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9780387954523

Introduction to Numerical Analysis

by ; ; ; ;
  • ISBN13:

    9780387954523

  • ISBN10:

    038795452X

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 2002-09-01
  • Publisher: Springer Nature

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Summary

This 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; and basics of multigrid methods. This new edition features expanded presentation of Hermite interpolation and B-splines, with a new section on multi-resolution methods and B-splines. New material on differential equations and the iterative solution of linear equations include: solving differential equations in the presence of discontinuities whose locations are not known at the outset; techniques for sensitivity analyses of differential equations dependent on additional parameters; new advanced techniques in multiple shooting; and Krylov space methods for non-symmetric systems of linear equations.

Table of Contents

Preface to the Third Edition vii
Preface to the Second Edition ix
Error Analysis
1(36)
Representation of Numbers
2(2)
Round off Errors and Floating-Point Arithmetic
4(5)
Error Propagation
9(12)
Examples
21(6)
Interval Arithmetic; Statistical Round off Estimation
27(10)
Exercises for Chapter 1
33(3)
References for Chapter 1
36(1)
Interpolation
37(108)
Interpolation by Polynomials
38(21)
Theoretical Foundation: The Interpolation Formula of Lagrange
38(2)
Neville's Algorithm
40(3)
Newtons Interpolation Formula: Divided Differences
43(5)
The Error in Polynomial Interpolation
48(3)
Hermite Interpolation
51(8)
Interpolation by Rational Functions
59(15)
General Properties of Rational Interpolation
59(5)
Inverse and Reciprocal Differences. Thiele's Continued Fraction
64(4)
Algorithms of the Neville Type
68(5)
Comparing Rational and Polynomial Interpolation
73(1)
Trigonometric Interpolation
74(23)
Basic Facts
74(6)
Fast Fourier Transforms
80(8)
The Algorithms of Goertzel and Reinsch
88(4)
The Calculation of Fourier Coefficients. Attenuation Factors
92(5)
Interpolation by Spline Functions
97(48)
Theoretical Foundations
97(4)
Determining Interpolating Cubic Spline Functions
101(6)
Convergence Properties of Cubic Spline Functions
107(4)
B-Splines
111(6)
The Computation of B-Splines
117(4)
Multi-Resolution Methods and B-Splines
121(13)
Exercises for Chapter 2
134(9)
References for Chapter 2
143(2)
Topics in Integration
145(45)
The Integration Formulas of Newton and Cotes
146(5)
Peano's Error Representation
151(5)
The Euler-Maclaurin Summation Formula
156(4)
Integration by Extrapolation
160(5)
About Extrapolation Methods
165(6)
Gaussian Integration Methods
171(10)
Integrals with Singularities
181(9)
Exercises for Chapter 3
184(4)
References for Chapter 3
188(2)
Systems of Linear Equations
190(99)
Gaussian Elimination. The Triangular Decomposition of a Matrix
190(10)
The Gauss-Jordan Algorithm
200(4)
The Choleski Decompostion
204(3)
Error Bounds
207(8)
Round off Error Analysis for Gaussian Elimination
215(6)
Round off Errors in Solving Triangular Systems
221(2)
Orthogonalization Techniques of Householder and Gram-Schmidt
223(8)
Data Fitting
231(16)
Linear Least Squares. The Normal Equations
232(3)
The Use of Orthogonalization in Solving Linear Least-Squares Problems
235(1)
The Condition of the Linear Least-Squares Problem
236(5)
Nonlinear Least-Squares Problems
241(2)
The Pseudoinverse of a Matrix
243(4)
Modification Techniques for Matrix Decompositions
247(9)
The Simplex Method
256(12)
Phase One of the Simplex Method
268(4)
Appendix: Elimination Methods for Sparse Matrices
272(17)
Exercises for Chapter 4
280(6)
References for Chapter 4
286(3)
Finding Zeros and Minimum Points by Iterative Methods
289(75)
The Development of Iterative Methods
290(3)
General Convergence Theorems
293(5)
The Convergence of Newton's Method in Several Variables
298(4)
A Modified Newton Method
302(14)
On the Convergence of Minimization Methods
303(5)
Application of the Convergence Criteria to the Modified Newton Method
308(5)
Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden
313(3)
Roots of Polynomials. Application of Newton's Method
316(12)
Sturm Sequences and Bisection Methods
328(5)
Bairstow's Method
333(2)
The Sensitivity of Polynomial Roots
335(3)
Interpolation Methods for Determining Roots
338(6)
The Δ2-Method of Aitken
344(5)
Minimization Problems without Constraints
349(15)
Exercises for Chapter 5
358(3)
References for Chapter 5
361(3)
Eigenvalue Problems
364(101)
Introduction
364(2)
Basic Facts on Eigenvalues
366(3)
The Jordan Normal Form of a Matrix
369(6)
The Fiobenius Normal Form of a Matrix
375(4)
The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrixes
379(7)
Reduction of Matrices to Simpler Form
386(19)
Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Householder
388(6)
Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi
394(4)
Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Lanczos
398(4)
Reduction to Hessenberg Form
402(3)
Methods for Determining the Eigenvalues and Eigenvectors
405(31)
Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix
405(2)
Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman
407(1)
Simple Vector Iteration and Inverse Iteration of Wielandt
408(7)
The LR and QR Methods
415(10)
The Practical Implementation of the QR Method
425(11)
Computation of the Singular Values of a Matrix
436(4)
Generalized Eigenvalue Problems
440(1)
Estimation of Eigenvalues
441(24)
Exercises for Chapter 6
455(7)
References for Chapter 6
462(3)
Ordinary Differential Equations
465(154)
Introduction
465(2)
Some Theorems from tile Theory of Ordinary Differential Equations
467(4)
Initial-Value Problems
471(68)
One-Step Methods: Basic Concepts
471(6)
Convergence of One-Step Methods
477(3)
Asymptotic Expansions for the Global Discretization Error of One-Step Methods
480(3)
The Influence of Rounding Errors in One-Step Methods
483(2)
Practical Implementation of One-Step Methods
485(7)
Multistep Methods: Examples
492(3)
General Multistep Methods
495(3)
An Example of Divergence
498(3)
Linear Difference Equations
501(3)
Convergence of Multistep Methods
504(4)
Linear Multistep Methods
508(5)
Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods
513(4)
Practical Implementation of Multistep Methods
517(4)
Extrapolation Methods for the Solution of the Initial-Value Problem
521(3)
Comparison of Methods for Solving Initial-Value Problems
524(1)
Stiff Differential Equations
525(6)
Implicit Differential Equations. Differential-Algebraic Equations
531(5)
Handling Discontinuities in Differential Equations
536(2)
Sensitivity Analysis of Initial-Value Problems
538(1)
Boundary-Value Problems
539(43)
Introduction
539(3)
The Simple Shooting Method
542(6)
The Simple Shooting Method for Linear Boundary-Value Problems
548(2)
An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems
550(2)
Difficulties in the Execution of the Simple Shooting Method
552(5)
The Multiple Shooting Method
557(4)
Hints for the Practical Implementation of the Multiple Shooting Method
561(4)
An Example: Optimal Control Program for a Lifting Reentry Space Vehicle
565(7)
Advanced Techniques in Multiple Shooting
572(5)
The Limiting Case m →∞ of the Multiple Shooting Method (General Newton's Method, Quasilinearization)
577(5)
Difference Methods
582(4)
Variational Methods
586(10)
Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations
596(4)
Variational Methods for Partial Differential Equations The Finite-Element Method
600(19)
Exercises for Chapter 7
607(6)
References for Chapter 7
613(6)
Iterative Methods for the Solution of Large Systems of Linear Equations. Additional Methods
619(111)
Introduction
619(2)
General Procedures for the Construction of Iterative Methods
621(2)
Convergence Theorems
623(6)
Relaxation Methods
629(10)
Applications to Difference Methods--An Example
639(6)
Block Iterative Methods
645(2)
The ADI-Method of Peaceman and Rachford
647(10)
Krylov Space Methods for Solving Linear Equations
657(34)
The Conjugate-Gradient Method of Hestenes and Stiefel
658(9)
The GMRES Algorithm
667(13)
The Biorthogonalization Method of Lanczos and the QMR algorithm
680(6)
The Bi-CG and BI-CGSTAB Algorithms
686(5)
Buneman's Algorithm and Fourier Methods for Solving the Discretized Poisson Equation
691(11)
Multigrid Methods
702(10)
Comparison of Iterative Methods
712(18)
Exercises for Chapter 8
719(8)
References for Chapter 8
727(3)
General Literature on Numerical Methods 730(2)
Index 732

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