Enhanced descriptions from Syndetics:

Emphasis on "cause and effect" in numerical mathematics.
* Flexibility with computing languages--the book is not specific to any one computing language.

Table of contents provided by Syndetics

• Preface (p. vii)
• Acknowledgments (p. x)
• Chapter 1 Introductory Concepts and Calculus Review (p. 1)
• 1.1 Basic Tools of Calculus (p. 2)
• 1.1.1 Taylor's Theorem (p. 2)
• 1.1.2 Mean Value and Extreme Value Theorems (p. 9)
• 1.2 Error, Approximate Equality, and Asymptotic Order Notation (p. 14)
• 1.2.1 Error (p. 14)
• 1.2.2 Notation: Approximate Equality (p. 15)
• 1.2.3 Notation: Asymptotic Order (p. 16)
• 1.3 A Primer on Computer Arithmetic (p. 20)
• 1.4 A Word on Computer Languages and Software (p. 28)
• 1.5 Simple Approximations (p. 29)
• 1.6 Application: Approximating the Natural Logarithm (p. 33)
• Chapter 2 A Survey of Simple Methods and Tools (p. 37)
• 2.1 Homer's Rule and Nested Multiplication (p. 37)
• 2.2 Difference Approximations to the Derivative (p. 40)
• 2.3 Application: Euler's Method for Initial Value Problems (p. 48)
• 2.4 Linear Interpolation (p. 53)
• 2.5 Application: The Trapezoid Rule (p. 60)
• 2.6 Solution of Tridiagonal Linear Systems (p. 69)
• 2.7 Application: Simple Two-Point Boundary Value Problems (p. 75)
• Chapter 3 Root Finding (p. 80)
• 3.1 The Bisection Method (p. 81)
• 3.2 Newton's Method: Derivation and Examples (p. 87)
• 3.3 How to Stop Newton's Method (p. 93)
• 3.4 Application: Division Using Newton's Method (p. 96)
• 3.5 The Newton Error Formula (p. 100)
• 3.6 Newton's Method: Theory and Convergence (p. 105)
• 3.7 Application: Computation of the Square Root (p. 109)
• 3.8 The Secant Method: Derivation and Examples (p. 111)
• 3.9 Fixed-Point Iteration (p. 116)
• 3.10 Special Topics in Root-Finding Methods (p. 126)
• 3.10.1 Extrapolation and Acceleration (p. 126)
• 3.10.2 Variants of Newton's Method (p. 131)
• 3.10.3 The Secant Method: Derivation and Examples (p. 135)
• 3.10.4 Multiple Roots (p. 139)
• 3.10.5 In Search of Fast Global Convergence: Hybrid Algorithm (p. 144)
• Literature and Software Discussion (p. 149)
• Chapter 4 Interpolation and Approximation (p. 150)
• 4.1 Lagrange Interpolation (p. 151)
• 4.2 Newton Interpolation and Divided Differences (p. 156)
• 4.3 Interpolation Error (p. 166)
• 4.4 Application: Muller's Method and Inverse Quadratic (p. 170)
• 4.5 Application: More Approximations to the Derivative (p. 174)
• 4.6 Hermite Interpolation (p. 177)
• 4.7 Piecewise Polynomial Interpolation (p. 182)
• 4.8 An Introduction to Splines (p. 189)
• 4.8.1 Definition of the Problem (p. 189)
• 4.8.2 Cubic B-Splines (p. 190)
• 4.9 Application: Solution of Boundary Value Problems (p. 204)
• 4.10 Least Squares Concepts in Approximation (p. 209)
• 4.10.1 An Introduction to Data Fitting (p. 209)
• 4.10.2 Least Squares Approximation and Orthogonal (p. 215)
• 4.11 Advanced Topics in Interpolation Error (p. 230)
• 4.11.1 Stability of Polynomial Interpolation (p. 230)
• 4.11.2 The Runge Example (p. 234)
• 4.11.3 The Chebyshev Nodes (p. 237)
• 4.12 Literature and Software Discussion (p. 243)
• Chapter 5 Numerical Intergration (p. 245)
• 5.1 A Review of the Definite Integral (p. 246)
• 5.2 Improving the Trapezoid Rule (p. 248)
• 5.3 Simpson's Rule and Degree of Precision (p. 253)
• 5.4 The Midpoint Rule (p. 265)
• 5.5 Application: Stirling's Formula (p. 268)
• 5.6 Gaussian Quadrature (p. 270)
• 5.7 Extrapolation Methods (p. 281)
• 5.8 Special Topics in Numerical Integration (p. 288)
• 5.8.1 Romberg Integration (p. #288)
• 5.8.2 Quadrature with Nonsmooth Integrands (p. 293)
• 5.8.3 Adaptive Integration (p. 299)
• 5.8.4 Peano Estimates for the Trapezoid Rule (p. 307)
• Literature and Software Discussion (p. 311)
• Chapter 6 Numerical Methods for Ordinary Differential Equations (p. 312)
• 6.1 The Initial Value Problem: Background (p. 313)
• 6.2 Euler's Method (p. 318)
• 6.3 Analysis of Euler's Method (p. 322)
• 6.4 Variants of Euler's Method (p. 326)
• 6.4.1 The Residual and Truncation Error (p. 329)
• 6.4.2 Implicit Methods and Predictor-Corrector Schemes (p. 331)
• 6.4.3 Starting Values and Multistep Methods (p. 337)
• 6.4.4 The Midpoint Method and Weak Stability (p. 339)
• 6.5 Single Step Methods: Runge-Kutta (p. 343)
• 6.6 Multistep Methods (p. 350)
• 6.6.1 The Adams Families (p. 350)
• 6.6.2 The BDF Family (p. 354)
• 6.7 Stability Issues (p. 356)
• 6.7.1 Stability Theory for Multistep Methods (p. 356)
• 6.7.2 Stability Regions (p. 360)
• 6.8 Application to Systems of Equations (p. 363)
• 6.8.1 Implementation Issues and Examples (p. 363)
• 6.8.2 Stiff Equations (p. 366)
• 6.8.3 A-Stability (p. 368)
• 6.9 Adaptive Solvers (p. 370)
• 6.10 Boundary Value Problems (p. 383)
• 6.10.1 Simple Difference Methods (p. 383)
• 6.10.2 Shooting Methods (p. 388)
• 6.11 Literature and Software Discussion (p. 392)
• Chapter 7 Numerical Methods for the Solution of Systems (p. 394)
• 7.1 Linear Algebra Review (p. 395)
• 7.2 Linear Systems and Gaussian Elimination (p. 397)
• 7.3 Operation Counts (p. 404)
• 7.4 The LU Factorization (p. 406)
• 7.5 Perturbation, Conditioning, and Stability (p. 416)
• 7.5.1 Vector and Matrix Norms of Equations (p. 417)
• 7.5.2 The Condition Number and Perturbations (p. 419)
• 7.5.3 Estimating the Condition Number (p. 427)
• 7.5.4 Interative Refinement (p. 430)
• 7.6 SPD Matrices and the Cholesky Decomposition (p. 434)
• 7.7 Iterative Methods for Linear Systems: A Brief Survey (p. 437)
• 7.8 Nonlinear Systems: Newton's Method and Related Ideas (p. 446)
• 7.8.1 Newton's Method (p. 447)
• 7.8.2 Fixed-Point Methods (p. 450)
• 7.9 Application: Numerical Solution of Nonlinear BVPs (p. 452)
• 7.10 Literature and Software Discussion (p. 454)
• Chapter 8 Approximate Solution of the Algebraic Eigenvalue Problem (p. 456)
• 8.1 Eigenvalue Review (p. 457)
• 8.2 Reduction to Hessenberg Form (p. 463)
• 8.3 Power Methods (p. 471)
• 8.4 An Overview of the QR Iteration (p. 490)
• Literature and Software Discussion (p. 499)
• Chapter 9 A Survey of Finite Difference Methods for Partial Differential Equations (p. 500)
• 9.1 Difference Methods for the Diffusion Equation (p. 501)
• 9.1.1 The Basic Problem (p. 501)
• 9.1.2 The Explicit Method and Stability (p. 502)
• 9.1.3 Implicit Methods and the Crank-Nicolson Method (p. 507)
• 9.2 Difference Methods for Poisson Equations (p. 517)
• 9.2.1 Discretization (p. 517)
• 9.2.2 Banded Cholesky Solvers (p. 520)
• 9.2.3 Iteration and the Method of Conjugate Gradients (p. 522)
• Literature and Software Discussion (p. 533)
• Appendix A Proofs of Selected Theorems, and Other Additional Material (p. 535)
• A.1 Proofs of the Interpolation Error Theorems (p. 535)
• A.2 Proof of Stability (p. 537)
• A.3 Stiff Systems of Differential Equations and Eigenvalues (p. 538)
• A.4 The Matrix Perturbation Theorem (p. 540)
• Appendix B Proofs of Selected Theorems, and Other Additional Material (p. 542)
• Index (p. 549)