Holdings: Numerical methods in scientific computing

Numerical methods in scientific computing: 1

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Bibliographic Details
Main Authors:	Dahlquist, Germund (Author), Björck, Åke 1934- (Author)
Format:	Book
Language:	English
Published:	Philadelphia SIAM, Society for Industrial and Applied Mathematics (2008)
Subjects:	Numerische Mathematik Wissenschaftliches Rechnen
Online Access:	Inhaltsverzeichnis Beschreibung für Leser Inhaltsverzeichnis
Physical Description:	XXVII, 717 S. graph. Darst.
ISBN:	9780898716443

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adam_text	Contents List of Figures xv List of Tables xix List of Conventions xxi Preface xxiii 1 Principles of Numerical Calculations 1 1.1 Common Ideas and Concepts ....................... 1 1.1.1 Fixed-Point Iteration ........................ 2 1.1.2 Newton s Method .......................... 5 1.1.3 Linearization and Extrapolation .................. 9 1.1.4 Finite Difference Approximations ................. 11 Review Questions ............................... 15 Problems and Computer Exercises ....................... 15 1.2 Some Numerical Algorithms ....................... 16 1.2.1 Solving a Quadratic Equation .................... 16 1.2.2 Recurrence Relations ........................ 17 1.2.3 Divide and Conquer Strategy .................... 20 1.2.4 Power Series Expansions ...................... 22 Review Questions ............................... 23 Problems and Computer Exercises ....................... 23 1.3 Matrix Computations ........................... 26 1.3.1 Matrix Multiplication ........................ 26 1.3.2 Solving Linear Systems by LU Factorization ............ 28 1.3.3 Sparse Matrices and Iterative Methods ............... 38 1.3.4 Software for Matrix Computations ................. 41 Review Questions ............................... 43 Problems and Computer Exercises ....................... 43 1.4 The Linear Least Squares Problem .................... 44 1.4.1 Basic Concepts in Probability and Statistics ............ 45 1.4.2 Characterization of Least Squares Solutions ............ 46 1.4.3 The Singular Value Decomposition ................. 50 1.4.4 The Numerical Rank of a Matrix .................. 52 Review Questions ............................... 54 vii viii Contents Problems and Computer Exercises ....................... 54 1.5 Numerical Solution of Differential Equations ............... 55 1.5.1 Euler s Method ........................... 55 1.5.2 An Introductory Example ...................... 56 1.5.3 Second Order Accurate Methods .................. 59 1.5.4 Adaptive Choice of Step Size .................... 61 Review Questions ............................... 63 Problems and Computer Exercises ....................... 63 1.6 Monte Carlo Methods ........................... 64 1.6.1 Origin of Monte Carlo Methods .................. 64 1.6.2 Generating and Testing Pseudorandom Numbers .......... 66 1.6.3 Random Deviates for Other Distributions ............. 73 1.6.4 Reduction of Variance ........................ 77 Review Questions ............................... 81 Problems and Computer Exercises ....................... 82 Notes and References .............................. 83 How to Obtain and Estimate Accuracy 87 2.1 Basic Concepts in Error Estimation .................... 87 2.1.1 Sources of Error ........................... 87 2.1.2 Absolute and Relative Errors .................... 90 2.1.3 Rounding and Chopping ...................... 91 Review Questions ............................... 93 2.2 Computer Number Systems ........................ 93 2.2.1 The Position System ........................ 93 2.2.2 Fixed-and Floating-Point Representation ............. 95 2.2.3 IEEE Floating-Point Standard ................... 99 2.2.4 Elementary Functions ........................ 102 2.2.5 Multiple Precision Arithmetic ................... 104 Review Questions ............................... 105 Problems and Computer Exercises ....................... 105 2.3 Accuracy and Rounding Errors ...................... 107 2.3.1 Floating-Point Arithmetic ...................... 107 2.3.2 Basic Rounding Error Results ................... 113 2.3.3 Statistical Models for Rounding Errors ............... 116 2.3.4 Avoiding Overflow and Cancellation ................ 118 Review Questions ............................... 122 Problems and Computer Exercises ....................... 122 2.4 Error Propagation ............................. 126 2.4.1 Numerical Problems, Methods, and Algorithms .......... 126 2.4.2 Propagation of Errors and Condition Numbers ........... 127 2.4.3 Perturbation Analysis for Linear Systems ............. 134 2.4.4 Error Analysis and Stability of Algorithms ............. 137 Review Questions ............................... 142 Problems and Computer Exercises ....................... 142 Contents ix 2.5 Automatic Control of Accuracy and Verified Computing ......... 145 2.5.1 Running Error Analysis ....................... 145 2.5.2 Experimental Perturbations ..................... 146 2.5.3 Interval Arithmetic ......................... 147 2.5.4 Range of Functions ......................... 150 2.5.5 Interval Matrix Computations ................... 153 Review Questions ............................... 154 Problems and Computer Exercises ....................... 155 Notes and References .............................. 155 3 Series, Operators, and Continued Fractions 157 3.1 Some Basic Facts about Series ...................... 157 3.1.1 Introduction ............................. 157 3.1.2 Taylor s Formula and Power Series ................. 162 3.1.3 Analytic Continuation ........................ 171 3.1.4 Manipulating Power Series ..................... 173 3.1.5 Formal Power Series ........................ 181 Review Questions ............................... 184 Problems and Computer Exercises ....................... 185 3.2 More about Series ............................. 191 3.2.1 Laurent and Fourier Series ..................... 191 3.2.2 The Cauchy-FFT Method ...................... 193 3.2.3 Chebyshev Expansions ....................... 198 3.2.4 Perturbation Expansions ...................... 203 3.2.5 Ill-Conditioned Series ........................ 206 3.2.6 Divergent or Semiconvergent Series ................ 212 Review Questions ............................... 215 Problems and Computer Exercises ....................... 215 3.3 Difference Operators and Operator Expansions .............. 220 3.3.1 Properties of Difference Operators ................. 220 3.3.2 The Calculus of Operators ..................... 225 3.3.3 The Peano Theorem ......................... 237 3.3.4 Approximation Formulas by Operator Methods .......... 242 3.3.5 Single Linear Difference Equations ................. 251 Review Questions ............................... 261 Problems and Computer Exercises ....................... 261 3.4 Acceleration of Convergence ....................... 271 3.4.1 Introduction ............................. 271 3.4.2 Comparison Series and Aitken Acceleration ............ 272 3.4.3 Euler s Transformation ....................... 278 3.4.4 Complete Monotonicity and Related Concepts ........... 284 3.4.5 Euler-Maclaurin s Formula ..................... 292 3.4.6 Repeated Richardson Extrapolation ................ 302 Review Questions ............................... 309 Problems and Computer Exercises ....................... 309 Contents 3.5 Continued Fractions and Padé Approximants ............... 321 3.5.1 Algebraic Continued Fractions ................... 321 3.5.2 Analytic Continued Fractions .................... 326 3.5.3 The Padé Table ........................... 329 3.5.4 The Epsilon Algorithm ....................... 336 3.5.5 The qdAlgorithm .......................... 339 Review Questions ............................... 345 Problems and Computer Exercises ....................... 345 Notes and References .............................. 348 4 Interpolation and Approximation 351 4.1 The Interpolation Problem ......................... 351 4.1.1 Introduction ............................. 351 4.1.2 Bases for Polynomial Interpolation ................. 352 4.1.3 Conditioning of Polynomial Interpolation ............. 355 Review Questions ............................... 357 Problems and Computer Exercises ....................... 357 4.2 Interpolation Formulas and Algorithms .................. 358 4.2.1 Newton s Interpolation Formula .................. 358 4.2.2 Inverse Interpolation ........................ 366 4.2.3 Barycentric Lagrange Interpolation ................. 367 4.2.4 Iterative Linear Interpolation .................... 371 4.2.5 Fast Algorithms for Vandermonde Systems ............ 373 4.2.6 The Runge Phenomenon ...................... 377 Review Questions ............................... 380 Problems and Computer Exercises ....................... 380 4.3 Generalizations and Applications ..................... 381 4.3.1 Hermite Interpolation ........................ 381 4.3.2 Complex Analysis in Polynomial Interpolation ........... 385 4.3.3 Rational Interpolation ........................ 389 4.3.4 Multidimensional Interpolation ................... 395 4.3.5 Analysis of a Generalized Runge Phenomenon ........... 398 Review Questions ............................... 407 Problems and Computer Exercises ....................... 407 4.4 Piecewise Polynomial Interpolation .................... 410 4.4.1 Bernstein Polynomials and Bézier Curves ............. 411 4.4.2 Spline Functions .......................... 417 4.4.3 The B-Spline Basis ......................... 426 4.4.4 Least Squares Splines Approximation ............... 434 Review Questions ............................... 436 Problems and Computer Exercises ....................... 437 4.5 Approximation and Function Spaces ................... 439 4.5.1 Distance and Norm ......................... 440 4.5.2 Operator Norms and the Distance Formula ............. 444 4.5.3 Inner Product Spaces and Orthogonal Systems ........... 450 Contents x¡ 4.5.4 Solution of the Approximation Problem.............. 454 4.5.5 Mathematical Properties of Orthogonal Polynomials ....... 457 4.5.6 Expansions in Orthogonal Polynomials .............. 466 4.5.7 Approximation in the Maximum Norm ............... 471 Review Questions ............................... 478 Problems and Computer Exercises ....................... 479 4.6 Fourier Methods .............................. 482 4.6.1 Basic Formulas and Theorems ................... 483 4.6.2 Discrete Fourier Analysis ...................... 487 4.6.3 Periodic Continuation of a Function ................ 491 4.6.4 Convergence Acceleration of Fourier Series ............ 492 4.6.5 The Fourier Integral Theorem ................... 494 4.6.6 Sampled Data and Aliasing ..................... 497 Review Questions ............................... 500 Problems and Computer Exercises ....................... 500 4.7 The Fast Fourier Transform ........................ 503 4.7.1 The FFT Algorithm ......................... 503 4.7.2 Discrete Convolution by FFT .................... 509 4.7.3 FFTs of Real Data .......................... 510 4.7.4 Fast Trigonometric Transforms ................... 512 4.7.5 The General Case FFT ....................... 515 Review Questions ............................... 516 Problems and Computer Exercises ....................... 517 Notes and References .............................. 518 Numerical Integration 521 5.1 Interpolatory Quadrature Rules ...................... 521 5.1.1 Introduction ............................. 521 5.1.2 Treating Singularities ........................ 525 5.1.3 Some Classical Formulas ...................... 527 5.1.4 Supercon vergence of the Trapezoidal Rule ............. 531 5.1.5 Higher-Order Newton-Cotes Formulas .............. 533 5.1.6 Fejér and Clensnaw-Curtis Rules .................. 538 Review Questions ............................... 542 Problems and Computer Exercises ....................... 542 5.2 Integration by Extrapolation ........................ 546 5.2.1 The Euler-Maclaurin Formula ................... 546 5.2.2 Romberg s Method ......................... 548 5.2.3 Oscillating Integrands ........................ 554 5.2.4 Adaptive Quadrature ........................ 560 Review Questions ............................... 564 Problems and Computer Exercises ....................... 564 5.3 Quadrature Rules with Free Nodes .................... 565 5.3.1 Method of Undetermined Coefficients ............... 565 5.3.2 Gauss-Christoffel Quadrature Rules ................ 568 xii Contents 5.3.3 Gauss Quadrature with Preassigned Nodes ............. 573 5.3.4 Matrices, Moments, and Gauss Quadrature ............. 576 5.3.5 Jacobi Matrices and Gauss Quadrature ............... 580 Review Questions ............................... 585 Problems and Computer Exercises ....................... 585 5.4 Multidimensional Integration ....................... 587 5.4.1 Analytic Techniques ......................... 588 5.4.2 Repeated One-Dimensional Integration .............. 589 5.4.3 ProductRules ............................ 590 5.4.4 Irregular Triangular Grids ...................... 594 5.4.5 Monte Carlo Methods ........................ 599 5.4.6 Quasi-Monte Carlo and Lattice Methods .............. 601 Review Questions ............................... 604 Problems and Computer Exercises ....................... 605 Notes and References .............................. 606 Solving Scalar Nonlinear Equations 609 6.1 Some Basic Concepts and Methods .................... 609 6.1.1 Introduction ............................. 609 6.1.2 The Bisection Method ....................... 610 6.1.3 Limiting Accuracy and Termination Criteria ............ 614 6.1.4 Fixed-Point Iteration ........................ 618 6.1.5 Convergence Order and Efficiency ................. 621 Review Questions ............................... 624 Problems and Computer Exercises ....................... 624 6.2 Methods Based on Interpolation ...................... 626 6.2.1 Method of False Position ...................... 626 6.2.2 The Secant Method ......................... 628 6.2.3 Higher-Order Interpolation Methods ................ 631 6.2.4 A Robust Hybrid Method ...................... 634 Review Questions ............................... 635 Problems and Computer Exercises ....................... 636 6.3 Methods Using Derivatives ........................ 637 6.3.1 Newton s Method .......................... 637 6.3.2 Newton s Method for Complex Roots ............... 644 6.3.3 An Interval Newton Method .................... 646 6.3.4 Higher-Order Methods ....................... 647 Review Questions ............................... 652 Problems and Computer Exercises ....................... 653 6.4 Finding a Minimum of a Function ..................... 656 6.4.1 Introduction ............................. 656 6.4.2 Unimodal Functions and Golden Section Search .......... 657 6.4.3 Minimization by Interpolation ................... 660 Review Questions ............................... 661 Problems and Computer Exercises ....................... 661 Contents x¡¡¡ 6.5 Algebraic Equations ............................ 662 6.5.1 Some Elementary Results ...................... 662 6.5.2 Ill-Conditioned Algebraic Equations ................ 665 6.5.3 Three Classical Methods ...................... 668 6.5.4 Deflation and Simultaneous Determination of Roots ........ 671 6.5.5 A Modified Newton Method .................... 675 6.5.6 Sturm Sequences .......................... 677 6.5.7 Finding Greatest Common Divisors ................ 680 Review Questions ............................... 682 Problems and Computer Exercises ....................... 683 Notes and References .............................. 685 Bibliography 687 Index 707 A Online Appendix: Introduction to Matrix Computations A-l A.I Vectors and Matrices ............................ A-l A. 1.1 Linear Vector Spaces ........................ A-l A. 1.2 Matrix and Vector Algebra ..................... A-3 A.1.3 Rank and Linear Systems ...................... A-5 A. 1.4 Special Matrices .......................... A-6 A.2 Submatrices and Block Matrices ..................... A-8 A.2.1 Block Gaussian Elimination .................... A-10 A.3 Permutations and Determinants ...................... A-12 A.4 Eigenvalues and Norms of Matrices .................... A-16 A.4.1 The Characteristic Equation .................... A-16 A.4.2 The Schur and Jordan Normal Forms ................ A-17 A.4.3 Norms of Vectors and Matrices ................... A-18 Review Questions ............................... A-21 Problems .................................... A-22 В Online Appendix: A MATLAB Multiple Precision Package B-l B.I The Mulprec Package ........................... B-l B.I.I Number Representation ....................... B-l B.1.2 The Mulprec Function Library ................... B-3 B.1.3 Basic Arithmetic Operations .................... B-3 B.1.4 Special Mulprec Operations .................... B-4 B.2 Function and Vector Algorithms ...................... B-4 B.2.1 Elementary Functions ........................ B-4 B.2.2 Mulprec Vector Algorithms ..................... B-5 B.2.3 Miscellaneous ............................ B-6 B.2.4 Using Mulprec ........................... B-6 Computer Exercises .............................. B-6 xiv Contents С Online Appendix: Guide to Literature C-I Cl Introduction ................................ C-l C.2 Textbooks in Numerical Analysis ..................... С -l С.З Handbooks and Collections ........................ C-5 C.4 Encyclopedias, Tables, and Formulas ................... C-6 C.5 Selected Journals ............................. C-8 C.6 Algorithms and Software ......................... C-9 C.7 Public Domain Software .......................... C-10
adam_txt	Contents List of Figures xv List of Tables xix List of Conventions xxi Preface xxiii 1 Principles of Numerical Calculations 1 1.1 Common Ideas and Concepts . 1 1.1.1 Fixed-Point Iteration . 2 1.1.2 Newton's Method . 5 1.1.3 Linearization and Extrapolation . 9 1.1.4 Finite Difference Approximations . 11 Review Questions . 15 Problems and Computer Exercises . 15 1.2 Some Numerical Algorithms . 16 1.2.1 Solving a Quadratic Equation . 16 1.2.2 Recurrence Relations . 17 1.2.3 Divide and Conquer Strategy . 20 1.2.4 Power Series Expansions . 22 Review Questions . 23 Problems and Computer Exercises . 23 1.3 Matrix Computations . 26 1.3.1 Matrix Multiplication . 26 1.3.2 Solving Linear Systems by LU Factorization . 28 1.3.3 Sparse Matrices and Iterative Methods . 38 1.3.4 Software for Matrix Computations . 41 Review Questions . 43 Problems and Computer Exercises . 43 1.4 The Linear Least Squares Problem . 44 1.4.1 Basic Concepts in Probability and Statistics . 45 1.4.2 Characterization of Least Squares Solutions . 46 1.4.3 The Singular Value Decomposition . 50 1.4.4 The Numerical Rank of a Matrix . 52 Review Questions . 54 vii viii Contents Problems and Computer Exercises . 54 1.5 Numerical Solution of Differential Equations . 55 1.5.1 Euler's Method . 55 1.5.2 An Introductory Example . 56 1.5.3 Second Order Accurate Methods . 59 1.5.4 Adaptive Choice of Step Size . 61 Review Questions . 63 Problems and Computer Exercises . 63 1.6 Monte Carlo Methods . 64 1.6.1 Origin of Monte Carlo Methods . 64 1.6.2 Generating and Testing Pseudorandom Numbers . 66 1.6.3 Random Deviates for Other Distributions . 73 1.6.4 Reduction of Variance . 77 Review Questions . 81 Problems and Computer Exercises . 82 Notes and References . 83 How to Obtain and Estimate Accuracy 87 2.1 Basic Concepts in Error Estimation . 87 2.1.1 Sources of Error . 87 2.1.2 Absolute and Relative Errors . 90 2.1.3 Rounding and Chopping . 91 Review Questions . 93 2.2 Computer Number Systems . 93 2.2.1 The Position System . 93 2.2.2 Fixed-and Floating-Point Representation . 95 2.2.3 IEEE Floating-Point Standard . 99 2.2.4 Elementary Functions . 102 2.2.5 Multiple Precision Arithmetic . 104 Review Questions . 105 Problems and Computer Exercises . 105 2.3 Accuracy and Rounding Errors . 107 2.3.1 Floating-Point Arithmetic . 107 2.3.2 Basic Rounding Error Results . 113 2.3.3 Statistical Models for Rounding Errors . 116 2.3.4 Avoiding Overflow and Cancellation . 118 Review Questions . 122 Problems and Computer Exercises . 122 2.4 Error Propagation . 126 2.4.1 Numerical Problems, Methods, and Algorithms . 126 2.4.2 Propagation of Errors and Condition Numbers . 127 2.4.3 Perturbation Analysis for Linear Systems . 134 2.4.4 Error Analysis and Stability of Algorithms . 137 Review Questions . 142 Problems and Computer Exercises . 142 Contents ix 2.5 Automatic Control of Accuracy and Verified Computing . 145 2.5.1 Running Error Analysis . 145 2.5.2 Experimental Perturbations . 146 2.5.3 Interval Arithmetic . 147 2.5.4 Range of Functions . 150 2.5.5 Interval Matrix Computations . 153 Review Questions . 154 Problems and Computer Exercises . 155 Notes and References . 155 3 Series, Operators, and Continued Fractions 157 3.1 Some Basic Facts about Series . 157 3.1.1 Introduction . 157 3.1.2 Taylor's Formula and Power Series . 162 3.1.3 Analytic Continuation . 171 3.1.4 Manipulating Power Series . 173 3.1.5 Formal Power Series . 181 Review Questions . 184 Problems and Computer Exercises . 185 3.2 More about Series . 191 3.2.1 Laurent and Fourier Series . 191 3.2.2 The Cauchy-FFT Method . 193 3.2.3 Chebyshev Expansions . 198 3.2.4 Perturbation Expansions . 203 3.2.5 Ill-Conditioned Series . 206 3.2.6 Divergent or Semiconvergent Series . 212 Review Questions . 215 Problems and Computer Exercises . 215 3.3 Difference Operators and Operator Expansions . 220 3.3.1 Properties of Difference Operators . 220 3.3.2 The Calculus of Operators . 225 3.3.3 The Peano Theorem . 237 3.3.4 Approximation Formulas by Operator Methods . 242 3.3.5 Single Linear Difference Equations . 251 Review Questions . 261 Problems and Computer Exercises . 261 3.4 Acceleration of Convergence . 271 3.4.1 Introduction . 271 3.4.2 Comparison Series and Aitken Acceleration . 272 3.4.3 Euler's Transformation . 278 3.4.4 Complete Monotonicity and Related Concepts . 284 3.4.5 Euler-Maclaurin's Formula . 292 3.4.6 Repeated Richardson Extrapolation . 302 Review Questions . 309 Problems and Computer Exercises . 309 Contents 3.5 Continued Fractions and Padé Approximants . 321 3.5.1 Algebraic Continued Fractions . 321 3.5.2 Analytic Continued Fractions . 326 3.5.3 The Padé Table . 329 3.5.4 The Epsilon Algorithm . 336 3.5.5 The qdAlgorithm . 339 Review Questions . 345 Problems and Computer Exercises . 345 Notes and References . 348 4 Interpolation and Approximation 351 4.1 The Interpolation Problem . 351 4.1.1 Introduction . 351 4.1.2 Bases for Polynomial Interpolation . 352 4.1.3 Conditioning of Polynomial Interpolation . 355 Review Questions . 357 Problems and Computer Exercises . 357 4.2 Interpolation Formulas and Algorithms . 358 4.2.1 Newton's Interpolation Formula . 358 4.2.2 Inverse Interpolation . 366 4.2.3 Barycentric Lagrange Interpolation . 367 4.2.4 Iterative Linear Interpolation . 371 4.2.5 Fast Algorithms for Vandermonde Systems . 373 4.2.6 The Runge Phenomenon . 377 Review Questions . 380 Problems and Computer Exercises . 380 4.3 Generalizations and Applications . 381 4.3.1 Hermite Interpolation . 381 4.3.2 Complex Analysis in Polynomial Interpolation . 385 4.3.3 Rational Interpolation . 389 4.3.4 Multidimensional Interpolation . 395 4.3.5 Analysis of a Generalized Runge Phenomenon . 398 Review Questions . 407 Problems and Computer Exercises . 407 4.4 Piecewise Polynomial Interpolation . 410 4.4.1 Bernstein Polynomials and Bézier Curves . 411 4.4.2 Spline Functions . 417 4.4.3 The B-Spline Basis . 426 4.4.4 Least Squares Splines Approximation . 434 Review Questions . 436 Problems and Computer Exercises . 437 4.5 Approximation and Function Spaces . 439 4.5.1 Distance and Norm . 440 4.5.2 Operator Norms and the Distance Formula . 444 4.5.3 Inner Product Spaces and Orthogonal Systems . 450 Contents x¡ 4.5.4 Solution of the Approximation Problem. 454 4.5.5 Mathematical Properties of Orthogonal Polynomials . 457 4.5.6 Expansions in Orthogonal Polynomials . 466 4.5.7 Approximation in the Maximum Norm . 471 Review Questions . 478 Problems and Computer Exercises . 479 4.6 Fourier Methods . 482 4.6.1 Basic Formulas and Theorems . 483 4.6.2 Discrete Fourier Analysis . 487 4.6.3 Periodic Continuation of a Function . 491 4.6.4 Convergence Acceleration of Fourier Series . 492 4.6.5 The Fourier Integral Theorem . 494 4.6.6 Sampled Data and Aliasing . 497 Review Questions . 500 Problems and Computer Exercises . 500 4.7 The Fast Fourier Transform . 503 4.7.1 The FFT Algorithm . 503 4.7.2 Discrete Convolution by FFT . 509 4.7.3 FFTs of Real Data . 510 4.7.4 Fast Trigonometric Transforms . 512 4.7.5 The General Case FFT . 515 Review Questions . 516 Problems and Computer Exercises . 517 Notes and References . 518 Numerical Integration 521 5.1 Interpolatory Quadrature Rules . 521 5.1.1 Introduction . 521 5.1.2 Treating Singularities . 525 5.1.3 Some Classical Formulas . 527 5.1.4 Supercon vergence of the Trapezoidal Rule . 531 5.1.5 Higher-Order Newton-Cotes'Formulas . 533 5.1.6 Fejér and Clensnaw-Curtis Rules . 538 Review Questions . 542 Problems and Computer Exercises . 542 5.2 Integration by Extrapolation . 546 5.2.1 The Euler-Maclaurin Formula . 546 5.2.2 Romberg's Method . 548 5.2.3 Oscillating Integrands . 554 5.2.4 Adaptive Quadrature . 560 Review Questions . 564 Problems and Computer Exercises . 564 5.3 Quadrature Rules with Free Nodes . 565 5.3.1 Method of Undetermined Coefficients . 565 5.3.2 Gauss-Christoffel Quadrature Rules . 568 xii Contents 5.3.3 Gauss Quadrature with Preassigned Nodes . 573 5.3.4 Matrices, Moments, and Gauss Quadrature . 576 5.3.5 Jacobi Matrices and Gauss Quadrature . 580 Review Questions . 585 Problems and Computer Exercises . 585 5.4 Multidimensional Integration . 587 5.4.1 Analytic Techniques . 588 5.4.2 Repeated One-Dimensional Integration . 589 5.4.3 ProductRules . 590 5.4.4 Irregular Triangular Grids . 594 5.4.5 Monte Carlo Methods . 599 5.4.6 Quasi-Monte Carlo and Lattice Methods . 601 Review Questions . 604 Problems and Computer Exercises . 605 Notes and References . 606 Solving Scalar Nonlinear Equations 609 6.1 Some Basic Concepts and Methods . 609 6.1.1 Introduction . 609 6.1.2 The Bisection Method . 610 6.1.3 Limiting Accuracy and Termination Criteria . 614 6.1.4 Fixed-Point Iteration . 618 6.1.5 Convergence Order and Efficiency . 621 Review Questions . 624 Problems and Computer Exercises . 624 6.2 Methods Based on Interpolation . 626 6.2.1 Method of False Position . 626 6.2.2 The Secant Method . 628 6.2.3 Higher-Order Interpolation Methods . 631 6.2.4 A Robust Hybrid Method . 634 Review Questions . 635 Problems and Computer Exercises . 636 6.3 Methods Using Derivatives . 637 6.3.1 Newton's Method . 637 6.3.2 Newton's Method for Complex Roots . 644 6.3.3 An Interval Newton Method . 646 6.3.4 Higher-Order Methods . 647 Review Questions . 652 Problems and Computer Exercises . 653 6.4 Finding a Minimum of a Function . 656 6.4.1 Introduction . 656 6.4.2 Unimodal Functions and Golden Section Search . 657 6.4.3 Minimization by Interpolation . 660 Review Questions . 661 Problems and Computer Exercises . 661 Contents x¡¡¡ 6.5 Algebraic Equations . 662 6.5.1 Some Elementary Results . 662 6.5.2 Ill-Conditioned Algebraic Equations . 665 6.5.3 Three Classical Methods . 668 6.5.4 Deflation and Simultaneous Determination of Roots . 671 6.5.5 A Modified Newton Method . 675 6.5.6 Sturm Sequences . 677 6.5.7 Finding Greatest Common Divisors . 680 Review Questions . 682 Problems and Computer Exercises . 683 Notes and References . 685 Bibliography 687 Index 707 A Online Appendix: Introduction to Matrix Computations A-l A.I Vectors and Matrices . A-l A. 1.1 Linear Vector Spaces . A-l A. 1.2 Matrix and Vector Algebra . A-3 A.1.3 Rank and Linear Systems . A-5 A. 1.4 Special Matrices . A-6 A.2 Submatrices and Block Matrices . A-8 A.2.1 Block Gaussian Elimination . A-10 A.3 Permutations and Determinants . A-12 A.4 Eigenvalues and Norms of Matrices . A-16 A.4.1 The Characteristic Equation . A-16 A.4.2 The Schur and Jordan Normal Forms . A-17 A.4.3 Norms of Vectors and Matrices . A-18 Review Questions . A-21 Problems . A-22 В Online Appendix: A MATLAB Multiple Precision Package B-l B.I The Mulprec Package . B-l B.I.I Number Representation . B-l B.1.2 The Mulprec Function Library . B-3 B.1.3 Basic Arithmetic Operations . B-3 B.1.4 Special Mulprec Operations . B-4 B.2 Function and Vector Algorithms . B-4 B.2.1 Elementary Functions . B-4 B.2.2 Mulprec Vector Algorithms . B-5 B.2.3 Miscellaneous . B-6 B.2.4 Using Mulprec . B-6 Computer Exercises . B-6 xiv Contents С Online Appendix: Guide to Literature C-I Cl Introduction . C-l C.2 Textbooks in Numerical Analysis . С -l С.З Handbooks and Collections . C-5 C.4 Encyclopedias, Tables, and Formulas . C-6 C.5 Selected Journals . C-8 C.6 Algorithms and Software . C-9 C.7 Public Domain Software . C-10
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spellingShingle	Dahlquist, Germund Björck, Åke 1934- Numerical methods in scientific computing Numerische Mathematik (DE-588)4042805-9 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd
subject_GND	(DE-588)4042805-9 (DE-588)4338507-2
title	Numerical methods in scientific computing
title_auth	Numerical methods in scientific computing
title_exact_search	Numerical methods in scientific computing
title_exact_search_txtP	Numerical methods in scientific computing
title_full	Numerical methods in scientific computing 1 Germund Dahlquist ; Åke Björck
title_fullStr	Numerical methods in scientific computing 1 Germund Dahlquist ; Åke Björck
title_full_unstemmed	Numerical methods in scientific computing 1 Germund Dahlquist ; Åke Björck
title_short	Numerical methods in scientific computing
title_sort	numerical methods in scientific computing
topic	Numerische Mathematik (DE-588)4042805-9 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd
topic_facet	Numerische Mathematik Wissenschaftliches Rechnen
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