Elimination Method | p. 104 |
| Gauss Elimination with Pivoting | p. 106 |
| Gauss-Jordan Elimination Method | p. 109 |
| LU Decomposition Method | p. 112 |
| LU Decomposition Using the Gauss Elimination Procedure | p. 114 |
| LU Decomposition Using Crout's Method | p. 115 |
| LU Decomposition with Pivoting | p. 122 |
| Inverse of a Matrix | p. 122 |
| Calculating the Inverse with the LU Decomposition Method | p. 123 |
| Calculating the Inverse Using the Gauss-Jordan Method | p. 125 |
| Iterative Methods | p. 126 |
| Jacobi Iterative Method | p. 127 |
| Gauss-Seidel Iterative Method | p. 127 |
| Use of MATLAB Built-In Functions for Solving a System of Linear Equations | p. 130 |
| Solving a System of Equations Using MATLAB's Left and Right Division | p. 130 |
| Solving a System of Equations Using MATLAB's Inverse Operation | p. 131 |
| MATLAB's Built-In Function for LU Decomposition | p. 132 |
| Additional MATLAB Built-In Functions | p. 133 |
| Tridiagonal Systems of Equations | p. 135 |
| Error, Residual, Norms, and Condition Number | p. 140 |
| Error and Residual | p. 140 |
| Norms and Condition Number | p. 142 |
| Ill-Conditioned Systems | p. 147 |
| Eigenvalues and Eigenvectors | p. 149 |
| The Basic Power Method | p. 152 |
| The Inverse Power Method | p. 156 |
| The Shifted Power Method | p. 157 |
| The QR Factorization and Iteration Method | p. 157 |
| Use of MATLAB Built-In Functions for Determining Eigenvalues and Eigenvectors | p. 167 |
| Problems | p. 169 |
| Curve Fitting and Interpolation | p. 179 |
| Background | p. 179 |
| Curve Fitting with a Linear Equation | p. 181 |
| Measuring How Good Is a Fit | p. 181 |
| Linear Least-Squares Regression | p. 183 |
| Curve Fitting with Nonlinear Equation by Writing the Equation in a Linear Form | p. 187 |
| Curve Fitting with Quadratic and Higher-Order Polynomials | p. 191 |
| Interpolation Using a Single Polynomial | p. 196 |
| Lagrange Interpolating Polynomials | p. 198 |
| Newton's Interpolating Polynomials | p. 202 |
| Piecewise (Spline) Interpolation | p. 209 |
| Linear Splines | p. 209 |
| Quadratic Splines | p. 211 |
| Cubic Splines | p. 215 |
| Use of MATLAB Built-In Functions for Curve Fitting and Interpolation | p. 222 |
| Curve Fitting with a Linear Combination of Nonlinear Functions | p. 224 |
| Problems | p. 227 |
| Numerical Differentiation | p. 233 |
| Background | p. 233 |
| Finite Difference Approximation of the Derivative | p. 235 |
| Finite Difference Formulas Using Taylor Series Expansion | p. 240 |
| Finite Difference Formulas of First Derivative | p. 240 |
| Finite Difference Formulas for the Second Derivative | p. 245 |
| Summary of Finite Difference Formulas for Numerical Differentiation | p. 247 |
| Differentiation Formulas Using Lagrange Polynomials | p. 249 |
| Differentiation Using Curve Fitting | p. 250 |
| Use of MATLAB Built-In Functions for Numerical Differentiation | p. 250 |
| Richardson's Extrapolation | p. 252 |
| Error in Numerical Differentiation | p. 255 |
| Numerical Partial Differentiation | p. 257 |
| Problems | p. 260 |
| Numerical Integration | p. 267 |
| Background | p. 267 |
| Overview of Approaches in Numerical Integration | p. 268 |
| Rectangle and Midpoint Methods | p. 270 |
| Trapezoidal Method | p. 272 |
| Composite Trapezoidal Method | p. 273 |
| Simpson's Methods | p. 276 |
| Simpson's 1/3 Method | p. 276 |
| Simpson's 3/8 Method | p. 279 |
| Gauss Quadrature | p. 281 |
| Evaluation of Multiple Integrals | p. 287 |
| Use of MATLAB Built-In Functions for Integration | p. 288 |
| Estimation of Error in Numerical Integration | p. 290 |
| Richardson's Extrapolation | p. 292 |
| Romberg Integration | p. 295 |
| Improper Integrals | p. 298 |
| Integrals with Singularities | p. 298 |
| Integrals with Unbounded Limits | p. 299 |
| Problems | p. 300 |
| Ordinary Differential Equations: Initial- Value Problems | p. 307 |
| Background | p. 307 |
| Euler's Methods | p. 312 |
| Euler's Explicit Method | p. 312 |
| Analysis of Truncation Error in Euler's Explicit Method | p. 316 |
| Euler's Implicit Method | p. 320 |
| Modified Euler's Method | p. 323 |
| Midpoint Method | p. 326 |
| Runge-Kutta Methods | p. 327 |
| Second-Order Runge-Kutta Methods | p. 328 |
| Third-Order Runge-Kutta Methods | p. 332 |
| Fourth-Order Runge-Kutta Methods | p. 333 |
| Multistep Methods | p. 339 |
| Adams-Bashforth Method | p. 340 |
| Adams-Moulton Method | p. 341 |
| Predictor-Corrector Methods | p. 342 |
| System of First-Order Ordinary Differential Equations | p. 344 |
| Solving a System of First-Order ODEs Using Euler's Explicit Method | p. 346 |
| Solving a System of First-Order ODEs Using Second-Order Runge-Kutta Method (Modified Euler Version) | p. 346 |
| Solving a System of First-Order ODEs Using the Classical Fourth-Order Runge-Kutta Method | p. 353 |
| Solving a Higher-Order Initial Value Problem | p. 354 |
| Use of MATLAB Built-In Functions for Solving Initial-Value Problems | p. 359 |
| Solving a Single First-Order ODE Using MATLAB | p. 360 |
| Solving a System of First-Order ODEs Using MATLAB | p. 366 |
| Local Truncation Error in Second-Order Range-Kutta Method | p. 369 |
| Step Size For Desired Accuracy | p. 370 |
| Stability | p. 374 |
| Stiff Ordinary Differential Equations | p. 376 |
| Problems | p. 379 |
| Ordinary Differential Equations: Boundary-Value Problems | p. 387 |
| Background | p. 387 |
| The Shooting Method | p. 390 |
| Finite Difference Method | p. 398 |
| Use of MATLAB Built-In Functions for Solving Boundary Value Problems | p. 408 |
| Error and Stability in Numerical Solution of Boundary Value Problems | p. 413 |
| Problems | p. 415 |
| Introductory MATLAB | p. 421 |
| Background | p. 421 |
| Starting with MATLAB | p. 421 |
| Arrays | p. 426 |
| Mathematical Operations with Arrays | p. 431 |
| Script Files | p. 435 |
| Function Files | p. 438 |
| Programming in MATLAB | p. 440 |
| Relational and Logical Operators | p. 440 |
| Conditional Statements, if-else Structures | p. 442 |
| Loops | p. 444 |
| Plotting | p. 445 |
| Problems | p. 447 |
| MATLAB Programs | p. 451 |
| Index | p. 455 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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