Enhanced descriptions from Syndetics:

This highly successful and scholarly book introduces students with diverse backgrounds to the various types of mathematical analysis that are commonly needed in scientific computing. The subject of numerical analysis is treated from a mathematical point of view, offering a complete analysis of methods for scientific computing with careful proofs and scientific background.An in-depth treatment of the topics of numerical analysis, a more scholarly approach, and a different menu of topics sets Numerical Analysis: Mathematics of Scientific Computing, Second Edition apart form the authors' well-respected and best-selling text: Numerical Mathematics and Computing, Fourth Edition.

Table of contents provided by Syndetics

• Numerical Analysis: What Is It? (p. 1)
• 1 Mathematical Preliminaries (p. 3)
• 1.0 Introduction (p. 3)
• 1.1 Basic Concepts and Taylor's Theorem (p. 3)
• 1.2 Orders of Convergence and Additional Basic Concepts (p. 15)
• 1.3 Difference Equations (p. 28)
• 2 Computer Arithmetic (p. 37)
• 2.0 Introduction (p. 37)
• 2.1 Floating-Point Numbers and Roundoff Errors (p. 37)
• 2.2 Absolute and Relative Errors: Loss of Significance (p. 55)
• 2.3 Stable and Unstable Computations: Conditioning (p. 64)
• 3 Solution of Nonlinear Equations (p. 73)
• 3.0 Introduction (p. 73)
• 3.1 Bisection (Interval Halving) Method (p. 74)
• 3.2 Newton's Method (p. 81)
• 3.3 Secant Method (p. 93)
• 3.4 Fixed Points and Functional Iteration (p. 100)
• 3.5 Computing Roots of Polynomials (p. 109)
• 3.6 Homotopy and Continuation Methods (p. 130)
• 4 Solving Systems of Linear Equations (p. 139)
• 4.0 Introduction (p. 139)
• 4.1 Matrix Algebra (p. 140)
• 4.2 LU and Cholesky Factorizations (p. 149)
• 4.3 Pivoting and Constructing an Algorithm (p. 163)
• 4.4 Norms and the Analysis of Errors (p. 186)
• 4.5 Neumann Series and Iterative Refinement (p. 197)
• 4.6 Solution of Equations by Iterative Methods (p. 207)
• 4.7 Steepest Descent and Conjugate Gradient Methods (p. 232)
• 4.8 Analysis of Roundoff Error in the Gaussian Algorithm (p. 245)
• 5 Selected Topics in Numerical Linear Algebra (p. 254)
• 5.0 Review of Basic Concepts (p. 254)
• 5.1 Matrix Eigenvalue Problem: Power Method (p. 257)
• 5.2 Schur's and Gershgorin's Theorems (p. 265)
• 5.3 Orthogonal Factorizations and Least-Squares Problems (p. 273)
• 5.4 Singular-Value Decomposition and Pseudoinverses (p. 287)
• 5.5 QR-Algorithm of Francis for the Eigenvalue Problem (p. 298)
• 6 Approximating Functions (p. 308)
• 6.0 Introduction (p. 308)
• 6.1 Polynomial Interpolation (p. 308)
• 6.2 Divided Differences (p. 327)
• 6.3 Hermite Interpolation (p. 338)
• 6.4 Spline Interpolation (p. 349)
• 6.5 B-Splines: Basic Theory (p. 366)
• 6.6 B-Splines: Applications (p. 377)
• 6.7 Taylor Series (p. 388)
• 6.8 Best Approximation: Least-Squares Theory (p. 392)
• 6.9 Best Approximation: Chebyshev Theory (p. 405)
• 6.10 Interpolation in Higher Dimensions (p. 420)
• 6.11 Continued Fractions (p. 438)
• 6.12 Trigonometric Interpolation (p. 445)
• 6.13 Fast Fourier Transform (p. 451)
• 6.14 Adaptive Approximation (p. 460)
• 7 Numerical Differentiation and Integration (p. 465)
• 7.1 Numerical Differentiation and Richardson Extrapolation (p. 465)
• 7.2 Numerical Integration Based on Interpolation (p. 478)
• 7.3 Gaussian Quadrature (p. 492)
• 7.4 Romberg Integration (p. 502)
• 7.5 Adaptive Quadrature (p. 507)
• 7.6 Sard's Theory of Approximating Functionals (p. 513)
• 7.7 Bernoulli Polynomials and the Euler-Maclaurin Formula (p. 519)
• 8 Numerical Solution of Ordinary Differential Equations (p. 524)
• 8.0 Introduction (p. 524)
• 8.1 The Existence and Uniqueness of Solutions (p. 524)
• 8.2 Taylor-Series Method (p. 530)
• 8.3 Runge-Kutta Methods (p. 539)
• 8.4 Multistep Methods (p. 549)
• 8.5 Local and Global Errors: Stability (p. 557)
• 8.6 Systems and Higher-Order Ordinary Differential Equations (p. 565)
• 8.7 Boundary-Value Problems (p. 572)
• 8.8 Boundary-Value Problems: Shooting Methods (p. 581)
• 8.9 Boundary-Value Problems: Finite-Differences (p. 589)
• 8.10 Boundary-Value Problems: Collocation (p. 593)
• 8.11 Linear Differential Equations (p. 597)
• 8.12 Stiff Equations (p. 608)
• 9 Numerical Solution of Partial Differential Equations (p. 615)
• 9.0 Introduction (p. 615)
• 9.1 Parabolic Equations: Explicit Methods (p. 615)
• 9.2 Parabolic Equations: Implicit Methods (p. 623)
• 9.3 Problems Without Time Dependence: Finite-Differences (p. 629)
• 9.4 Problems Without Time Dependence: Galerkin Methods (p. 634)
• 9.5 First-Order Partial Differential Equations: Characteristics (p. 642)
• 9.6 Quasilinear Second-Order Equations: Characteristics (p. 650)
• 9.7 Other Methods for Hyperbolic Problems (p. 660)
• 9.8 Multigrid Method (p. 667)
• 9.9 Fast Methods for Poisson's Equation (p. 676)
• 10 Linear Programming and Related Topics (p. 681)
• 10.1 Convexity and Linear Inequalities (p. 681)
• 10.2 Linear Inequalities (p. 689)
• 10.3 Linear Programming (p. 695)
• 10.4 The Simplex Algorithm (p. 700)
• 11 Optimization (p. 711)
• 11.0 Introduction (p. 711)
• 11.1 One-Variable Case (p. 712)
• 11.2 Descent Methods (p. 716)
• 11.3 Analysis of Quadratic Objective Functions (p. 719)
• 11.4 Quadratic-Fitting Algorithms (p. 721)
• 11.5 Nelder-Meade Algorithm (p. 722)
• 11.6 Simulated Annealing (p. 723)
• 11.7 Genetic Algorithms (p. 724)
• 11.8 Convex Programming (p. 725)
• 11.9 Constrained Minimization (p. 726)
• 11.10 Pareto Optimization (p. 727)
• Appendix A An Overview of Mathematical Software (p. 731)
• Bibliography (p. 745)
• Index (p. 771)