Numerical Analysis - Rainer Kress

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Résumé :
This volume is intended as an introduction into numerical analysis for students in mathematics, physics, and engineering. Instead of attempting to exhaustively cover all parts of numerical analysis, the goal is to guide the reader towards the basic ideas and general principles by way of considering main and important numerical algorithms. Given the rapid development of numerical methods, a reasonable introduction to numerical analysis has to confine itself to presenting a solid foundation by restricting the presentation to the basic principles and procedures....

Sommaire:
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.- 3 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 Polynomials.- 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 B?zier 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-Lindel?f Theorem.- 10.2 Euler's Method.- 10.3 Single-Step Methods.- 10.4 Multistep Methods.- Problems.- 11 Boundary Value Problems.- 11.1 Shooting Methods.- 11.2 Finite Difference Methods.- 11.3 The Riesz and Lax-Milgram Theorems.- 11.4 Weak Solutions.- 11.5 The Finite Element Method.- Problems.- 12 Integral Equations.- 12.1 The Riesz Theory.- 12.2 Operator Approximations.- 12.3 Nystr?m's Method.- 12.4 The Collocation Method.- 12.5 Stability.- Problems.- References.