Introduction to Numerical Analysis

ISBN-10: 038797878X
ISBN-13: 9780387978789
Edition: 2nd 1996 (Revised)
List price: $74.95
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Description: The book contains a large amount of information not found in standard textbooks. Written for the advanced undergraduate/beginning graduate student, it combines the modern mathematical standards of numerical analysis with an understanding of the  More...

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Book details

List price: $74.95
Edition: 2nd
Copyright year: 1996
Publisher: Springer
Publication date: 4/11/1996
Binding: Hardcover
Pages: 674
Size: 6.40" wide x 9.54" long x 1.39" tall
Weight: 2.662
Language: English

The book contains a large amount of information not found in standard textbooks. Written for the advanced undergraduate/beginning graduate student, it combines the modern mathematical standards of numerical analysis with an understanding of the needs of the computer scientist working on practical applications. Among its many particular features are: - fully worked-out examples; - many carefully selected and formulated problems; - fast Fourier transform methods; - a thorough discussion of some important minimization methods; - solution of stiff or implicit ordinary differential equations and of differential algebraic systems; - modern shooting techniques for solving two-point boundary-value problems; - basics of multigrid methods. Included are numerous references to contemporary research literature.

Preface to the Second Edition
Preface to the First Edition
Error Analysis
Representation of Numbers
Roundoff Errors and Floating-Point Arithmetic
Error Propagation
Interval Arithmetic; Statistical Roundoff Estimation
Interpolation by Polynomials
Theoretical Foundation: The Interpolation Formula of Lagrange
Neville's Algorithm
Newton's Interpolation Formula: Divided Differences
The Error in Polynomial Interpolation
Hermite Interpolation
Interpolation by Rational Functions
General Properties of Rational Interpolation
Inverse and Reciprocal Differences. Thiele's Continued Fraction
Algorithms of the Neville Type
Comparing Rational and Polynomial Interpolations
Trigonometric Interpolation
Basic Facts
Fast Fourier Transforms
The Algorithms of Goertzel and Reinsch
The Calculation of Fourier Coefficients. Attenuation Factors
Interpolation by Spline Functions
Theoretical Foundations
Determining Interpolating Cubic Spline Functions
Convergence Properties of Cubic Spline Functions
The Computation of B-Splines
Topics in Integration
The Integration Formulas of Newton and Cotes
Peano's Error Representation
The Euler-Maclaurin Summation Formula
Integrating by Extrapolation
About Extrapolation Methods
Gaussian Integration Methods
Integrals with Singularities
Systems of Linear Equations
Gaussian Elimination. The Triangular Decomposition of a Matrix
The Gauss-Jordan Algorithm
The Cholesky Decomposition
Error Bounds
Roundoff-Error Analysis for Gaussian Elimination
Roundoff Errors in Solving Triangular Systems
Orthogonalization Techniques of Householder and Gram-Schmidt
Data Fitting
Linear Least Squares. The Normal Equations
The Use of Orthogonalization in Solving Linear Least-Squares Problems
The Condition of the Linear Least-Squares Problem
Nonlinear Least-Squares Problems
The Pseudoinverse of a Matrix
Modification Techniques for Matrix Decompositions
The Simplex Method
Phase One of the Simplex Method
Appendix to Chapter 4
Elimination Methods for Sparse Matrices
Finding Zeros and Minimum Points by Iterative Methods
The Development of Iterative Methods
General Convergence Theorems
The Convergence of Newton's Method in Several Variables
A Modified Newton Method
On the Convergence of Minimization Methods
Application of the Convergence Criteria to the Modified Newton Method
Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden
Roots of Polynomials. Application of Newton's Method
Sturm Sequences and Bisection Methods
Bairstow's Method
The Sensitivity of Polynomial Roots
Interpolation Methods for Determining Roots
The [Delta][superscript 2]-Method of Aitken
Minimization Problems without Constraints
Eigenvalue Problems
Basic Facts on Eigenvalues
The Jordan Normal Form of a Matrix
The Frobenius Norma] Form of a Matrix
The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrices
Reduction of Matrices to Simpler Form
Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Householder
Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi
Reduction ofa Hermitian Matrix to Tridiagonal Form: The Method of Lanczos
Reduction to Hessenberg Form
Methods for Determining the Eigenvalues and Eigenvectors
Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix
Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman
Simple Vector Iteration and Inverse Iteration of Wielandt
The LR and QR Methods
The Practical Implementation of the QR Method
Computation of the Singular Values of a Matrix
Generalized Eigenvalue Problems
Estimation of Eigenvalues
Ordinary Differential Equations
Some Theorems from the Theory of Ordinary Differential Equations
Initial-Value Problems
One-Step Methods: Basic Concepts
Convergence of One-Step Methods
Asymptotic Expansions for the Global Discretization Error of One-Step Methods
The Influence of Rounding Errors in One-Step Methods
Practical Implementation of One-Step Methods
Multistep Methods: Examples
General Multistep Methods
An Example of Divergence
Linear Difference Equations
Convergence of Multistep Methods
Linear Multistep Methods
Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods
Practical Implementation of Multistep Methods
Extrapolation Methods for the Solution of the Initial-Value Problem
Comparison of Methods for Solving Initial-Value Problems
Stiff Differential Equations
Implicit Differential Equations. Differential-Algebraic Equations
Boundary-Value Problems
The Simple Shooting Method
The Simple Shooting Method for Linear Boundary-Value Problems
An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems
Difficulties in the Execution of the Simple Shooting Method
The Multiple Shooting Method
Hints for the Practical Implementation of the Multiple Shooting Method
An Example: Optimal Control Program for a Lifting Reentry Space Vehicle
The Limiting Case m [actual symbol not reproducible] of the Multiple Shooting Method (General Newton's Method, Quasilinearization)
Difference Methods
Variational Methods
Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations
Variational Methods for Partial Differential Equations. The Finite-Element Method
Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods
General Procedures for the Construction of Iterative Methods
Convergence Theorems
Relaxation Methods
Applications to Difference Methods - An Example
Block Iterative Methods
The ADI-Method of Peaceman and Rachford
The Conjugate-Gradient Method of Hestenes and Stiefel
The Algorithm of Buneman for the Solution of the Discretized Poisson Equation
Multigrid Methods
Comparison of Iterative Methods
General Literature on Numerical Methods

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