Enhanced descriptions from Syndetics:

Uses introductory problems from particular applications that are easy to understand and show the reader that there is a need for a particular mathematical technique. Numerical techniques are explained from basics with an emphasis on why they work. Discusses the contexts and reasons for selection of each problem and solution method. Worked-out examples are very realistic and not contrived.

Table of contents provided by Syndetics

• Preface (p. xi)
• 1 Foundations (p. 1)
• 1.1 Introductory Examples (p. 4)
• 1.1.1 Nonlinear Equations (p. 4)
• 1.1.2 Linear Systems (p. 6)
• 1.1.3 Numerical Integration (p. 8)
• 1.2 Useful Background (p. 10)
• 1.2.1 Results from Calculus (p. 10)
• 1.2.2 Results from Linear Algebra (p. 11)
• 1.2.3 A Little Information about Computers (p. 13)
• 1.3 Some Basic Issues (p. 16)
• 1.3.1 Error (p. 16)
• 1.3.2 Convergence (p. 22)
• 1.3.3 Getting Better Results (p. 26)
• 1.4 Using MATLAB (p. 31)
• 1.4.1 Command Window Computations (p. 31)
• 1.4.2 M-Files (p. 35)
• 1.4.3 Programming in MATLAB (p. 37)
• 1.4.4 Matrix Multiplication (p. 39)
• 1.5 Chapter Wrap-Up (p. 41)
• 2 Functions of One Variable (p. 47)
• 2.1 Bisection Method (p. 50)
• 2.2 Secant-Type Methods (p. 54)
• 2.2.1 Regula Falsi (p. 55)
• 2.2.2 Secant Method (p. 58)
• 2.2.3 Analysis (p. 61)
• 2.3 Newton's Method (p. 64)
• 2.4 Muller's Method (p. 71)
• 2.5 Minimization (p. 76)
• 2.5.1 Golden-Section Search (p. 76)
• 2.5.2 Brent's Method (p. 79)
• 2.6 Beyond the Basics (p. 80)
• 2.6.1 Using MATLAB's Functions (p. 80)
• 2.6.2 Laguerre's Method (p. 82)
• 2.6.3 Zeros of a Nonlinear Function (p. 85)
• 2.7 Chapter Wrap-Up (p. 88)
• 3 Solving Linear Systems: Direct Methods (p. 95)
• 3.1 Gaussian Elimination (p. 98)
• 3.1.1 Basic Method (p. 98)
• 3.1.2 Row Pivoting (p. 107)
• 3.2 Gauss-Jordan (p. 112)
• 3.2.1 Inverse of a Matrix (p. 113)
• 3.3 Tridiagonal Systems (p. 114)
• 3.4 Further Topics (p. 119)
• 3.4.1 MATLAB's Methods (p. 119)
• 3.4.2 Condition of a Matrix (p. 121)
• 3.4.3 Iterative Refinement (p. 123)
• 3.5 Chapter Wrap-Up (p. 125)
• 4 LU and QR Factorization (p. 135)
• 4.1 LU Factorization (p. 138)
• 4.1.1 Using Gaussian Elimination (p. 138)
• 4.1.2 Direct LU Factorization (p. 146)
• 4.1.3 Applications (p. 150)
• 4.2 Matrix Transformations (p. 154)
• 4.2.1 Householder Transformation (p. 155)
• 4.2.2 Givens Rotations (p. 162)
• 4.3 QR Factorization (p. 164)
• 4.3.1 Using Householder Transformations (p. 164)
• 4.3.2 Using Givens Rotations (p. 166)
• 4.4 Beyond the Basics (p. 168)
• 4.4.1 LU Factorization with Implicit Row Pivoting (p. 168)
• 4.4.2 Efficient Conversion to Hessenberg Form (p. 170)
• 4.4.3 Using MATLAB's Functions (p. 171)
• 4.5 Chapter Wrap-Up (p. 172)
• 5 Eigenvalues and Eigenvectors (p. 179)
• 5.1 Power Method (p. 182)
• 5.1.1 Basic Power Method (p. 183)
• 5.1.2 Rayleigh Quotient (p. 186)
• 5.1.3 Shifted Power Method (p. 188)
• 5.1.4 Accelerating Convergence (p. 189)
• 5.2 Inverse Power Method (p. 190)
• 5.2.1 General Inverse Power Method (p. 192)
• 5.2.2 Convergence (p. 193)
• 5.3 QR Method (p. 194)
• 5.3.1 Basic QR Method (p. 194)
• 5.3.2 Better QR Method (p. 196)
• 5.3.3 Finding Eigenvectors (p. 198)
• 5.3.4 Accelerating Convergence (p. 199)
• 5.4 Further Topics (p. 202)
• 5.4.1 Singular Value Decomposition (p. 202)
• 5.4.2 MATLAB's Methods (p. 203)
• 5.5 Chapter Wrap-Up (p. 204)
• 6 Solving Linear Systems: Iterative Methods (p. 213)
• 6.1 Jacobi Method (p. 217)
• 6.2 Gauss-Seidel Method (p. 224)
• 6.3 Successive Over-Relaxation (p. 228)
• 6.4 Beyond the Basics (p. 232)
• 6.4.1 MATLAB's Built-in Functions (p. 232)
• 6.4.2 Conjugate Gradient Methods (p. 234)
• 6.4.3 GMRES (p. 238)
• 6.4.4 Simplex Method (p. 240)
• 6.5 Chapter Wrap-Up (p. 243)
• 7 Nonlinear Functions of Several Variables (p. 251)
• 7.1 Nonlinear Systems (p. 254)
• 7.1.1 Newton's Method (p. 254)
• 7.1.2 Secant Methods (p. 260)
• 7.1.3 Fixed-Point Iteration (p. 262)
• 7.2 Minimization (p. 264)
• 7.2.1 Descent Methods (p. 264)
• 7.2.2 Quasi-Newton Methods (p. 266)
• 7.3 Further Topics (p. 268)
• 7.3.1 Levenberg-Marquardt Method (p. 268)
• 7.3.2 Nelder-Mead Simplex Search (p. 269)
• 7.4 Chapter Wrap-Up (p. 270)
• 8 Interpolation (p. 275)
• 8.1 Polynomial Interpolation (p. 278)
• 8.1.1 Lagrange Form (p. 278)
• 8.1.2 Newton Form (p. 284)
• 8.1.3 Difficulties (p. 290)
• 8.2 Hermite Interpolation (p. 294)
• 8.3 Piecewise Polynomial Interpolation (p. 299)
• 8.3.1 Piecewise Linear Interpolation (p. 300)
• 8.3.2 Piecewise Quadratic Interpolation (p. 301)
• 8.3.3 Piecewise Cubic Hermite Interpolation (p. 304)
• 8.3.4 Cubic Spline Interpolation (p. 305)
• 8.4 Beyond the Basics (p. 312)
• 8.4.1 Rational-Function Interpolation (p. 312)
• 8.4.2 Using MATLAB's Functions (p. 316)
• 8.5 Chapter Wrap-Up (p. 323)
• 9 Approximation (p. 333)
• 9.1 Least-Squares Approximation (p. 336)
• 9.1.1 Approximation by a Straight Line (p. 336)
• 9.1.2 Approximation by a Parabola (p. 342)
• 9.1.3 General Least-Squares Approximation (p. 346)
• 9.1.4 Approximation for Other Functional Forms (p. 348)
• 9.2 Continuous Least-Squares Approximation (p. 350)
• 9.2.1 Approximation Using Powers of x (p. 350)
• 9.2.2 Orthogonal Polynomials (p. 352)
• 9.2.3 Legendre Polynomials (p. 354)
• 9.2.4 Chebyshev Polynomials (p. 356)
• 9.3 Function Approximation at a Point (p. 358)
• 9.3.1 Pade Approximation (p. 358)
• 9.3.2 Taylor Approximation (p. 361)
• 9.4 Further Topics (p. 362)
• 9.4.1 Bezier Curves (p. 362)
• 9.4.2 Using MATLAB's Functions (p. 364)
• 9.5 Chapter Wrap-Up (p. 366)
• 10 Fourier Methods (p. 373)
• 10.1 Fourier Approximation and Interpolation (p. 376)
• 10.1.1 Derivation (p. 381)
• 10.1.2 Data on Other Intervals (p. 384)
• 10.2 Radix-2 Fourier Transforms (p. 386)
• 10.2.1 Discrete Fourier Transform (p. 386)
• 10.2.2 Fast Fourier Transform (p. 387)
• 10.2.3 Matrix Form of FFT (p. 388)
• 10.2.4 Algebraic Form of FFT (p. 389)
• 10.3 Mixed-Radix FFT (p. 392)
• 10.4 Using MATLAB's Functions (p. 396)
• 10.5 Chapter Wrap-Up (p. 400)
• 11 Numerical Differentiation and Integration (p. 405)
• 11.1 Differentiation (p. 408)
• 11.1.1 First Derivatives (p. 408)
• 11.1.2 Higher Derivatives (p. 412)
• 11.1.3 Partial Derivatives (p. 413)
• 11.1.4 Richardson Extrapolation (p. 414)
• 11.2 Numerical Integration (p. 416)
• 11.2.1 Trapezoid Rule (p. 417)
• 11.2.2 Simpson's Rule (p. 420)
• 11.2.3 Newton-Cotes Open Formulas (p. 426)
• 11.2.4 Extrapolation Methods (p. 428)
• 11.3 Quadrature (p. 431)
• 11.3.1 Gaussian Quadrature (p. 431)
• 11.3.2 Other Gauss-Type Quadratures (p. 435)
• 11.4 MATLAB's Methods (p. 437)
• 11.4.1 Differentiation (p. 437)
• 11.4.2 Integration (p. 437)
• 11.5 Chapter Wrap-Up (p. 438)
• 12 Ordinary Differential Equations: Fundamentals (p. 445)
• 12.1 Euler's Method (p. 447)
• 12.1.1 Geometric Introduction (p. 447)
• 12.1.2 Approximating the Derivative (p. 448)
• 12.1.3 Approximating the Integral (p. 449)
• 12.1.4 Using Taylor Series (p. 451)
• 12.2 Runge-Kutta Methods (p. 452)
• 12.2.1 Second-Order Runge-Kutta Methods (p. 452)
• 12.2.2 Third-Order Runge-Kutta Methods (p. 457)
• 12.2.3 Classic Runge-Kutta Method (p. 459)
• 12.2.4 Fourth-Order Runge-Kutta Methods (p. 462)
• 12.2.5 Fifth-Order Runge-Kutta Methods (p. 464)
• 12.2.6 Runge-Kutta-Fehlberg Methods (p. 465)
• 12.3 Multistep Methods (p. 474)
• 12.3.1 Adams-Bashforth Methods (p. 476)
• 12.3.2 Adams-Moulton Methods (p. 479)
• 12.3.3 Adams Predictor-Corrector Methods (p. 480)
• 12.3.4 Other Predictor-Corrector Methods (p. 485)
• 12.4 Further Topics (p. 487)
• 12.4.1 MATLAB's Methods (p. 487)
• 12.4.2 Consistency and Convergence (p. 488)
• 12.5 Chapter Wrap-Up (p. 490)
• 13 ODE: Systems, Stiffness, Stability (p. 499)
• 13.1 Systems (p. 502)
• 13.1.1 Systems of Two ODE (p. 504)
• 13.1.2 Euler's Method for Systems (p. 510)
• 13.1.3 Runge-Kutta Methods for Systems (p. 512)
• 13.1.4 Multistep Methods for Systems (p. 516)
• 13.1.5 Second-Order ODE (p. 522)
• 13.2 Stiff ODE (p. 526)
• 13.2.1 BDF Methods (p. 527)
• 13.2.2 Implicit Runge-Kutta Methods (p. 529)
• 13.3 Stability (p. 530)
• 13.3.1 A-Stable and Stiffly Stable Methods (p. 532)
• 13.3.2 Stability in the Limit (p. 533)
• 13.4 Further Topics (p. 536)
• 13.4.1 MATLAB's Methods for Stiff ODE (p. 536)
• 13.4.2 Extrapolation Methods (p. 537)
• 13.4.3 Rosenbrock Methods (p. 539)
• 13.4.4 Multivalue Methods (p. 539)
• 13.5 Chapter Wrap-Up (p. 552)
• 14 ODE: Boundary-Value Problems (p. 561)
• 14.1 Shooting Method (p. 565)
• 14.1.1 Linear ODE (p. 565)
• 14.1.2 Nonlinear ODE (p. 570)
• 14.2 Finite-Difference Method (p. 576)
• 14.2.1 Linear ODE (p. 576)
• 14.2.2 Nonlinear ODE (p. 580)
• 14.3 Function Space Methods (p. 582)
• 14.3.1 Collocation (p. 582)
• 14.3.2 Rayleigh-Ritz (p. 586)
• 14.4 Chapter Wrap-Up (p. 588)
• 15 Partial Differential Equations (p. 593)
• 15.1 Heat Equation: Parabolic PDE (p. 598)
• 15.1.1 Explicit Method (p. 599)
• 15.1.2 Implicit Method (p. 604)
• 15.1.3 Crank-Nicolson Method (p. 608)
• 15.1.4 Insulated Boundary (p. 611)
• 15.2 Wave Equation: Hyperbolic PDE (p. 612)
• 15.2.1 Explicit Method (p. 614)
• 15.2.2 Implicit Method (p. 616)
• 15.3 Poisson Equation: Elliptic PDE (p. 618)
• 15.4 Finite-Element Method for Elliptic PDE (p. 622)
• 15.4.1 Defining the Subregions (p. 623)
• 15.4.2 Defining the Basis Functions (p. 624)
• 15.4.3 Computing the Coefficients (p. 626)
• 15.4.4 Using MATLAB (p. 629)
• 15.5 Chapter Wrap-Up (p. 634)
• Bibliography (p. 643)
• Answers (p. 653)
• Index (p. 667)