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Numerical Analysis, 1998 Graduate Texts in Mathematics Series, Vol. 181

Langue : Anglais

Auteur :

Couverture de l’ouvrage Numerical Analysis
An introduction into numerical analysis for students in mathematics, physics, and engineering. Instead of attempting to exhaustively cover everything, the goal is to guide readers towards the basic ideas and general principles by way of the main and important numerical methods. The book includes the necessary basic functional analytic tools for the solid mathematical foundation of numerical analysis -- indispensable for any deeper study and understanding of numerical methods, in particular, for differential equations and integral equations. The text is presented in a concise and easily understandable fashion so as to be successfully mastered in a one-year course.
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 Bézier 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—Lindelöf 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 Nyström’s Method.- 12.4 The Collocation Method.- 12.5 Stability.- Problems.- References.
* Good introduction to an area experiencing rapid development for those in math, physics, and engineering. * Gives a solid foundation by restricting the presentation to the basic principles and procedures, as well as the primary numerical algorithms. * Includes the necessary functional analytic framework for a solid mathematical foundation in the subject. * Gives particular emphasis to the question of stability. * Presented in a concise and easily understandable fashion.

Date de parution :

Ouvrage de 326 p.

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Prix indicatif 63,29 €

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Date de parution :

Ouvrage de 336 p.

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