Numerical Mathematics and Computing (6TH)

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Numerical Mathematics and Computing (6TH)

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  • 製本 Hardcover:ハードカバー版/ページ数 763 p.
  • 言語 ENG,ENG
  • 商品コード 9780495114758
  • DDC分類 518

Table of Contents

    Introduction                                   1  (42)
Preliminary Remarks 1 (19)
Significant Digits of Precision: 3 (2)
Examples
Errors: Absolute and Relative 5 (1)
Accuracy and Precision 5 (1)
Rounding and Chopping 6 (1)
Nested Multiplication 7 (2)
Pairs of Easy/Hard Problems 9 (1)
First Programming Experiment 9 (1)
Mathematical Software 10 (1)
Summary 11 (1)
Additional References 11 (1)
Problems 1.1 12 (2)
Computer Problems 1.1 14 (6)
Review of Taylor Series 20 (23)
Taylor Series 20 (3)
Complete Horner's Algorithm 23 (1)
Taylor's Theorem in Terms of (x -- c) 24 (2)
Mean-Value Theorem 26 (1)
Taylor's Theorem in Terms of h 26 (2)
Alternating Series 28 (2)
Summary 30 (1)
Additional References 31 (1)
Problems 1.2 31 (5)
Computer Problems 1.2 36 (7)
Floating-Point Representation and Errors 43 (33)
Floating-Point Representation 43 (1)
Normalized Floating-Point Representation 44 (2)
Floating-Point Representation 46 (1)
Single-Precision Floating-Point Form 46 (2)
Double-Precision Floating-Point Form 48 (2)
Computer Errors in Representing Numbers 50 (1)
Notation fl(x) and Backward Error 51 (3)
Analysis
Historical Notes 54 (1)
Summary 54 (1)
Problems 2.1 55 (4)
Computer Problems 2.1 59 (2)
Loss of Significance 61 (15)
Significant Digits 61 (1)
Computer-Caused Loss of Significance 62 (1)
Theorem on Loss of Precision 63 (1)
Avoiding Loss of Significance in 64 (3)
Subtraction
Range Reduction 67 (1)
Summary 68 (1)
Additional References 68 (1)
Problems 2.2 68 (3)
Computer Problems 2.2 71 (5)
Locating Roots of Equations 76 (48)
Bisection Method 76 (13)
Introduction 76 (2)
Bisection Algorithm and Pseudocode 78 (1)
Examples 79 (2)
Convergence Analysis 81 (2)
False Position (Regula Falsi) Method 83 (2)
and Modifications
Summary 85 (1)
Problems 3.1 85 (2)
Computer Problems 3.1 87 (2)
Newton's Method 89 (22)
Interpretations of Newton's Method 90 (2)
Pseudocode 92 (1)
Illustration 92 (1)
Convergence Analysis 93 (3)
Systems of Nonlinear Equations 96 (3)
Fractal Basins of Attraction 99 (1)
Summary 100(1)
Additional References 100(1)
Problems 3.2 101(4)
Computer Problems 3.2 105(6)
Secant Method 111(13)
Secant Algorithm 112(2)
Convergence Analysis 114(3)
Comparison of Methods 117(1)
Hybrid Schemes 117(1)
Fixed-Point Iteration 117(1)
Summary 118(1)
Additional References 119(1)
Problems 3.3 119(2)
Computer Problems 3.3 121(3)
Interpolation and Numerical Differentiation 124(56)
Polynomial Interpolation 124(29)
Preliminary Remarks 124(1)
Polynomial Interpolation 125(1)
Interpolating Polynomial: Lagrange Form 126(2)
Existence of Interpolating Polynomial 128(1)
Interpolating Polynomial: Newton Form 128(2)
Nested Form 130(1)
Calculating Coefficients ai Using 131(5)
Divided Differences
Algorithms and Pseudocode 136(3)
Vandermonde Matrix 139(2)
Inverse Interpolation 141(1)
Polynomial Interpolation by Neville's 142(2)
Algorithm
Interpolation of Bivariate Functions 144(1)
Summary 145(1)
Problems 4.1 146(6)
Computer Problems 4.1 152(1)
Errors in Polynomial Interpolation 153(11)
Dirichlet Function 154(1)
Runge Function 154(2)
Theorems on Interpolation Errors 156(4)
Summary 160(1)
Problems 4.2 161(2)
Computer Problems 4.2 163(1)
Estimating Derivatives and Richardson 164(16)
Extrapolation
First-Derivative Formulas via Taylor 164(2)
Series
Richardson Extrapolation 166(4)
First-Derivative Formulas via 170(3)
Interpolation Polynomials
Second-Derivative Formulas via Taylor 173(1)
Series
Noise in Computation 174(1)
Summary 174(1)
Additional References for Chapter 4 175(1)
Problems 4.3 175(3)
Computer Problems 4.3 178(2)
Numerical Integration 180(36)
Lower and Upper Sums 180(10)
Definite and Indefinite Integrals 180(1)
Lower and Upper Sums 181(2)
Riemann-Integrable Functions 183(1)
Examples and Pseudocode 184(3)
Summary 187(1)
Problems 5.1 187(1)
Computer Problems 5.1 188(2)
Trapezoid Rule 190(14)
Uniform Spacing 191(1)
Error Analysis 192(3)
Applying the Error Formula 195(1)
Recursive Trapezoid Formula for Equal 196(2)
Subintervals
Multidimensional Integration 198(1)
Summary 199(1)
Problems 5.2 200(3)
Computer Problems 5.2 203(1)
Romberg Algorithm 204(12)
Description 204(1)
Pseudocode 205(1)
Euler-Maclaurin Formula 206(3)
General Extrapolation 209(2)
Summary 211(1)
Additional References 211(1)
Problems 5.3 212(2)
Computer Problems 5.3 214(2)
Additional Topics on Numerical Integration 216(29)
Simpson's Rule and Adaptive Simpson's Rule 216(14)
Basic Simpson's Rule 216(3)
Simpson's Rule 219(1)
Composite Simpson's Rule 220(1)
An Adaptive Simpson's Scheme 221(3)
Example Using Adaptive Simpson Procedure 224(1)
Newton-Cotes Rules 225(1)
Summary 226(1)
Problems 6.1 227(2)
Computer Problems 6.1 229(1)
Gaussian Quadrature Formulas 230(15)
Description 230(1)
Change of Intervals 231(1)
Gaussian Nodes and Weights 232(2)
Legendre Polynomials 234(3)
Integrals with Singularities 237(1)
Summary 237(2)
Additional References 239(1)
Problems 6.2 239(2)
Computer Problems 6.2 241(4)
Systems of Linear Equations 245(48)
Naive Gaussian Elimination 245(14)
A Larger Numerical Example 247(1)
Algorithm 248(2)
Pseudocode 250(3)
Testing the Pseudocode 253(1)
Residual and Error Vectors 254(1)
Summary 255(1)
Problems 7.1 255(2)
Computer Problems 7.1 257(2)
Gaussian Elimination with Scaled Partial 259(21)
Pivoting
Naive Gaussian Elimination Can Fail 259(2)
Partial Pivoting and Complete Partial 261(1)
Pivoting
Gaussian Elimination with Scaled 262(3)
Partial Pivoting
A Larger Numerical Example 265(1)
Pseudocode 266(3)
Long Operation Count 269(2)
Numerical Stability 271(1)
Scaling 271(1)
Summary 271(1)
Problems 7.2 272(4)
Computer Problems 7.2 276(4)
Tridiagonal and Banded Systems 280(13)
Tridiagonal Systems 281(1)
Strictly Diagonal Dominance 282(1)
Pentadiagonal Systems 283(2)
Block Pentadiagonal Systems 285(1)
Summary 286(1)
Additional References 287(1)
Problems 7.3 287(1)
Computer Problems 7.3 288(5)
Additional Topics Concerning Systems of 293(78)
Linear Equations
Matrix Factorizations 293(26)
Numerical Example 294(2)
Formal Derivation 296(4)
Pseudocode 300(1)
Solving Linear Systems Using LU 300(2)
Factorization
LDLT Factorization 302(3)
Cholesky Factorization 305(1)
Multiple Right-Hand Sides 306(1)
Computing A--1 307(1)
Example Using Software Packages 307(2)
Summary 309(2)
Problems 8.1 311(5)
Computer Problems 8.1 316(3)
Iterative Solutions of Linear Systems 319(23)
Vector and Matrix Norms 319(2)
Condition Number and Ill-Conditioning 321(1)
Basic Iterative Methods 322(5)
Pseudocode 327(1)
Convergence Theorems 328(3)
Matrix Formulation 331(1)
Another View of Overrelaxation 332(1)
Conjugate Gradient Method 332(3)
Summary 335(2)
Problems 8.2 337(2)
Computer Problems 8.2 339(3)
Eigenvalues and Eigenvectors 342(18)
Calculating Eigenvalues and Eigenvectors 343(1)
Mathematical Software 344(1)
Properties of Eigenvalues 345(2)
Gershgorin's Theorem 347(1)
Singular Value Decomposition 348(3)
Numerical Examples of Singular Value 351(2)
Decomposition
Application: Linear Differential 353(1)
Equations
Application: A Vibration Problem 354(1)
Summary 355(1)
Problems 8.3 356(2)
Computer Problems 8.3 358(2)
Power Method 360(11)
Power Method Algorithms 361(2)
Aitken Acceleration 363(1)
Inverse Power Method 364(1)
Software Examples: Inverse Power Method 365(1)
Shifted (Inverse) Power Method 365(1)
Example: Shifted Inverse Power Method 366(1)
Summary 366(1)
Additional References 367(1)
Problems 8.4 367(1)
Computer Problems 8.4 368(3)
Approximation by Spline Functions 371(55)
First-Degree and Second-Degree Splines 371(14)
First-Degree Spline 372(2)
Modulus of Continuity 374(2)
Second-Degree Splines 376(1)
Interpolating Quadratic Spline Q(x) 376(2)
Subbotin Quadratic Spline 378(2)
Summary 380(1)
Problems 9.1 381(3)
Computer Problems 9.1 384(1)
Natural Cubic Splines 385(19)
Introduction 385(1)
Natural Cubic Spline 386(2)
Algorithm for Natural Cubic Spline 388(4)
Pseudocode for Natural Cubic Splines 392(1)
Using Pseudocode for Interpolating and 393(1)
Curve Fitting
Space Curves 394(2)
Smoothness Property 396(2)
Summary 398(1)
Problems 9.2 399(4)
Computer Problems 9.2 403(1)
B Splines: Interpolation and Approximation 404(22)
Interpolation and Approximation by B 410(2)
Splines
Pseudocode and a Curve-Fitting Example 412(2)
Schoenberg's Process 414(1)
Pseudocode 414(2)
Bezier Curves 416(2)
Summary 418(1)
Additional References 419(1)
Problems 9.3 420(3)
Computer Problems 9.3 423(3)
Ordinary Differential Equations 426(39)
Taylor Series Methods 426(13)
Initial-Value Problem: Analytical 426(2)
versus Numerical Solution
An Example of a Practical Problem 428(1)
Solving Differential Equations and 428(1)
Integration
Vector Fields 429(2)
Taylor Series Methods 431(1)
Euler's Method Pseudocode 432(1)
Taylor Series Method of Higher Order 433(2)
Types of Errors 435(1)
Taylor Series Method Using Symbolic 435(1)
Computations
Summary 435(1)
Problems 10.1 436(2)
Computer Problems 10.1 438(1)
Runge-Kutta Methods 439(11)
Taylor Series for f(x, y) 440(1)
Runge-Kutta Method of Order 2 441(1)
Runge-Kutta Method of Order 4 442(1)
Pseudocode 443(1)
Summary 444(1)
Problems 10.2 445(2)
Computer Problems 10.2 447(3)
Stability and Adaptive Runge-Kutta and 450(15)
Multistep Methods
An Adaptive Runge-Kutta-Fehlberg Method 450(4)
An Industrial Example 454(1)
Adams-Bashforth-Moulton Formulas 455(1)
Stability Analysis 456(3)
Summary 459(1)
Additional References 460(1)
Problems 10.3 460(1)
Computer Problems 10.3 461(4)
Systems of Ordinary Differential Equations 465(30)
Methods for First-Order Systems 465(12)
Uncoupled and Coupled Systems 465(1)
Taylor Series Method 466(1)
Vector Notation 467(1)
Systems of ODEs 468(1)
Taylor Series Method: Vector Notation 468(1)
Runge-Kutta Method 469(2)
Autonomous ODE 471(2)
Summary 473(1)
Problems 11.1 474(1)
Computer Problems 11.1 475(2)
Higher-Order Equations and Systems 477(6)
Higher-Order Differential Equations 477(2)
Systems of Higher-Order Differential 479(1)
Equations
Autonomous ODE Systems 479(1)
Summary 480(1)
Problems 11.2 480(2)
Computer Problems 11.2 482(1)
Adams-Bashforth-Moulton Methods 483(12)
A Predictor-Corrector Scheme 483(1)
Pseudocode 484(4)
An Adaptive Scheme 488(1)
An Engineering Example 488(1)
Some Remarks about Stiff Equations 489(2)
Summary 491(1)
Additional References 492(1)
Problems 11.3 492(1)
Computer Problems 11.3 492(3)
Smoothing of Data and the Method of Least 495(37)
Squares
Method of Least Squares 495(10)
Linear Least Squares 495(3)
Linear Example 498(1)
Nonpolynomial Example 499(1)
Basis Functions {g0, g1, . . . , gn} 500(1)
Summary 501(1)
Problems 12.1 502(3)
Computer Problems 12.1 505(1)
Orthogonal Systems and Chebyshev 505(13)
Polynomials
Orthonormal Basis Functions {g0, g1, . 505(3)
. . , gn}
Outline of Algorithm 508(2)
Smoothing Data: Polynomial Regression 510(5)
Summary 515(1)
Problems 12.2 516(1)
Computer Problems 12.2 517(1)
Other Examples of the Least-Squares 518(14)
Principle
Use of a Weight Function w (x) 519(1)
Nonlinear Example 520(1)
Linear and Nonlinear Example 521(1)
Additional Details on SVD 522(2)
Using the Singular Value Decomposition 524(3)
Summary 527(1)
Additional References 527(1)
Problems 12.3 527(3)
Computer Problems 12.3 530(2)
Monte Carlo Methods and Simulation 532(31)
Random Numbers 532(12)
Random-Number Algorithms and Generators 533(2)
Examples 535(2)
Uses of Pseudocode Random 537(4)
Summary 541(1)
Problems 13.1 541(1)
Computer Problems 13.1 542(2)
Estimation of Areas and Volumes by Monte 544(8)
Carlo Techniques
Numerical Integration 544(1)
Example and Pseudocode 545(2)
Computing Volumes 547(1)
Ice Cream Cone Example 548(1)
Summary 549(1)
Problems 13.2 549(1)
Computer Problems 13.2 549(3)
Stimulation 552(11)
Loaded Die Problem 552(1)
Birthday Problem 553(2)
Buffon's Needle Problem 555(1)
Two Dice Problem 556(1)
Neutron Shielding 557(1)
Summary 558(1)
Additional References 558(1)
Computer Problems 13.3 559(4)
Boundary-Value Problems for Ordinary 563(19)
Differential Equations
Shooting Method 563(7)
Shooting Method Algorithm 565(2)
Modifications and Refinements 567(1)
Summary 567(1)
Problems 14.1 568(2)
Computer Problems 14.1 570(12)
A Discretization Method 570(1)
Finite-Difference Approximations 570(1)
The Linear Case 571(1)
Pseudocode and Numerical Example 572(2)
Shooting Method in the Linear Case 574(1)
Pseudocode and Numerical Example 575(2)
Summary 577(1)
Additional References 578(1)
Problems 14.2 578(2)
Computer Problems 14.2 580(2)
Partial Differential Equations 582(43)
Parabolic Problems 582(14)
Some Partial Differential Equations 582(3)
from Applied Problems
Heat Equation Model Problem 585(1)
Finite-Difference Method 585(2)
Pseudocode for Explicit Method 587(1)
Crank-Nicolson Method 588(1)
Pseudocode for the Crank-Nicolson Method 589(1)
Alternative Version of the 590(1)
Crank-Nicolson Method
Stability 591(2)
Summary 593(1)
Problems 15.1 594(2)
Computer Problems 15.1 596(9)
Hyperbolic Problems 596(1)
Wave Equation Model Problem 596(1)
Analytic Solution 597(1)
Numerical Solution 598(2)
Pseudocode 600(1)
Advection Equation 601(1)
Lax Method 602(1)
Upwind Method 602(1)
Lax-Wendroff Method 602(1)
Summary 603(1)
Problems 15.2 604(1)
Computer Problems 15.2 604(1)
Elliptic Problems 605(20)
Helmholtz Equation Model Problem 605(1)
Finite-Difference Method 606(4)
Gauss-Seidel Iterative Method 610(1)
Numerical Example and Pseudocode 610(3)
Finite-Element Methods 613(4)
More on Finite Elements 617(2)
Summary 619(1)
Additional References 620(1)
Problems 15.3 620(2)
Computer Problems 15.3 622(3)
Minimization of Functions 625(32)
One-Variable Case 625(14)
Unconstrained and Constrained 625(1)
Minimization Problems
One-Variable Case 626(1)
Unimodal Functions F 627(1)
Fibonacci Search Algorithm 628(3)
Golden Section Search Algorithm 631(2)
Quadratic Interpolation Algorithm 633(2)
Summary 635(1)
Problems 16.1 635(2)
Computer Problems 16.1 637(2)
Multivariate Case 639(18)
Taylor Series for F: Gradient Vector 640(1)
and Hessian Matrix
Alternative Form of Taylor Series 641(2)
Steepest Descent Procedure 643(1)
Contour Diagrams 644(1)
More Advanced Algorithms 644(2)
Minimum, Maximum, and Saddle Points 646(1)
Positive Definite Matrix 647(1)
Quasi-Newton Methods 647(1)
Nelder-Mead Algorithm 647(1)
Method of Simulated Annealing 648(2)
Summary 650(1)
Additional References 651(1)
Problems 16.2 651(3)
Computer Problems 16.2 654(3)
Linear Programming 657(27)
Standard Forms and Duality 657(13)
First Primal Form 657(1)
Numerical Example 658(2)
Transforming Problems into First Primal 660(1)
Form
Dual Problem 661(2)
Second Primal Form 663(1)
Summary 664(1)
Problems 17.1 665(4)
Computer Problems 17.1 669(1)
Simplex Method 670(5)
Vertices in K and Linearly Independent 671(1)
Columns of A
Simplex Method 672(2)
Summary 674(1)
Problems 17.2 674(1)
Computer Problems 17.2 675(1)
Approximate Solution of Inconsistent 675(9)
Linear Systems
l1 Problem 676(2)
l∞ Problem 678(2)
Summary 680(2)
Additional References 682(1)
Problems 17.3 682(1)
Computer Problems 17.3 682(2)
Appendix A Advice on Good Programming 684(8)
Practices
Programming Suggestions 684(8)
Case Studies 687(4)
On Developing Mathematical Software 691(1)
Appendix B Representation of Numbers in 692(11)
Different Bases
Representation of Numbers in Different 692(11)
Bases
Base β Numbers 693(1)
Conversion of Integer Parts 693(2)
Conversion of Fractional Parts 695(1)
Base Conversion 10 ↔ 8 ↔ 2 696(2)
Base 16 698(1)
More Examples 698(1)
Summary 699(1)
Problems B.1 699(2)
Computer Problems B.1 701(2)
Appendix C Additional Details on IEEE 703(3)
Floating-Point Arithmetic
More on IEEE Standard Floating-Point 703(3)
Arithmetic
Appendix D Linear Algebra Concepts and 706(18)
Notation
Elementary Concepts 706(10)
Vectors 706(2)
Matrices 708(3)
Matrix-Vector Product 711(1)
Matrix Product 711(2)
Other Concepts 713(2)
Cramer's Rule 715(1)
Abstract Vector Spaces 716(8)
Subspaces 717(1)
Linear Independence 717(1)
Bases 718(1)
Linear Transformations 718(1)
Eigenvalues and Eigenvectors 719(1)
Change of Basis and Similarity 719(1)
Orthogonal Matrices and Spectral Theorem 720(1)
Norms 721(1)
Gram-Schmidt Process 722(2)
Answers for Selected Problems 724(21)
Bibliography 745(9)
Index 754