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9780486414546

A First Course in Numerical Analysis Second Edition

by ;
  • ISBN13:

    9780486414546

  • ISBN10:

    048641454X

  • Edition: 2nd
  • Format: Paperback
  • Copyright: 2001-02-06
  • Publisher: Dover Publications

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Summary

Outstanding text treats numerical analysis with mathematical rigor, but relatively few theorems and proofs. Oriented toward computer solutions of problems, it stresses errors in methods and computational efficiency. Problems some strictly mathematical, others requiring a computer appear at the end of each chapter.

Table of Contents

Preface to the Dover Edition xii
Preface to the Second Edition xiii
Notation xvi
Introduction and Preliminaries
1(30)
What Is Numerical Analysis?
1(1)
Sources of Error
2(2)
Error Definitions and Related Matters
4(5)
Significant Digits
Error in Functional Evaluation
Norms
Roundoff Error
9(3)
The Probabilistic Approach to Roundoff: A Particular Example
Computer Arithmetic
12(8)
Fixed-Point Arithmetic
Floating-Point Numbers
Floating-Point Arithmetic
Overflow and Underflow
Single- and Double-Precision Arithmetic
Error Analysis
20(2)
Backward Error Analysis
Condition and Stability
22(9)
Bibliographic Notes
24(1)
Bibliography
24(1)
Problems
24(7)
Approximation and Algorithms
31(21)
Approximation
31(8)
Classes of Approximating Functions
Types of Approximations
The Case for Polynomial Approximation
Numerical Algorithms
39(3)
Functionals and Error Analysis
42(2)
The Method of Undetermined Coefficients
44(8)
Bibliographic Notes
46(1)
Bibliography
46(1)
Problems
47(5)
Interpolation
52(37)
Introduction
52(2)
Lagrangian Interpolation
54(2)
Interpolation at Equal Intervals
56(7)
Lagrangian Interpolation at Equal Intervals
Finite Differences
The Use of Interpolation Formulas
63(3)
Iterated Interpolation
66(2)
Inverse Interpolation
68(2)
Hermite Interpolation
70(3)
Spline Interpolation
73(5)
Other Methods of Interpolation; Extrapolation
78(11)
Bibliographic Notes
79(1)
Bibliography
80(1)
Problems
81(8)
Numerical Differentiation, Numerical Quadrature, and Summation
89(75)
Numerical Differentiation of Data
89(4)
Numerical Differentiation of Functions
93(3)
Numerical Quadrature: The General Problem
96(2)
Numerical Integration of Data
Gaussian Quadrature
98(4)
Weight Functions
102(2)
Orthogonal Polynomials and Gaussian Quadrature
104(1)
Gaussian Quadrature over Infinite Intervals
105(3)
Particular Gaussian Quadrature Formulas
108(5)
Gauss-Jacobi Quadrature
Gauss-Chebyshev Quadrature
Singular Integrals
Composite Quadrature Formulas
113(5)
Newton-Cotes Quadrature Formulas
118(8)
Composite Newton-Cotes Formulas
Romberg Integration
Adaptive Integration
126(4)
Choosing a Quadrature Formula
130(6)
Summation
136(28)
The Euler-Maclaurin Sum Formula
Summation of Rational Functions; Factorial Functions
The Euler Transformation
Bibliographic Notes
145(1)
Bibliography
146(2)
Problems
148(16)
The Numerical Solution of Ordinary Differential Equations
164(83)
Statement of the Problem
164(2)
Numerical Integration Methods
166(5)
The Method of Undetermined Coefficients
Truncation Error in Numerical Integration Methods
171(2)
Stability of Numerical Integration Methods
173(10)
Convergence and Stability
Propagated-Error Bounds and Estimates
Predictor-Corrector Methods
183(12)
Convergence of the Iterations
Predictors and Correctors
Error Estimation
Stability
Starting the Solution and Changing the Interval
195(3)
Analytic Methods
A Numerical Method
Changing the Interval
Using Predictor-Corrector Methods
198(10)
Variable-Order-Variable-Step Methods
Some Illustrative Examples
Runge-Kutta Methods
208(16)
Errors in Runge-Kutta Methods
Second-Order Methods
Third-Order Methods
Fourth-Order Methods
Higher-Order Methods
Practical Error Estimation
Step-Size Strategy
Stability
Comparison of Runge-Kutta and Predictor-Corrector Methods
Other Numerical Integration Methods
224(4)
Methods Based on Higher Derivatives
Extrapolation Methods
Stiff Equations
228(19)
Bibliographic Notes
233(1)
Bibliography
234(2)
Problems
236(11)
Functional Approximation: Least-Squares Techniques
247(38)
Introduction
247(1)
The Principle of Least Squares
248(3)
Polynomial Least-Squares Approximations
251(3)
Solution of the Normal Equations
Choosing the Degree of the Polynomial
Orthogonal-Polynomial Approximations
254(6)
An Example of the Generation of Least-Squares Approximations
260(3)
The Fourier Approximation
263(22)
The Fast Fourier Transform
Least-Squares Approximations and Trigonometric Interpolation
Bibliographic Notes
274(1)
Bibliography
275(1)
Problems
276(9)
Functional Approximation: Minimum maximum Error Techniques
285(47)
General Remarks
285(2)
Rational Functions, Polynomials, and Continued Fractions
287(6)
Pade Approximations
293(2)
An Example
295(4)
Chebyshev Polynomials
299(2)
Chebyshev Expansions
301(6)
Economization of Rational Functions
307(4)
Economization of Power Series
Generalization to Rational Functions
Chebyshev's Theorem on Minimax Approximations
311(4)
Constructing Minimax Approximations
315(17)
The Second Algorithm of Remes
The Differential Correction Algorithm
Bibliographic Notes
320(1)
Bibliography
320(2)
Problems
322(10)
The Solution of Nonlinear Equations
332(78)
Introduction
332(2)
Functional Iteration
334(4)
Computational Efficiency
The Secant Method
338(6)
One-Point Iteration Formulas
344(3)
Multipoint Iteration Formulas
347(6)
Iteration Formulas Using General Inverse Interpolation
Derivative Estimated Iteration Formulas
Functional Iteration at a Multiple Root
353(3)
Some Computational Aspects of Functional Iteration
356(3)
The δ Process
Systems of Nonlinear Equations
359(8)
The Zeros of Polynomials: The Problem
367(4)
Sturm Sequences
Classical Methods
371(12)
Bairstow's Method
Graeffe's Root-squaring Method
Bernoulli's Method
Laguerre's Method
The Jenkins-Traub Method
383(9)
A Newton-based Method
392(3)
The Effect of Coefficient Errors on the Roots; Ill-conditioned Polynomials
395(15)
Bibliographic Notes
397(2)
Bibliography
399(1)
Problems
400(10)
The Solution of Simultaneous Linear Equations
410(73)
The Basic Theorem and the Problem
410(2)
General Remarks
412(2)
Direct Methods
414(16)
Gaussian Elimination
Compact Forms of Gaussian Elimination
The Doolittle, Crout, and Cholesky Algorithms
Pivoting and Equilibration
Error Analysis
430(7)
Roundoff-Error Analysis
Iterative Refinement
437(3)
Matrix Iterative Methods
440(3)
Stationary Iterative Processes and Related Matters
443(7)
The Jacobi Iteration
The Gauss-Seidel Method
Roundoff Error in Iterative Methods
Acceleration of Stationary Iterative Processes
Matrix Inversion
450(1)
Overdetermined Systems of Linear Equations
451(6)
The Simplex Method for Solving Linear Programming Problems
457(8)
Miscellaneous Topics
465(18)
Bibliographic Notes
468(2)
Bibliography
470(2)
Problems
472(11)
The Calculation of Eigenvalues and Eigenvectors of Matrices
483(66)
Basic Relationships
483(9)
Basic Theorems
The Characteristic Equation
The Location of, and Bounds on, the Eigenvalues
Canonical Forms
The Largest Eigenvalue in Magnitude by the Power Method
492(9)
Acceleration of Convergence
The Inverse Power Method
The Eigenvalues and Eigenvectors of Symmetric Matrices
501(12)
The Jacobi Method
Given's Method
Householder's Method
Methods for Nonsymmetric Matrices
513(8)
Lanczos' Method
Supertriangularization
Jacobi-Type Methods
The LR and QR Algorithms
521(15)
The Simple QR Algorithm
The Double QR Algorithm
Errors in Computed Eigenvalues and Eigenvectors
536(13)
Bibliographic Notes
538(1)
Bibliography
539(2)
Problems
541(8)
Index 549(8)
Hints and Answers to Problems 557

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