| 1 Introduction 2 Linear Systems 2.1 Examples for Systems of Equations 2.2 Gaussian Elimination 2.3 LR Decomposition 2.4 QR Decomposition Problems Basic Functional Analysis 3.1 Normed Spaces 3.2 Scalar Products 3.3 Bounded Linear Operators 3.4 Matrix Norms 3.5 Completeness 3.6 The Banach Fixed Point Theorem 3.7 Best Approximation Problems 4 Iterative Methods for Linear Systems 4.1 Jacobi and Gauss-Seidel Iterations 4.2 Relaxation Methods 4.3 Two-Grid Methods Problems 5 Ill-Conditioned Linear Systems 5.1 Condition Number 5.2 Singular Value Decomposition 5.3 Tikhonov Regularization Problems 6 Iterative Methods for Nonlinear Systems 6.1 Successive Approximations 6.2 Newton's Method 6.3 Zeros of Polynonfials 6.4 Least Squares Problems Problems 7 Matrix Eigenvalue Problems 7.1 Examples 7.2 Estimates for the Eigenvalues 7.3 The Jacobi Method 7.4 The QR Algorithm 7.5 Hessenberg Matrices Problems 8 Interpolation 8.1 Polynomial Interpolation 8.2 Trigonometric Interpolation 8.3 Spline Interpolation 8.4 Bdzier Polynomials Problems 9 Numerical Integration 9.1 Interpolatory Quadratures 9.2 Convergence of Quadrature Formulae 9.3 Gaussian Quadrature Formulae 9.4 Quadrature of Periodic Functions 9.5 Romberg Integration 9.6 Improper Integrals Problems 10 Initial Value Problems 10.1 The Picard-LindelSf Theorem 10.2 Euler's Method 10.3 Single-Step Methods 10.4 Multistep Methods Problems …… 11 Boundary Value Problems 12 Integral Equations References Index |
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