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Preface. 1. Functional Analysis Ideas. 1.1 Introduction. 1.2 Some Sets. 1.3 Some Special Mappings: Metrics, Norms and Inner Products. 1.4 The Discrete Fourier Series (DFS). 2. Number Representations. 2.1 Introduction. 2.2 Fixed-point Representations. 2.3 Floating-point Representations. 2.4 Rounding Effects in Dot Product Computation. 2.5 Matching Epsilon. 3. Sequence and Series. 3.1 Introduction. 3.2 Cauchy Sequences and Complete Spaces. 3.3 Pointwise Convergence and Uniform Convergence. 3.4 Fourier Series. 3.5 Taylor Series. 3.6 Asymptotic Series. 3.7 More on the Dirichlet Kernel. 3.8 Final Remarks. 4. Linear Systems of Equations. 4.1 Introduction. 4.2 Lease-squares Approximation and Linear Systems. 4.3 Least-squares Approximation and Ill-conditioned Linear Systems. 4.4 Condition Numbers. 4.5 LU Decomposition. 4.6 Least-squares Problems and QR Decomposition. 4.7 Iterative Methods for Linear Systems. 4.8 Final Remarks. 5. Orthogonal Polynomials. 5.1 Introduction. 5.2 General Properties of Orthogonal Polynomials. 5.3 Chebyshev Polynomials. 5.4 Hermite Polynomials. 5.5 Legendre Polynomials. 5.6 An Example of Orethogonal Polynomial Least-Squares Approximation. 5.7 Uniform Approximation. 6. Interpolation. 6.1 Introduction. 6.2 Lagrange Interpolation. 6.3 Newton Interpolation. 6.4 Hermite Interpolation. 6.5 Spline Interpolation. 7. Nonlinear Systems of Equations. 7.1 Introduction. 7.2 Bisection Method. 7.3 Fixed Point Method. 7.4 Newton-Raphson Method. 7.5 Systems of Nonlinear Equations. 7.6 Chaotic Phenomena and a Cryptography Application. 8. Unconstrained Optimization. 8.1 Introduction. 8.2 Problem Statement and Preliminaries. 8.3 Line Searches. 8.4 Newton’s Method. 8.5 Equality Constraints and Lagrange Multipliers. 9. Numerical Integration and Differentiation. 9.1 Introduction. 9.2 Trapezoidal Rule. 9.3 Simpson’s Rule. 9.4 Gausussian Quadrature. 9.5 Romberg Integration. 9.6 Numerical Differentiation. 10. Numerical Solution of Ordinary Differential Equations. 10.1 Introduction. 10.2 First Order ODEs. 10.3 Systems of First Order ODEs. 10.4 Multistep Methods for ODEs. 10.5 Variable Step-size (Adaptive) Methods for ODEs. 10.6 Stiff Systems. 10.7 Final Remarks. 11. Numerical Methods for Eigenproblems. 11.1 Introduction. 11.2 Review of Eigenvalues and Eigenvectors. 11.3 The Matrix Exponential. 11.4 The Power Methods. 11.5 QR Iterations. 12. Numerical Solution of Partial Differential Equations. 12.1 Introduction. 12.2 A Brief Overview of Partial Differential Equations. 12.3 Applications of Hyperbolic PDEs. 12.4 The Finite-Difference (FD) Method. 12.5 The Finite-Difference Time-Domain (FDTD) Method. 13. An Introduction to MATLAB. 13.1 Introduction. 13.2 Startup. 13.3 Some Basic Operators, Operations and Functions. 13.4 Working with Polynomials. 13.5 Loops. 13.6 Plotting and M-files. _

Library of Congress subject headings for this publication: Electric engineering Mathematics, Computer science Mathematics, Numerical analysis