This volume studies algorithms for approximating mathematical problems. It emphasizes stability, convergence, and computational efficiency.
Part I. Foundations
Chapter 1. Numerical Computation
1.1 Floating-point arithmetic 1.2 Rounding errors 1.3 Stability concepts 1.4 Conditioning 1.5 Examples
Chapter 2. Error Analysis
2.1 Absolute and relative error 2.2 Propagation of errors 2.3 Backward and forward error 2.4 Sensitivity analysis 2.5 Examples
Chapter 3. Linear Algebra Review
3.1 Vectors and matrices 3.2 Norms and condition numbers 3.3 Eigenvalues overview 3.4 Applications 3.5 Examples
Part II. Linear Systems
Chapter 4. Direct Methods
4.1 Gaussian elimination 4.2 LU decomposition 4.3 Pivoting strategies 4.4 Applications 4.5 Examples
Chapter 5. Iterative Methods
5.1 Jacobi method 5.2 Gauss–Seidel method 5.3 Conjugate gradient method 5.4 Convergence analysis 5.5 Examples
Chapter 6. Sparse Systems
6.1 Sparse matrix structures 6.2 Storage methods 6.3 Iterative solvers 6.4 Applications 6.5 Examples
Part III. Approximation and Interpolation
Chapter 7. Polynomial Interpolation
7.1 Lagrange interpolation 7.2 Newton interpolation 7.3 Error analysis 7.4 Applications 7.5 Examples
Chapter 8. Numerical Approximation
8.1 Least squares 8.2 Orthogonal polynomials 8.3 Approximation theory links 8.4 Applications 8.5 Examples
Chapter 9. Splines
9.1 Piecewise polynomials 9.2 Cubic splines 9.3 Smoothing methods 9.4 Applications 9.5 Examples
Part IV. Numerical Integration and Differentiation
Chapter 10. Numerical Integration
10.1 Trapezoidal rule 10.2 Simpson’s rule 10.3 Gaussian quadrature 10.4 Error estimates 10.5 Examples
Chapter 11. Numerical Differentiation
11.1 Finite differences 11.2 Error analysis 11.3 Stability 11.4 Applications 11.5 Examples
Chapter 12. Adaptive Methods
12.1 Adaptive quadrature 12.2 Error control 12.3 Efficiency 12.4 Applications 12.5 Examples
Part V. Nonlinear Equations
Chapter 13. Root Finding
13.1 Bisection method 13.2 Newton’s method 13.3 Secant method 13.4 Convergence analysis 13.5 Examples
Chapter 14. Systems of Nonlinear Equations
14.1 Newton’s method for systems 14.2 Fixed point methods 14.3 Applications 14.4 Examples 14.5 Connections
Chapter 15. Optimization Methods
15.1 Gradient methods 15.2 Line search 15.3 Newton-type methods 15.4 Applications 15.5 Examples
Part VI. Differential Equations
Chapter 16. ODE Solvers
16.1 Euler method 16.2 Runge–Kutta methods 16.3 Stability 16.4 Applications 16.5 Examples
Chapter 17. PDE Solvers
17.1 Finite difference methods 17.2 Finite element methods 17.3 Stability and convergence 17.4 Applications 17.5 Examples
Chapter 18. Time-Stepping Methods
18.1 Explicit methods 18.2 Implicit methods 18.3 Stability regions 18.4 Applications 18.5 Examples
Part VII. Advanced Topics
Chapter 19. Eigenvalue Problems
19.1 Power method 19.2 QR algorithm 19.3 Applications 19.4 Examples 19.5 Connections
Chapter 20. High-Performance Computing
20.1 Parallel algorithms 20.2 Memory considerations 20.3 Performance optimization 20.4 Applications 20.5 Examples
Chapter 21. Randomized Numerical Methods
21.1 Monte Carlo methods 21.2 Randomized linear algebra 21.3 Applications 21.4 Examples 21.5 Connections
Part VIII. Research Directions
Chapter 22. Advanced Topics
22.1 Numerical methods for large-scale systems 22.2 Machine learning connections 22.3 Uncertainty quantification 22.4 Modern developments 22.5 Emerging areas
Chapter 23. Open Problems
23.1 Stability challenges 23.2 High-dimensional computation 23.3 Algorithmic complexity 23.4 Error estimation limits 23.5 Future directions
Chapter 24. Historical and Conceptual Notes
24.1 Development of numerical analysis 24.2 Key contributors 24.3 Evolution of algorithms 24.4 Cross-disciplinary impact 24.5 Summary
Appendix
A. Numerical method tables B. Error bound formulas C. Proof techniques checklist D. Algorithm templates E. Cross-reference to other MSC branches