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.

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.

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
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