This text is appropriate for undergraduate courses on numerical methods and numerical analysis found in engineering, mathematics & computer science departments.   Each chapter uses introductory problems from specific applications. These easy-to-understand problems clarify for the reader the need for a particular mathematical technique. Numerical techniques are explained with an emphasis on why they work.

Table of Contents:
Contents   Preface   1 Foundations 1 1.1 Introductory Examples 1.1.1 Nonlinear Equations 1.1.2 Linear Systems 1.1.3 Numerical Integration 1.2 Useful Background 1.2.1 Results from Calculus 1.2.2 Results from Linear Algebra 1.2.3 A Little Information 1.3.1 Error 1.3.2 Convergence 1.3.3 Getting Better Results 1.4 Using MATLAB 1.4.1 Command Window Computations 1.4.2 M-Files 1.4.3 Programming in MATLAB 1.4.4 Matrix Multiplication 1.5 Chapter Wrap-Up   2 Functions of One Variable 47 2.1 Bisection Method 2.2 Secant-Type Methods 2.2.1 Regula Falsi 2.2.2 Secant Method 2.2.3 Analysis 2.3 Newton’s Method 2.4 Muller’s Method 2.5 Minimization 2.5.1 Golden-Section Search 2.5.2 Brent’s Method 2.6 Beyond the Basics 2.6.1 Using MATLAB’s Functions 2.6.2 Laguerre’s Method 2.6.3 Zeros of a Nonlinear Function 2.7 Chapter Wrap-Up   3 Solving Linear Systems: Direct Methods 95 3.1 Gaussian Elimination 3.1.1 Basic Method 3.1.2 Row Pivoting . 3.2 Gauss-Jordan 3.2.1 Inverse of a Matrix 3.3 Tridiagonal Systems 3.4 Further Topics 3.4.1 MATLAB’s Methods 3.4.2 Condition of a Matrix 3.4.3 Iterative Refinement 3.5 Chapter Wrap-Up   4 LU and QR Factorization 135 4.1 LU Factorization 4.1.1 Using Gaussian Elimination 4.1.2 Direct LU Factorization 4.1.3 Applications 4.2 Matrix Transformations 4.2.1 Householder Transformation 4.2.2 Givens Rotations 4.3 QR Factorization 4.3.1 Using Householder Transformations 4.3.2 Using Givens Rotations 4.4 Beyond the Basics 4.4.1 LU Factorization with Implicit Row Pivoting 4.4.2 Efficient Conversion to Hessenberg Form 4.4.3 Using MATLAB’s Functions 4.5 Chapter Wrap-Up   5 Eigenvalues and Eigenvectors 179 5.1 Power Method 5.1.1 Basic Power Method 5.1.2 Rayleigh Quotient 5.1.3 Shifted Power Method 5.1.4 Accelerating Convergence 5.2 Inverse Power Method 5.2.1 General Inverse Power Method 5.2.2 Convergence 5.3 QR Method 5.3.1 Basic QR Method 5.3.2 Better QR Method 5.3.3 Finding Eigenvectors 5.3.4 Accelerating Convergence 5.4 Further Topics 5.4.1 Singular Value Decomposition 5.4.2 MATLAB’s Methods 5.5 Chapter Wrap-Up   6 Solving Linear Systems: Iterative Methods 213 6.1 Jacobi Method 6.2 Gauss-Seidel Method 6.3 Successive Over-Relaxation 6.4 Beyond the Basics 6.4.1 MATLAB’s Built-In Functions 6.4.2 Conjugate Gradient Methods 6.4.3 GMRES 6.4.4 Simplex Method 6.5 Chapter Wrap-Up   7 Nonlinear Functions of Several Variables 251 7.1 Nonlinear Systems 7.1.1 Newton’s Method 7.1.2 Secant Methods 7.1.3 Fixed-Point Iteration 7.2 Minimization 7.2.1 Descent Methods 7.2.2 Quasi-Newton Methods 7.3 Further Topics 7.3.1 Levenberg-Marquardt Method 7.3.2 Nelder-Mead Simplex Search 7.4 Chapter Wrap-Up   8 Interpolation 275 8.1 Polynomial Interpolation 8.1.1 Lagrange Form 8.1.2 Newton Form 8.1.3 Difficulties 8.2 Hermite Interpolation 8.3 Piecewise Polynomial Interpolation 8.3.1 Piecewise Linear Interpolation 8.3.2 Piecewise Quadratic Interpolation 8.3.3 Piecewise Cubic Hermite Interpolation 8.3.4 Cubic Spline Interpolation 8.4 Beyond the Basics 8.4.1 Rational-Function Interpolation 8.4.2 Using MATLAB’s Functions 8.5 Chapter Wrap-Up   9 Approximation 333 9.1 Least-Squares Approximation 9.1.1 Approximation by a Straight Line 9.1.2 Approximation by a Parabola 9.1.3 General Least-Squares Approximation 9.1.4 Approximation for Other Functional Forms 9.2 Continuous Least-Squares Approximation 9.2.1 Approximation Using Powers of x 9.2.2 Orthogonal Polynomials 9.2.3 Legendre Polynomials 9.2.4 Chebyshev Polynomials 9.3 Function Approximation at a Point 9.3.1 Pad´e Approximation 9.3.2 Taylor Approximation 9.4 Further Topics 9.4.1 Bezier Curves 9.4.2 Using MATLAB’s Functions 9.5 Chapter Wrap-Up   10 Fourier Methods 373 10.1 Fourier Approximation and Interpolation 10.1.1 Derivation 10.1.2 Data on Other Intervals 10.2 Radix-2 Fourier Transforms 10.2.1 Discrete Fourier Transform 10.2.2 Fast Fourier Transform 10.2.3 Matrix Form of FFT 10.2.4 Algebraic Form of FFT 10.3 Mixed-Radix FFT 10.4 Using MATLAB’s Functions 10.5 Chapter Wrap-Up   11 Numerical Differentiation and Integration 405 11.1 Differentiation 11.1.1 First Derivatives 11.1.2 Higher Derivatives 11.1.3 Partial Derivatives 11.1.4 Richardson Extrapolation 11.2 Numerical Integration 11.2.1 Trapezoid Rule 11.2.2 Simpson’s Rule 11.2.3 Newton-Cotes Open Formulas 11.2.4 Extrapolation Methods 11.3 Quadrature 11.3.1 Gaussian Quadrature 11.3.2 Other Gauss-Type Quadratures 11.4 MATLAB’s Methods 11.4.1 Differentiation 11.4.2 Integration 11.5 Chapter Wrap-Up   12 Ordinary Differential Equations: Fundamentals 445 12.1 Euler’s Method 12.1.1 Geometric Introduction 12.1.2 Approximating the Derivative 12.1.3 Approximating the Integral 12.1.4 Using Taylor Series 12.2 Runge-Kutta Methods 12.2.1 Second-Order Runge-Kutta Methods 12.2.2 Third-Order Runge-Kutta Methods 12.2.3 Classic Runge-Kutta Method 12.2.4 Fourth-Order Runge-Kutta Methods 12.2.5 Fifth-Order Runge-Kutta Methods 12.2.6 Runge-Kutta-Fehlberg Methods 12.3 Multistep Methods 12.3.1 Adams-Bashforth Methods 12.3.2 Adams-Moulton Methods 12.3.3 Adams Predictor-Corrector Methods 12.3.4 Other Predictor-Corrector Methods 12.4 Further Topics 12.4.1 MATLAB’s Methods 12.4.2 Consistency and Convergence 12.5 Chapter Wrap-Up   13 ODE: Systems, Stiffness, Stability 499 13.1 Systems 13.1.1 Systems of Two ODE 13.1.2 Euler’s Method for Systems 13.1.3 Runge-Kutta Methods for Systems 13.1.4 Multistep Methods for Systems 13.1.5 Second-Order ODE 13.2 Stiff ODE 13.2.1 BDF Methods 13.2.2 Implicit Runge-Kutta Methods 13.3 Stability 13.3.1 A-Stable and Stiffly Stable Methods 13.3.2 Stability in the Limit 13.4 Further Topics 13.4.1 MATLAB’s Methods for Stiff ODE 13.4.2 Extrapolation Methods 13.4.3 Rosenbrock Methods 13.4.4 Multivalue Methods 13.5 Chapter Wrap-Up   14 ODE: Boundary-Value Problems 561 14.1 Shooting Method 14.1.1 Linear ODE 14.1.2 Nonlinear ODE 14.2 Finite-Difference Method 14.2.1 Linear ODE 14.2.2 Nonlinear ODE 14.3 Function Space Methods 14.3.1 Collocation 14.3.2 Rayleigh-Ritz 14.4 Chapter Wrap-Up   15 Partial Differential Equations 593 15.1 Heat Equation: Parabolic PDE 15.1.1 Explicit Method 15.1.2 Implicit Method 15.1.3 Crank-Nicolson Method 15.1.4 Insulated Boundary 15.2 Wave Equation: Hyperbolic PDE 15.2.1 Explicit Method 15.2.2 Implicit Method 15.3 Poisson Equation: Elliptic PDE 15.4 Finite-Element Method for Elliptic PDE 15.4.1 Defining the Subregions 15.4.2 Defining the Basis Functions 15.4.3 Computing the Coefficients 15.4.4 Using MATLAB 15.5 Chapter Wrap-Up   Bibliography 643   Answers 653   Index 667