Front Cover -- Theory and Applications of Numerical Analysis -- Copyright Page -- Contents -- Preface -- From the preface to the first edition -- Chapter 1. Introduction -- 1.1 What is numerical analysis? -- 1.2 Numerical algorithms -- 1.3 Properly posed and well-conditioned problems -- Problems -- Chapter 2. Basic analysis -- 2.1 Functions -- 2.2 Limits and derivatives -- 2.3 Sequences and series -- 2.4 Integration -- 2.5 Logarithmic and exponential functions -- Problems -- Chapter 3. Taylor's polynomial and series -- 3.1 Function approximation -- 3.2 Taylor's theorem -- 3.3 Convergence of Taylor series -- 3.4 Taylor series in two variables -- 3.5 Power series -- Problems -- Chapter 4. The interpolating polynomial -- 4.1 Linear interpolation -- 4.2 Polynomial interpolation -- 4.3 Accuracy of interpolation -- 4.4 The Neville-Aitken algorithm -- 4.5 Inverse interpolation -- 4.6 Divided differences -- 4.7 Equally spaced points -- 4.8 Derivatives and differences -- 4.9 Effect of rounding error -- 4.10 Choice of interpolating points -- 4.11 Examples of Bemstein and Runge -- Problems -- Chapter 5. 'Best' approximation -- 5.1 Norms of functions -- 5.2 Best approximations -- 5.3 Least squares approximation -- 5.4 Orthogonal functions -- 5.5 Orthogonal polynomials -- 5.6 Minimax approximation -- 5.7 Chebyshev series -- 5.8 Economization of power series -- 5.9 The Remez algorithms -- 5.10 Further results on minimax approximation -- Problems -- Chapter 6. Splines and other approximations -- 6.1 Introduction -- 6.2 B-splines -- 6.3 Equally spaced knots -- 6.4 Hermite interpolation -- 6.5 Padé and rational approximation -- Problems -- Chapter 7. Numerical integration and differentiation -- 7.1 Numerical integration -- 7.2 Romberg integration -- 7.3 Gaussian integration -- 7.4 Indefinite integrals -- 7.5 Improper integrals -- 7.6 Multiple integrals.

7.7 Numerical differentiation -- 7.8 Effect of errors -- Problems -- Chapter 8. Solution of algebraic equations of one variable -- 8.1 Introduction -- 8.2 The bisection method -- 8.3 Interpolation methods -- 8.4 One-point iterative methods -- 8.5 Faster convergence -- 8.6 Higher order processes -- 8.7 The contraction mapping theorem -- Problems -- Chapter 9. Linear equations -- 9.1 Introduction -- 9.2 Matrices -- 9.3 Linear equations -- 9.4 Pivoting -- 9.5 Analysis of elimination method -- 9.6 Matrix factorization -- 9.7 Compact elimination methods -- 9.8 Symmetric matrices -- 9.9 Tridiagonal matrices -- 9.10 Rounding errors in solving linear equations -- Problems -- Chapter 10. Matrix norms and applications -- 10.1 Determinants, eigenvalues and eigenvectors -- 10.2 Vector norms -- 10.3 Matrix norms -- 10.4 Conditioning -- 10.5 Iterative correction from residual vectors -- 10.6 Iterative methods -- Problems -- Chapter 11. Matrix eigenvalues and eigenvectors -- 11.1 Relations between matrix norms and eigenvalues -- Gerschgorin theorems -- 11.2 Simple and inverse iterative method -- 11.3 Sturm sequence method -- 11.4 The QR algorithm -- 11.5 Reduction to tridiagonal form: Householder's method -- Problems -- Chapter 12. Systems of non-linear equations -- 12.1 Contraction mapping theorem -- 12.2 Newton's method -- Problems -- 13. Ordinary differential equations -- 13.1 Introduction -- 13.2 Difference equations and inequalities -- 13.3 One-step methods -- 13.4 Truncation errors of one-step methods -- 13.5 Convergence of one-step methods -- 13.6 Effect of rounding errors on one-step methods -- 13.7 Methods based on numerical integration -- explicit methods -- 13.8 Methods based on numerical integration -- implicit methods -- 13.9 Iterating with the corrector -- 13.10 Milne's method of estimating truncation errors -- 13.11 Numerical stability.

13.12 Systems and higher order equations -- 13.13 Comparison of step-by-step methods -- Problems -- Chapter 14. Boundary value and other methods for ordinary differential equations -- 14.1 Shooting method for boundary value problems -- 14.2 Boundary value method -- 14.3 Extrapolation to the limit -- 14.4 Deferred correction -- 14.5 Chebyshev series method -- Problems -- Appendix: Computer arithmetic -- A.1 Binary numbers -- A.2 Integers and fixed point fractions -- A.3 Floating point arithmetic -- Problems -- Solutions to selected problems -- References and further reading -- Index.