Contents
1 Mathematical Preliminaries and Error Analysis
1.1 Review of calculus
1.2 Round-off errors and computer arithmetic
1.3 Algorithms and convergence
1.4 Practice Problems
2 Solutions of Equations in One Variable
2.1 The bisection method
2.2 The method of false position
2.3 Fixed-point iteration
2.4 Newton’s method and the secant method
2.5 Multiple roots and accelerating convergence
2.6 Practice Problems
3 Zeros of Polynomials and Complex Roots
3.1 Complex numbers and the roots of polynomials
3.2 Horner’s method and deflation
3.3 Müller’s method
3.4 Newton’s method in the complex plane
3.5 Practice Problems
4 Direct Methods for Solving Linear Systems
4.1 Linear systems and matrices
4.2 Gaussian elimination and backward substitution
4.3 Pivoting strategies
4.4 Matrix factorization
4.5 Special types of matrices
4.6 Practice Problems
5 Iterative Techniques in Matrix Algebra
5.1 Norms of vectors and matrices
5.2 Eigenvalues and the spectral radius
5.3 The Jacobi and Gauss–Seidel methods
5.4 Relaxation methods
5.5 Error bounds and iterative refinement
5.6 Practice Problems
6 Interpolation and Polynomial Approximation
6.1 Interpolation and the Lagrange polynomial
6.2 Neville’s method
6.3 Divided differences and Newton’s interpolation formulas
6.4 Hermite interpolation
6.5 Practice Problems
7 Cubic Splines and Least-Squares Approximation
7.1 Cubic spline interpolation
7.2 Discrete least-squares approximation
7.3 Practice Problems
8 Numerical Differentiation and Integration
8.1 Numerical differentiation
8.2 Elements of numerical integration
8.3 Composite numerical integration
8.4 Romberg integration
8.5 Gaussian quadrature
8.6 Practice Problems
9 Initial-Value Problems for Ordinary Differential Equations
9.1 The theory of initial-value problems
9.2 Euler’s method
9.3 Higher-order Taylor methods
9.4 Runge–Kutta methods
9.5 Systems and higher-order equations
9.6 Stability
9.7 Practice Problems
References
References