First Course in Numerical Analysis

Name: First Course in Numerical Analysis
Brand: DOVER
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Second Edition

By: ANTHONY RALSTON

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Paperback | 6 February 2001 | Edition Number 2

At a Glance

Paperback
576 Pages

Edition Type
Revised

Dimensions(cm)
20.9 x 13.6 x 3.18

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$49.99

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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.

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You Can Find This Book In

Non-Fiction Science Physics Classical Mathematics Dynamics & Statics Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Solids Dynamics & Vibration Mathematics Calculus & Mathematical Analysis

Optimisation Linear Programming

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

First Course in Numerical Analysis

Second Edition

At a Glance

Paperback

Shipping

How to return your order

Defective items

You Can Find This Book In

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