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| Preface | |
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| Functional Analysis Ideas | |
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| Introduction | |
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| Some Sets | |
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| Some Special Mappings: Metrics, Norms, and Inner Products | |
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| Metrics and Metric Spaces | |
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| Norms and Normed Spaces | |
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| Inner Products and Inner Product Spaces | |
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| The Discrete Fourier Series (DFS) | |
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| Complex Arithmetic | |
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| Elementary Logic | |
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| References | |
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| Problems | |
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| Number Representations | |
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| Introduction | |
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| Fixed-Point Representations | |
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| Floating-Point Representations | |
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| Rounding Effects in Dot Product Computation | |
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| Machine Epsilon | |
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| Review of Binary Number Codes | |
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| References | |
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| Problems | |
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| Sequences and Series | |
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| Introduction | |
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| Cauchy Sequences and Complete Spaces | |
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| Pointwise Convergence and Uniform Convergence | |
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| Fourier Series | |
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| Taylor Series | |
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| Asymptotic Series | |
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| More on the Dirichlet Kernel | |
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| Final Remarks | |
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| COordinate Rotation DIgital Computing (CORDIC) | |
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| Introduction | |
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| The Concept of a Discrete Basis | |
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| Rotating Vectors in the Plane | |
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| Computing Arctangents | |
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| Final Remarks | |
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| Mathematical Induction | |
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| Catastrophic Cancellation | |
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| References | |
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| Problems | |
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| Linear Systems of Equations | |
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| Introduction | |
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| Least-Squares Approximation and Linear Systems | |
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| Least-Squares Approximation and Ill-Conditioned Linear Systems | |
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| Condition Numbers | |
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| LU Decomposition | |
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| Least-Squares Problems and QR Decomposition | |
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| Iterative Methods for Linear Systems | |
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| Final Remarks | |
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| Hilbert Matrix Inverses | |
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| SVD and Least Squares | |
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| References | |
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| Problems | |
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| Orthogonal Polynomials | |
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| Introduction | |
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| General Properties of Orthogonal Polynomials | |
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| Chebyshev Polynomials | |
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| Hermite Polynomials | |
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| Legendre Polynomials | |
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| An Example of Orthogonal Polynomial Least-Squares Approximation | |
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| Uniform Approximation | |
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| References | |
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| Problems | |
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| Interpolation | |
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| Introduction | |
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| Lagrange Interpolation | |
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| Newton Interpolation | |
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| Hermite Interpolation | |
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| Spline Interpolation | |
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| References | |
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| Problems | |
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| Nonlinear Systems of Equations | |
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| Introduction | |
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| Bisection Method | |
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| Fixed-Point Method | |
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| Newton-Raphson Method | |
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| The Method | |
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| Rate of Convergence Analysis | |
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| Breakdown Phenomena | |
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| Systems of Nonlinear Equations | |
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| Fixed-Point Method | |
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| Newton-Raphson Method | |
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| Chaotic Phenomena and a Cryptography Application | |
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| References | |
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| Problems | |
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| Unconstrained Optimization | |
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| Introduction | |
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| Problem Statement and Preliminaries | |
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| Line Searches | |
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| Newton's Method | |
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| Equality Constraints and Lagrange Multipliers | |
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| MATLAB Code for Golden Section Search | |
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| References | |
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| Problems | |
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| Numerical Integration and Differentiation | |
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| Introduction | |
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| Trapezoidal Rule | |
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| Simpson's Rule | |
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| Gaussian Quadrature | |
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| Romberg Integration | |
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| Numerical Differentiation | |
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| References | |
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| Problems | |
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| Numerical Solution of Ordinary Differential Equations | |
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| Introduction | |
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| First-Order ODEs | |
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| Systems of First-Order ODEs | |
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| Multistep Methods for ODEs | |
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| Adams-Bashforth Methods | |
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| Adams-Moulton Methods | |
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| Comments on the Adams Families | |
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| Variable-Step-Size (Adaptive) Methods for ODEs | |
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| Stiff Systems | |
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| Final Remarks | |
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| MATLAB Code for Example 10.8 | |
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| MATLAB Code for Example 10.13 | |
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| References | |
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| Problems | |
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| Numerical Methods for Eigenproblems | |
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| Introduction | |
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| Review of Eigenvalues and Eigenvectors | |
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| The Matrix Exponential | |
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| The Power Methods | |
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| QR Iterations | |
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| References | |
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| Problems | |
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| Numerical Solution of Partial Differential Equations | |
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| Introduction | |
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| A Brief Overview of Partial Differential Equations | |
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| Applications of Hyperbolic PDEs | |
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| The Vibrating String | |
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| Plane Electromagnetic Waves | |
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| The Finite-Difference (FD) Method | |
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| The Finite-Difference Time-Domain (FDTD) Method | |
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| MATLAB Code for Example 12.5 | |
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| References | |
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| Problems | |
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| An Introduction to MATLAB | |
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| Introduction | |
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| Startup | |
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| Some Basic Operators, Operations, and Functions | |
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| Working with Polynomials | |
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| Loops | |
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| Plotting and M-Files | |
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| References | |
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| Index | |