A Theoretical Introduction to Numerical Analysis - Ryaben'kii, Victor S.; Tsynkov, Semyon V.

A Theoretical Introduction to Numerical Analysis

Victor S. Ryaben'kii Semyon V. Tsynkov 

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A Theoretical Introduction to Numerical Analysis

Table of contents:
Interpolation of Functions. Quadratures. Systems of Algebraic Equations. The Method of Finite Differences for the Numerical Solution of Differential Equations. The Methods of Boundary Equations for the Numerical Solution of Boundary Value Problems. References. Index.

Illustrated by sampling numerical methods from mathematical analysis, linear algebra, and differential equations, A Theoretical Introduction to Numerical Analysis presents the general methodology and principles of numerical analysis. The book emphasizes fundamentals and links between sub-disciplines. It provides coverage of topics such as Lebesgue constants, Chebyshev polynomials, stability on initial boundary value problems, interpolation in two variables, and splines.. Carefully structured for learning at different levels of depth, this text is organized so that chapters can be read independently, allowing instructors to tailor the course to students' needs, with enough material for three semester long courses.

Illustrated by sampling numerical methods from mathematical analysis, linear algebra, and differential equations, A Theoretical Introduction to Numerical Analysis presents the general methodology and principles of numerical analysis. The book emphasizes fundamentals and links between sub-disciplines. It provides coverage of topics such as Lebesgue constants, Chebyshev polynomials, stability on initial boundary value problems, interpolation in two variables, and splines.. Carefully structured for learning at different levels of depth, this text is organized so that chapters can be read independently, allowing instructors to tailor the course to students' needs, with enough material for three semester long courses.


Produktinformation

  • Verlag: CRC PR INC
  • 2006
  • Ausstattung/Bilder: 544 pages - 50 b/w ill.
  • Seitenzahl: 537
  • Englisch
  • Abmessung: 235mm x 166mm x 34mm
  • Gewicht: 889g
  • ISBN-13: 9781584886075
  • ISBN-10: 1584886072
  • Best.Nr.: 21303297
Keldysh Institute for Applied Math, Moscow, Russia Moscow 125047 North Carolina State University, Raleigh, NC, USA RALEIGH
Keldysh Institute for Applied Math, Moscow, Russia Moscow 125047 North Carolina State University, Raleigh, NC, USA RALEIGH

Inhaltsangabe

PREFACE
ACKNOWLEDGMENTS
INTRODUCTION
Discretization
Conditioning
Error
On Methods of Computation
INTERPOLATION OF FUNCTIONS. QUADRATURES
ALGEBRAIC INTERPOLATION
Existence and Uniqueness of Interpolating Polynomial
Classical Piecewise Polynomial Interpolation
Smooth Piecewise Polynomial Interpolation (Splines)
Interpolation of Functions of Two Variables
TRIGONOMETRIC INTERPOLATION
Interpolation of Periodic Functions
Interpolation of Functions on an Interval. Relation between Algebraic and Trigonometric Interpolation
COMPUTATION OF DEFINITE INTEGRALS. QUADRATURES
Trapezoidal Rule, Simpson's Formula, and the Like
Quadrature Formulae with No Saturation. Gaussian Quadratures
Improper Integrals. Combination of Numerical and Analytical Methods
Multiple Integrals
SYSTEMS OF SCALAR EQUATIONS
SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS: DIRECT METHODS
Different Forms of Consistent Linear Systems
Linear Spaces, Norms, and Operators
Conditioning of Linear Systems
Gaussian Elimination and Its Tri-Diagonal Version
Minimization of Quadratic Functions and Its Relation to Linear Systems
The Method of Conjugate Gradients
Finite Fourier Series
ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS
Richardson Iterations and the Like
Chebyshev Iterations and Conjugate Gradients
Krylov Subspace Iterations
Multigrid Iterations
OVERDETERMINED LINEAR SYSTEMS. THE METHOD OF LEAST SQUARES
Examples of Problems that Result in Overdetermined Systems
Weak Solutions of Full Rank Systems. QR Factorization
Rank Deficient Systems. Singular Value Decomposition
NUMERICAL SOLUTION OF NONLINEAR EQUATIONS AND SYSTEMS
Commonly Used Methods of Rootfinding
Fixed Point Iterations
Newton's Method
THE METHOD OF FINITE DIFFERENCES FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS
NUMERCAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
Examples of Finite-Difference Schemes. Convergence
Approximation of Continuous Problem by a Difference Scheme. Consistency
Stability of Finite-Difference Schemes
The Runge-Kutta Methods
Solution of Boundary Value Problems
Saturation of Finite-Difference Methods
The Notion of Spectral Methods
FINITE-DIFFERENCE SCHEMES FOR PARTIAL DIFFERENTIAL EQUATIONS
Key Definitions and Illustrating Examples
Construction of Consistent Difference Schemes
Spectral Stability Criterion for Finite-Difference Cauchy Problems
Stability for Problems with Variable Coefficients
Stability for Initial Boundary Value Problems
Explicit and Implicit Schemes for the Heat Equation
DISCONTINUOUS SOLUTIONS AND METHODS OF THEIR COMPUTATION
Differential Form of an Integral Conservation Law
Construction of Difference Schemes
DISCRETE METHODS FOR ELLIPTIC PROBLEMS
A Simple Finite-Difference Scheme. The Maximum Principle
The Notion of Finite Elements. Ritz and Galerkin Approximations
THE METHODS OF BOUNDARY EQUATIONS FOR THE NUMERICAL SOLUTION OF BOUNDARY VALUE PROBLEMS
BOUNDARY INTEGRAL EQUATIONS AND THE METHOD OF BOUNDARY ELEMENTS
Reduction of Boundary Value Problems to Integral Equations
Discretization of Integral Equations and Boundary Elements
The Range of Applicability for Boundary Elements
BOUNDARY EQUATIONS WITH PROJECTIONS AND THE METHOD OF DIFFERENCE POTENTIALS
Formulation of Model Problems
Difference Potentials
Solution of Model Problems
LIST OF FIGURES
REFERENCED BOOKS
REFERENCED JOURNAL ARTICLES
INDEX