Verfügbarkeit: Numerical mathematics and computing

Numerical mathematics and computing:

Gespeichert in:

Bibliographische Detailangaben
Hauptverfasser:	Cheney, Elliott W. 1929- (VerfasserIn), Kincaid, David 1942- (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Belmont, Calif. Thomson Brooks/Cole 2008
Ausgabe:	6. ed., internat. student ed.
Schlagworte:	Datenverarbeitung Numerical analysis > Data processing Numerische Mathematik Algorithmus
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XX, 763 S. Ill., graph. Darst.
ISBN:	0495384720 9780495384724 0495114758 9780495114758

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Datensatz im Suchindex

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adam_text	Contents 1 Introduction 1 1.1 Preliminary Remarks 1 Significant Digits of Precision: Examples Errors: Absolute and Relative 5 Accuracy and Precision 5 Rounding and Chopping 6 Nested Multiplication 7 Pairs of Easy/Hard Problems 9 First Programming Experiment 9 Mathematical Software 10 Summary 11 Additional References 11 Problems 1.1 12 Computer Problems 1.1 14 1.2 Review of Taylor Series 20 Taylor Series 20 Complete Homer's Algorithm 23 Taylor's Theorem in Terms of (x — c) 24 Mean-Value Theorem 26 Taylor's Theorem in Terms of h 26 Alternating Series 28 Summary 30 Additional References 31 Problems 1.2 31 Computer Problems 1.2 36 2 Floating-Point Representation and Errors 43 2.1 Floating-Point Representation 43 Normalized Floating-Point Representation 44 Floating-Point Representation 46 Single-Precision Floating-Point Form 46 Double-Precision Floating-point Form 48 Computer Errors in Representing Numbers 50 Notation fl(x) and Backward Error Analysis 51 Historical Notes 54 Summary 54 Problems 2.1 55 Computer Problems 2.1 59 2.2 Loss of Significance 61 Significant Digits 61 Computer-Caused Loss of Significance 62 Theorem on Loss of Precision 63 Avoiding Loss of Significance in Subtraction 64 Range Reduction 67 Summary 68 Additional References 68 Problems 2.2 68 Computer Problems 2.2 71 3 Locating Roots of Equations 76 3.1 Bisection Method 76 Introduction 76 Bisection Algorithm and Pseudocode 78 Examples 79 Convergence Analysis 81 False Position (Regula Falsi) Method and Modifications 83 Summary 85 Problems 3.1 85 Computer Problems 3.1 87 3.2 Newton's Method 89 Interpretations of Newton's Method 90 Pseudocode 92 Illustration 92 Convergence Analysis 93 Systems of Nonlinear Equations 96 Fractal Basins of Attraction 99 Summary 100 Additional References 100 Problems 3.2 101 Computer Problems 3.2 105 3.3 Secant Method 111 Secant Algorithm 112 Convergence Analysis 114 Comparison of Methods 117 Hybrid Schemes 117 Fixed-Point Iteration 117 Summary 118 Additional References 119 Problems 3.3 119 Computer Problems 3.3 121 4 Interpolation and Numerical Differentiation 124 4.1 Polynomial Interpolation 124 Preliminary Remarks 124 Polynomial Interpolation 125 Interpolating Polynomial: Lagrange Form 126 Existence of Interpolating Polynomial 128 Interpolating Polynomial: Newton Form 128 Nested Form 130 Calculating Coefficients a¡ Using Divided Differences 131 Algorithms and Pseudocode 136 Vandermonde Matrix 139 Inverse Interpolation 141 Polynomial Interpolation by Neville's Algorithm 142 Interpolation of Bivariate Functions 144 Summary 145 Problems 4.1 146 Computer Problems 4.1 152 4.2 Errors in Polynomial Interpolation 153 Dirichlet Function 154 Runge Function 154 Theorems on Interpolation Errors 156 Summary 160 Problems 4.2 161 Computer Problems 4.2 163 4.3 Estimating Derivatives and Richardson Extrapolation 164 First-Derivative Formulas via Taylor Series 164 Richardson Extrapolation 166 First-Derivative Formulas via Interpolation Polynomials 170 Second-Derivative Formulas via Taylor Series 173 Noise in Computation 174 Summary 174 Additional References for Chapter 4 175 Problems 4.3 175 Computer Problems 4.3 178 5 Numerical Integration 180 5.1 Lower and Upper Sums 180 Definite and Indefinite Integrals 180 Lower and Upper Sums 181 Riemann-Integrable Functions 183 Examples and Pseudocode 184 Summary 187 Problems 5.1 187 Computer Problems 5.1 188 5.2 Trapezoid Rule 190 Uniform Spacing 191 Error Analysis 192 Applying the Error Formula 195 Recursive Trapezoid Formula for Equal Subintervals 196 Multidimensional Integration 198 Summary 199 Problems 5.2 200 Computer Problems 5.2 203 5.3 Romberg Algorithm 204 Description 204 Pseudocode 205 Euler-Maclaurin Formula 206 General Extrapolation 209 Summary 211 Additional References 211 Problems 5.3 212 Computer Problems 5.3 214 6 Additional Topics on Numerical Integration 216 6.1 Simpson's Rule and Adaptive Simpson's Rule 216 Basic Simpson's Rule 216 Simpson's Rule 219 Composite Simpson's Rule 220 An Adaptive Simpson's Scheme 221 Example Using Adaptive Simpson Procedure 224 Newton-Cotes Rules 225 Summary 226 Problems 6.1 227 Computer Problems 6.1 229 6.2 Gaussian Quadrature Formulas 230 Description 230 Change of Intervals 231 Gaussian Nodes and Weights 232 Legendre Polynomials 234 Integrals with Singularities 237 Summary 237 Additional References 239 Problems 6.2 239 Computer Problems 6.2 241 7 Systems of Linear Equations 245 7.1 Naive Gaussian Elimination 245 A Larger Numerical Example 247 Algorithm 248 Pseudocode 250 Testing the Pseudocode 253 Residual and Error Vectors 254 Summary 255 Problems 7.1 255 Computer Problems 7.1 257 7.2 Gaussian Elimination with Scaled Partial Pivoting 259 Naive Gaussian Elimination Can Fail 259 Partial Pivoting and Complete Partial Pivoting 261 Gaussian Elimination with Scaled Partial Pivoting 262 A Larger Numerical Example 265 Pseudocode 266 Long Operation Count 269 Numerical Stability 271 Scaling 271 Summary 271 Problems 7.2 272 Computer Problems 7.2 276 7.3 Tridiagonal and Banded Systems 280 Tridiagonal Systems 281 Strictly Diagonal Dominance 282 Pentadiagonal Systems 283 Block Pentadiagonal Systems 285 Summary 286 Additional References 287 Problems 7.3 287 Computer Problems 7.3 288 8 Additional Topics Concerning Systems of Linear Equations 293 8.1 Matrix Factorizations 293 Numerical Example 294 Formal Derivation 296 Pseudocode 300 Solving Linear Systems Using LU Factorization 300 LDLT Factorization 302 Cholesky Factorization 305 Multiple Right-Hand Sides 306 Computing A'1 307 Example Using Software Packages 307 Summary 309 Problems 8.1 311 Computer Problems 8.1 316 8.2 Iterative Solutions of Linear Systems 319 Vector and Matrix Norms 319 Condition Number and Ill-Conditioning 321 Basic Iterative Methods 322 Pseudocode 327 Convergence Theorems 328 Matrix Formulation 331 Another View of Overrelaxation 332 Conjugate Gradient Method 332 Summary 335 Problems 8.2 337 Computer Problems 8.2 339 8.3 Eigenvalues and Eigenvectors 342 Calculating Eigenvalues and Eigenvectors 343 Mathematical Software 344 Properties of Eigenvalues 345 Gershgorin's Theorem 347 Singular Value Decomposition 348 Numerical Examples of Singular Value Decomposition 351 Application: Linear Differential Equations 353 Application: A Vibration Problem 354 Summary 355 Problems 8.3 356 Computer Problems 8.3 358 8.4 Power Method 360 Power Method Algorithms 361 Aitken Acceleration 363 Inverse Power Method 364 Software Examples: Inverse Power Method 365 Shifted (Inverse) Power Method 365 Example: Shifted Inverse Power Method 366 Summary 366 Additional References 367 Problems 8.4 367 Computer Problems 8.4 368 9 Approximation by Spline Functions 371 9.1 First-Degree and Second-Degree Splines 371 First-Degree Spline 372 Modulus of Continuity 374 Second-Degree Splines 376 Interpolating Quadratic Spline Q(x) 376 Subbotin Quadratic Spline 378 Summary 380 Problems 9.1 381 Computer Problems 9.1 384 9.2 Natural Cubic Splines 385 Introduction 385 Natural Cubic Spline 386 Algorithm for Natural Cubic Spline 388 Pseudocode for Natural Cubic Splines 392 Using Pseudocode for Interpolating and Curve Fitting 393 Space Curves 394 Smoothness Property 396 Summary 398 Problems 9.2 399 Computer Problems 9.2 403 9.3 В Splines: Interpolation and Approximation 404 Interpolation and Approximation by В Splines 410 Pseudocode and a Curve-Fitting Example 412 Schoenberg's Process 414 Pseudocode 414 Bézier Curves 416 Summary 418 Additional References 419 Problems 9.3 420 Computer Problems 9.3 423 10 Ordinary Differential Equations 426 10.1 Taylor Series Methods 426 Initial-Value Problem: Analytical versus Numerical Solution 426 An Example of a Practical Problem 428 Solving Differential Equations and Integration 428 Vector Fields 429 Taylor Series Methods 431 Euler's Method Pseudocode 432 Taylor Series Method of Higher Order 433 Types of Errors 435 Taylor Series Method Using Symbolic Computations 435 Summary 435 Problems 10.1 436 Computer Problems 10.1 438 10.2 Runge-Kutta Methods 439 Taylor Series for ƒ (x, y) 440 Runge-Kutta Method of Order 2 441 Runge-Kutta Method of Order 4 442 Pseudocode 443 Summary 444 Problems 10.2 445 Computer Problems 10.2 447 10.3 Stability and Adaptive Runge-Kutta and Multistep Methods 450 An Adaptive Runge-Kutta-Fehlberg Method 450 An Industrial Example 454 Adams-Bashforth-Moulton Formulas 455 Stability Analysis 456 Summary 459 Additional References 460 Problems 10.3 460 Computer Problems 10.3 461 11 Systems of Ordinary Differential Equations 465 11.1 Methods for First-Order Systems 465 Uncoupled and Coupled Systems 465 Taylor Series Method 466 Vector Notation 467 Systems of ODEs 468 Taylor Series Method: Vector Notation 468 Runge-Kutta Method 469 Autonomous ODE 471 Summary 473 Problems 11.1 474 Computer Problems 11.1 475 11.2 Higher-Order Equations and Systems 477 Higher-Order Differential Equations 477 Systems of Higher-Order Differential Equations 479 Autonomous ODE Systems 479 Summary 480 Problems 11.2 480 Computer Problems 11.2 482 11.3 Adams-Bashforth-Moulton Methods 483 A Predictor-Corrector Scheme 483 Pseudocode 484 An Adaptive Scheme 488 An Engineering Example 488 Some Remarks about Stiff Equations 489 Summary 491 Additional References 492 Problems 11.3 492 Computer Problems 11.3 492 12 Smoothing of Data and the Method of Least Squares 495 12.1 Method of Least Squares 495 Linear Least Squares 495 Linear Example 498 Nonpolynomial Example 499 Basis Functions {go, gu ., gn] 500 Summary 501 Problems 12.1 502 Computer Problems 12.1 505 12.2 Orthogonal Systems and Chebyshev Polynomials 505 Orthonormal Basis Functions [go, gi, ■ ■ ■, gn} 505 Outline of Algorithm 508 Smoothing Data: Polynomial Regression 510 Summary 515 Problems 12.2 516 Computer Problems 12.2 517 12.3 Other Examples of the Least-Squares Principle 518 Use of a Weight Function w (χ) 519 Nonlinear Example 520 Linear and Nonlinear Example 521 Additional Details on SVD 522 Using the Singular Value Decomposition 524 Summary 527 Additional References 527 Problems 12.3 527 Computer Problems 12.3 530 13 Monte Carlo Methods and Simulation 532 13.1 Random Numbers 532 Random-Number Algorithms and Generators 533 Examples 535 Uses of Pseudocode Random 537 Summary 541 Problems 13.1 541 Computer Problems 13.1 542 13.2 Estimation of Areas and Volumes by Monte Carlo Techniques 544 Numerical Integration 544 Example and Pseudocode 545 Computing Volumes 547 Ice Cream Cone Example 548 Summary 549 Problems 13.2 549 Computer Problems 13.2 549 13.3 Simulation 552 Loaded Die Problem 552 Birthday Problem 553 Buffon's Needle Problem 555 Two Dice Problem 556 Neutron Shielding 557 Summary 558 Additional References 558 Computer Problems 13.3 559 14 Boundary-Value Problems for Ordinary Differential Equations 563 14.1 Shooting Method 563 Shooting Method Algorithm 565 Modifications and Refinements 567 Summary 567 Problems 14.1 568 Computer Problems 14.1 570 14.2 A Discretization Method 570 Finite-Difference Approximations 570 The Linear Case 571 Pseudocode and Numerical Example 572 Shooting Method in the Linear Case 574 Pseudocode and Numerical Example 575 Summary 577 Additional References 578 Problems 14.2 578 Computer Problems 14.2 580 15 Partial Differential Equations 582 15.1 Parabolic Problems 582 Some Partial Differential Equations from Applied Problems 582 Heat Equation Model Problem 585 Finite-Difference Method 585 Pseudocode for Explicit Method 587 Crank-Nicolson Method 588 Pseudocode for the Crank-Nicolson Method 589 Alternative Version of the Crank-Nicolson Method 590 Stability 591 Summary 593 Problems 15.1 594 Computer Problems 15.1 596 15.2 Hyperbolic Problems 596 Wave Equation Model Problem 596 Analytic Solution 597 Numerical Solution 598 Pseudocode 600 Advection Equation 601 Lax Method 602 Upwind Method 602 Lax-Wendroff Method 602 Summary 603 Problems 15.2 604 Computer Problems 15.2 604 15.3 Elliptic Problems 605 Helmholtz Equation Model Problem 605 Finite-Difference Method 606 Gauss-Seidel Iterative Method 610 Numerical Example and Pseudocode 610 Finite-Element Methods 613 More on Finite Elements 617 Summary 619 Additional References 620 Problems 15.3 620 Computer Problems 15.3 622 16 Minimization of Functions 625 16.1 One-Variable Case 625 Unconstrained and Constrained Minimization Problems 625 One-Variable Case 626 Unimodal Functions F 627 Fibonacci Search Algorithm 628 Golden Section Search Algorithm 631 Quadratic Interpolation Algorithm 633 Summary 635 Problems 16.1 635 Computer Problems 16.1 637 16.2 Multivariate Case 639 Taylor Series for F: Gradient Vector and Hessian Matrix 640 Alternative Form of Taylor Series 641 Steepest Descent Procedure 643 Contour Diagrams 644 More Advanced Algorithms 644 Minimum, Maximum, and Saddle Points 646 Positive Definite Matrix 647 Quasi-Newton Methods 647 Nelder-Mead Algorithm 647 Method of Simulated Annealing 648 Summary 650 Additional References 651 Problems 16.2 651 Computer Problems 16.2 654 17 Linear Programming 657 17.1 Standard Forms and Duality 657 First Primal Form 657 Numerical Example 658 Transforming Problems into First Primal Form 660 Dual Problem 661 Second Primal Form 663 Summary 664 Problems 17.1 665 Computer Problems 17.1 669 17.2 Simplex Method 670 Vertices in К and Linearly Independent Columns of A 671 Simplex Method 672 Summary 674 Problems 17.2 674 Computer Problems 17.2 675 17.3 Approximate Solution of Inconsistent Linear Systems 675 lx Problem 676 €oo Problem 678 Summary 680 Additional References 682 Problems 17.3 682 Computer Problems 17.3 682 Appendix A Advice on Good Programming Practices 684 A.I Programming Suggestions 684 Case Studies 687 On Developing Mathematical Software 691 Appendix В Representation of Numbers in Different Bases 692 B.I Representation of Numbers in Different Bases 692 Base β Numbers 693 Conversion of Integer Parts 693 Conversion of Fractional Parts 695 Base Conversion 10 о 8 <*■ 2 696 Base 16 698 More Examples 698 Summary 699 Problems B.I 699 Computer Problems B.I 701 Appendix С Additional Details on IEEE Floating-Point Arithmetic 703 C.I More on IEEE Standard Floating-Point Arithmetic 703 Appendix D Linear Algebra Concepts and Notation 706 D.I Elementary Concepts 706 Vectors 706 Matrices 708 Matrix- Vector Product 711 Matrix Product 711 Other Concepts 713 Cramer's Rule 715 D.2 Abstract Vector Spaces 716 Subspaces 717 Linear Independence 717 Bases 718 Linear Transformations 718 Eigenvalues and Eigenvectors 719 Change of Basis and Similarity 719 Orthogonal Matrices and Spectral Theorem 720 Norms 721 Gram-Schmidt Process 722 Answers for Selected Problems 724 Bibliography 745 Index 754
adam_txt	Contents 1 Introduction 1 1.1 Preliminary Remarks 1 Significant Digits of Precision: Examples Errors: Absolute and Relative 5 Accuracy and Precision 5 Rounding and Chopping 6 Nested Multiplication 7 Pairs of Easy/Hard Problems 9 First Programming Experiment 9 Mathematical Software 10 Summary 11 Additional References 11 Problems 1.1 12 Computer Problems 1.1 14 1.2 Review of Taylor Series 20 Taylor Series 20 Complete Homer's Algorithm 23 Taylor's Theorem in Terms of (x — c) 24 Mean-Value Theorem 26 Taylor's Theorem in Terms of h 26 Alternating Series 28 Summary 30 Additional References 31 Problems 1.2 31 Computer Problems 1.2 36 2 Floating-Point Representation and Errors 43 2.1 Floating-Point Representation 43 Normalized Floating-Point Representation 44 Floating-Point Representation 46 Single-Precision Floating-Point Form 46 Double-Precision Floating-point Form 48 Computer Errors in Representing Numbers 50 Notation fl(x) and Backward Error Analysis 51 Historical Notes 54 Summary 54 Problems 2.1 55 Computer Problems 2.1 59 2.2 Loss of Significance 61 Significant Digits 61 Computer-Caused Loss of Significance 62 Theorem on Loss of Precision 63 Avoiding Loss of Significance in Subtraction 64 Range Reduction 67 Summary 68 Additional References 68 Problems 2.2 68 Computer Problems 2.2 71 3 Locating Roots of Equations 76 3.1 Bisection Method 76 Introduction 76 Bisection Algorithm and Pseudocode 78 Examples 79 Convergence Analysis 81 False Position (Regula Falsi) Method and Modifications 83 Summary 85 Problems 3.1 85 Computer Problems 3.1 87 3.2 Newton's Method 89 Interpretations of Newton's Method 90 Pseudocode 92 Illustration 92 Convergence Analysis 93 Systems of Nonlinear Equations 96 Fractal Basins of Attraction 99 Summary 100 Additional References 100 Problems 3.2 101 Computer Problems 3.2 105 3.3 Secant Method 111 Secant Algorithm 112 Convergence Analysis 114 Comparison of Methods 117 Hybrid Schemes 117 Fixed-Point Iteration 117 Summary 118 Additional References 119 Problems 3.3 119 Computer Problems 3.3 121 4 Interpolation and Numerical Differentiation 124 4.1 Polynomial Interpolation 124 Preliminary Remarks 124 Polynomial Interpolation 125 Interpolating Polynomial: Lagrange Form 126 Existence of Interpolating Polynomial 128 Interpolating Polynomial: Newton Form 128 Nested Form 130 Calculating Coefficients a¡ Using Divided Differences 131 Algorithms and Pseudocode 136 Vandermonde Matrix 139 Inverse Interpolation 141 Polynomial Interpolation by Neville's Algorithm 142 Interpolation of Bivariate Functions 144 Summary 145 Problems 4.1 146 Computer Problems 4.1 152 4.2 Errors in Polynomial Interpolation 153 Dirichlet Function 154 Runge Function 154 Theorems on Interpolation Errors 156 Summary 160 Problems 4.2 161 Computer Problems 4.2 163 4.3 Estimating Derivatives and Richardson Extrapolation 164 First-Derivative Formulas via Taylor Series 164 Richardson Extrapolation 166 First-Derivative Formulas via Interpolation Polynomials 170 Second-Derivative Formulas via Taylor Series 173 Noise in Computation 174 Summary 174 Additional References for Chapter 4 175 Problems 4.3 175 Computer Problems 4.3 178 5 Numerical Integration 180 5.1 Lower and Upper Sums 180 Definite and Indefinite Integrals 180 Lower and Upper Sums 181 Riemann-Integrable Functions 183 Examples and Pseudocode 184 Summary 187 Problems 5.1 187 Computer Problems 5.1 188 5.2 Trapezoid Rule 190 Uniform Spacing 191 Error Analysis 192 Applying the Error Formula 195 Recursive Trapezoid Formula for Equal Subintervals 196 Multidimensional Integration 198 Summary 199 Problems 5.2 200 Computer Problems 5.2 203 5.3 Romberg Algorithm 204 Description 204 Pseudocode 205 Euler-Maclaurin Formula 206 General Extrapolation 209 Summary 211 Additional References 211 Problems 5.3 212 Computer Problems 5.3 214 6 Additional Topics on Numerical Integration 216 6.1 Simpson's Rule and Adaptive Simpson's Rule 216 Basic Simpson's Rule 216 Simpson's Rule 219 Composite Simpson's Rule 220 An Adaptive Simpson's Scheme 221 Example Using Adaptive Simpson Procedure 224 Newton-Cotes Rules 225 Summary 226 Problems 6.1 227 Computer Problems 6.1 229 6.2 Gaussian Quadrature Formulas 230 Description 230 Change of Intervals 231 Gaussian Nodes and Weights 232 Legendre Polynomials 234 Integrals with Singularities 237 Summary 237 Additional References 239 Problems 6.2 239 Computer Problems 6.2 241 7 Systems of Linear Equations 245 7.1 Naive Gaussian Elimination 245 A Larger Numerical Example 247 Algorithm 248 Pseudocode 250 Testing the Pseudocode 253 Residual and Error Vectors 254 Summary 255 Problems 7.1 255 Computer Problems 7.1 257 7.2 Gaussian Elimination with Scaled Partial Pivoting 259 Naive Gaussian Elimination Can Fail 259 Partial Pivoting and Complete Partial Pivoting 261 Gaussian Elimination with Scaled Partial Pivoting 262 A Larger Numerical Example 265 Pseudocode 266 Long Operation Count 269 Numerical Stability 271 Scaling 271 Summary 271 Problems 7.2 272 Computer Problems 7.2 276 7.3 Tridiagonal and Banded Systems 280 Tridiagonal Systems 281 Strictly Diagonal Dominance 282 Pentadiagonal Systems 283 Block Pentadiagonal Systems 285 Summary 286 Additional References 287 Problems 7.3 287 Computer Problems 7.3 288 8 Additional Topics Concerning Systems of Linear Equations 293 8.1 Matrix Factorizations 293 Numerical Example 294 Formal Derivation 296 Pseudocode 300 Solving Linear Systems Using LU Factorization 300 LDLT Factorization 302 Cholesky Factorization 305 Multiple Right-Hand Sides 306 Computing A'1 307 Example Using Software Packages 307 Summary 309 Problems 8.1 311 Computer Problems 8.1 316 8.2 Iterative Solutions of Linear Systems 319 Vector and Matrix Norms 319 Condition Number and Ill-Conditioning 321 Basic Iterative Methods 322 Pseudocode 327 Convergence Theorems 328 Matrix Formulation 331 Another View of Overrelaxation 332 Conjugate Gradient Method 332 Summary 335 Problems 8.2 337 Computer Problems 8.2 339 8.3 Eigenvalues and Eigenvectors 342 Calculating Eigenvalues and Eigenvectors 343 Mathematical Software 344 Properties of Eigenvalues 345 Gershgorin's Theorem 347 Singular Value Decomposition 348 Numerical Examples of Singular Value Decomposition 351 Application: Linear Differential Equations 353 Application: A Vibration Problem 354 Summary 355 Problems 8.3 356 Computer Problems 8.3 358 8.4 Power Method 360 Power Method Algorithms 361 Aitken Acceleration 363 Inverse Power Method 364 Software Examples: Inverse Power Method 365 Shifted (Inverse) Power Method 365 Example: Shifted Inverse Power Method 366 Summary 366 Additional References 367 Problems 8.4 367 Computer Problems 8.4 368 9 Approximation by Spline Functions 371 9.1 First-Degree and Second-Degree Splines 371 First-Degree Spline 372 Modulus of Continuity 374 Second-Degree Splines 376 Interpolating Quadratic Spline Q(x) 376 Subbotin Quadratic Spline 378 Summary 380 Problems 9.1 381 Computer Problems 9.1 384 9.2 Natural Cubic Splines 385 Introduction 385 Natural Cubic Spline 386 Algorithm for Natural Cubic Spline 388 Pseudocode for Natural Cubic Splines 392 Using Pseudocode for Interpolating and Curve Fitting 393 Space Curves 394 Smoothness Property 396 Summary 398 Problems 9.2 399 Computer Problems 9.2 403 9.3 В Splines: Interpolation and Approximation 404 Interpolation and Approximation by В Splines 410 Pseudocode and a Curve-Fitting Example 412 Schoenberg's Process 414 Pseudocode 414 Bézier Curves 416 Summary 418 Additional References 419 Problems 9.3 420 Computer Problems 9.3 423 10 Ordinary Differential Equations 426 10.1 Taylor Series Methods 426 Initial-Value Problem: Analytical versus Numerical Solution 426 An Example of a Practical Problem 428 Solving Differential Equations and Integration 428 Vector Fields 429 Taylor Series Methods 431 Euler's Method Pseudocode 432 Taylor Series Method of Higher Order 433 Types of Errors 435 Taylor Series Method Using Symbolic Computations 435 Summary 435 Problems 10.1 436 Computer Problems 10.1 438 10.2 Runge-Kutta Methods 439 Taylor Series for ƒ (x, y) 440 Runge-Kutta Method of Order 2 441 Runge-Kutta Method of Order 4 442 Pseudocode 443 Summary 444 Problems 10.2 445 Computer Problems 10.2 447 10.3 Stability and Adaptive Runge-Kutta and Multistep Methods 450 An Adaptive Runge-Kutta-Fehlberg Method 450 An Industrial Example 454 Adams-Bashforth-Moulton Formulas 455 Stability Analysis 456 Summary 459 Additional References 460 Problems 10.3 460 Computer Problems 10.3 461 11 Systems of Ordinary Differential Equations 465 11.1 Methods for First-Order Systems 465 Uncoupled and Coupled Systems 465 Taylor Series Method 466 Vector Notation 467 Systems of ODEs 468 Taylor Series Method: Vector Notation 468 Runge-Kutta Method 469 Autonomous ODE 471 Summary 473 Problems 11.1 474 Computer Problems 11.1 475 11.2 Higher-Order Equations and Systems 477 Higher-Order Differential Equations 477 Systems of Higher-Order Differential Equations 479 Autonomous ODE Systems 479 Summary 480 Problems 11.2 480 Computer Problems 11.2 482 11.3 Adams-Bashforth-Moulton Methods 483 A Predictor-Corrector Scheme 483 Pseudocode 484 An Adaptive Scheme 488 An Engineering Example 488 Some Remarks about Stiff Equations 489 Summary 491 Additional References 492 Problems 11.3 492 Computer Problems 11.3 492 12 Smoothing of Data and the Method of Least Squares 495 12.1 Method of Least Squares 495 Linear Least Squares 495 Linear Example 498 Nonpolynomial Example 499 Basis Functions {go, gu ., gn] 500 Summary 501 Problems 12.1 502 Computer Problems 12.1 505 12.2 Orthogonal Systems and Chebyshev Polynomials 505 Orthonormal Basis Functions [go, gi, ■ ■ ■, gn} 505 Outline of Algorithm 508 Smoothing Data: Polynomial Regression 510 Summary 515 Problems 12.2 516 Computer Problems 12.2 517 12.3 Other Examples of the Least-Squares Principle 518 Use of a Weight Function w (χ) 519 Nonlinear Example 520 Linear and Nonlinear Example 521 Additional Details on SVD 522 Using the Singular Value Decomposition 524 Summary 527 Additional References 527 Problems 12.3 527 Computer Problems 12.3 530 13 Monte Carlo Methods and Simulation 532 13.1 Random Numbers 532 Random-Number Algorithms and Generators 533 Examples 535 Uses of Pseudocode Random 537 Summary 541 Problems 13.1 541 Computer Problems 13.1 542 13.2 Estimation of Areas and Volumes by Monte Carlo Techniques 544 Numerical Integration 544 Example and Pseudocode 545 Computing Volumes 547 Ice Cream Cone Example 548 Summary 549 Problems 13.2 549 Computer Problems 13.2 549 13.3 Simulation 552 Loaded Die Problem 552 Birthday Problem 553 Buffon's Needle Problem 555 Two Dice Problem 556 Neutron Shielding 557 Summary 558 Additional References 558 Computer Problems 13.3 559 14 Boundary-Value Problems for Ordinary Differential Equations 563 14.1 Shooting Method 563 Shooting Method Algorithm 565 Modifications and Refinements 567 Summary 567 Problems 14.1 568 Computer Problems 14.1 570 14.2 A Discretization Method 570 Finite-Difference Approximations 570 The Linear Case 571 Pseudocode and Numerical Example 572 Shooting Method in the Linear Case 574 Pseudocode and Numerical Example 575 Summary 577 Additional References 578 Problems 14.2 578 Computer Problems 14.2 580 15 Partial Differential Equations 582 15.1 Parabolic Problems 582 Some Partial Differential Equations from Applied Problems 582 Heat Equation Model Problem 585 Finite-Difference Method 585 Pseudocode for Explicit Method 587 Crank-Nicolson Method 588 Pseudocode for the Crank-Nicolson Method 589 Alternative Version of the Crank-Nicolson Method 590 Stability 591 Summary 593 Problems 15.1 594 Computer Problems 15.1 596 15.2 Hyperbolic Problems 596 Wave Equation Model Problem 596 Analytic Solution 597 Numerical Solution 598 Pseudocode 600 Advection Equation 601 Lax Method 602 Upwind Method 602 Lax-Wendroff Method 602 Summary 603 Problems 15.2 604 Computer Problems 15.2 604 15.3 Elliptic Problems 605 Helmholtz Equation Model Problem 605 Finite-Difference Method 606 Gauss-Seidel Iterative Method 610 Numerical Example and Pseudocode 610 Finite-Element Methods 613 More on Finite Elements 617 Summary 619 Additional References 620 Problems 15.3 620 Computer Problems 15.3 622 16 Minimization of Functions 625 16.1 One-Variable Case 625 Unconstrained and Constrained Minimization Problems 625 One-Variable Case 626 Unimodal Functions F 627 Fibonacci Search Algorithm 628 Golden Section Search Algorithm 631 Quadratic Interpolation Algorithm 633 Summary 635 Problems 16.1 635 Computer Problems 16.1 637 16.2 Multivariate Case 639 Taylor Series for F: Gradient Vector and Hessian Matrix 640 Alternative Form of Taylor Series 641 Steepest Descent Procedure 643 Contour Diagrams 644 More Advanced Algorithms 644 Minimum, Maximum, and Saddle Points 646 Positive Definite Matrix 647 Quasi-Newton Methods 647 Nelder-Mead Algorithm 647 Method of Simulated Annealing 648 Summary 650 Additional References 651 Problems 16.2 651 Computer Problems 16.2 654 17 Linear Programming 657 17.1 Standard Forms and Duality 657 First Primal Form 657 Numerical Example 658 Transforming Problems into First Primal Form 660 Dual Problem 661 Second Primal Form 663 Summary 664 Problems 17.1 665 Computer Problems 17.1 669 17.2 Simplex Method 670 Vertices in К and Linearly Independent Columns of A 671 Simplex Method 672 Summary 674 Problems 17.2 674 Computer Problems 17.2 675 17.3 Approximate Solution of Inconsistent Linear Systems 675 lx Problem 676 €oo Problem 678 Summary 680 Additional References 682 Problems 17.3 682 Computer Problems 17.3 682 Appendix A Advice on Good Programming Practices 684 A.I Programming Suggestions 684 Case Studies 687 On Developing Mathematical Software 691 Appendix В Representation of Numbers in Different Bases 692 B.I Representation of Numbers in Different Bases 692 Base β Numbers 693 Conversion of Integer Parts 693 Conversion of Fractional Parts 695 Base Conversion 10 о 8 <*■ 2 696 Base 16 698 More Examples 698 Summary 699 Problems B.I 699 Computer Problems B.I 701 Appendix С Additional Details on IEEE Floating-Point Arithmetic 703 C.I More on IEEE Standard Floating-Point Arithmetic 703 Appendix D Linear Algebra Concepts and Notation 706 D.I Elementary Concepts 706 Vectors 706 Matrices 708 Matrix- Vector Product 711 Matrix Product 711 Other Concepts 713 Cramer's Rule 715 D.2 Abstract Vector Spaces 716 Subspaces 717 Linear Independence 717 Bases 718 Linear Transformations 718 Eigenvalues and Eigenvectors 719 Change of Basis and Similarity 719 Orthogonal Matrices and Spectral Theorem 720 Norms 721 Gram-Schmidt Process 722 Answers for Selected Problems 724 Bibliography 745 Index 754
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dewey-tens	510 - Mathematics
discipline	Mathematik
discipline_str_mv	Mathematik
edition	6. ed., internat. student ed.
format	Book
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id	DE-604.BV035008723
illustrated	Illustrated
index_date	2024-07-02T21:43:08Z
indexdate	2024-07-20T07:20:32Z
institution	BVB
isbn	0495384720 9780495384724 0495114758 9780495114758
language	English
lccn	2007922553
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-016678007
oclc_num	224057851
open_access_boolean
owner	DE-739 DE-898 DE-BY-UBR DE-824
owner_facet	DE-739 DE-898 DE-BY-UBR DE-824
physical	XX, 763 S. Ill., graph. Darst.
publishDate	2008
publishDateSearch	2008
publishDateSort	2008
publisher	Thomson Brooks/Cole
record_format	marc
spelling	Cheney, Elliott W. 1929- Verfasser (DE-588)122199545 aut Numerical mathematics and computing Ward Cheney ; David Kincaid 6. ed., internat. student ed. Belmont, Calif. Thomson Brooks/Cole 2008 XX, 763 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Datenverarbeitung (DE-588)4011152-0 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s Algorithmus (DE-588)4001183-5 s Datenverarbeitung (DE-588)4011152-0 s 1\p DE-604 Kincaid, David 1942- Verfasser (DE-588)121384152 aut Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016678007&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Cheney, Elliott W. 1929- Kincaid, David 1942- Numerical mathematics and computing Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd Datenverarbeitung (DE-588)4011152-0 gnd Algorithmus (DE-588)4001183-5 gnd
subject_GND	(DE-588)4042805-9 (DE-588)4011152-0 (DE-588)4001183-5
title	Numerical mathematics and computing
title_auth	Numerical mathematics and computing
title_exact_search	Numerical mathematics and computing
title_exact_search_txtP	Numerical mathematics and computing
title_full	Numerical mathematics and computing Ward Cheney ; David Kincaid
title_fullStr	Numerical mathematics and computing Ward Cheney ; David Kincaid
title_full_unstemmed	Numerical mathematics and computing Ward Cheney ; David Kincaid
title_short	Numerical mathematics and computing
title_sort	numerical mathematics and computing
topic	Datenverarbeitung Numerical analysis Data processing Numerische Mathematik (DE-588)4042805-9 gnd Datenverarbeitung (DE-588)4011152-0 gnd Algorithmus (DE-588)4001183-5 gnd
topic_facet	Datenverarbeitung Numerical analysis Data processing Numerische Mathematik Algorithmus
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016678007&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT cheneyelliottw numericalmathematicsandcomputing AT kincaiddavid numericalmathematicsandcomputing

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