Numerical analysis

867 Pages·2005·44.6 MB·English

by  Richard L Burden

Index of Algorithms
Bisection2.1 47 LU Factorization6.4 392
Fixed-PointIteration2.2 57 LDLt Factorization6.5 404
Newton’s2.3 64 Choleski’s6.6 404
Secant2.4 68 CroutFactorizationforTridiagonalLinearSystems
FalsePosition2.5 70 6.7 408
Steffensen’s2.6 85 JacobiIterative7.1 439
Horner’s2.7 90 Gauss-SeidelIterative7.2 441
Mu¨ller’s2.8 93 SOR7.3 449
Neville’sIteratedInterpolation3.1 114 IterativeRefinement7.4 459
Newton’sInterpolatoryDivided-Difference3.2 121 PreconditionedConjugateGradient7.5 473
HermiteInterpolation3.3 135 Pade´ RationalApproximation8.1 515
NaturalCubicSpline3.4 142 ChebyshevRationalApproximation8.2 520
ClampedCubicSpline3.5 147 FastFourierTransform8.3 539
Be´zierCurve3.6 162 Power9.1 560
CompositeSimpson’sRule4.1 199 SymmetricPower9.2 562
Romberg4.2 209 InversePower9.3 566
AdaptiveQuadrature4.3 216 WielandtDeflation9.4 570
Simpson’sDoubleIntegral4.4 233 Householder’s9.5 579
GaussianDoubleIntegral4.5 234 QR9.6 588
GaussianTripleIntegral4.6 236 Newton’sforSystems10.1 610
Euler’s5.1 257 Broyden10.2 619
Runge-Kutta(OrderFour)5.2 278 SteepestDescent10.3 628
Runge-Kutta-Fehlberg5.3 287 Continuation10.4 637
AdamsFourth-OrderPredictor-Corrector5.4 299 LinearShooting11.1 644
AdamsVariableStep-SizePredictor-Corrector5.5 NonlinearShootingwithNewton’s
304 Method11.2 652
Extrapolation5.6 310 LinearFinite-Difference11.3 658
Runge-KuttaforSystemsofDifferentialEquations NonlinearFinite-Difference11.4 664
5.7 316 PiecewiseLinearRayleigh-Ritz11.5 674
TrapezoidalwithNewtonIteration5.8 339 CubicSplineRayleigh-Ritz11.6 680
GaussianEliminationwithBackwardSubstitution6.1 PoissonEquationFinite-Difference12.1 694
352 HeatEquationBackward-Difference12.2 705
GaussianEliminationwithPartialPivoting6.2 362 Crank-Nicolson12.3 708
GaussianEliminationwithScaledPartialPivoting6.3 WaveEquationFinite-Difference12.4 716
364 Finite-Element12.5 728
Glossary of Notation
C(X) Setofallfunctionscontinuouson X 3
Cn(X) Setofallfunctionshavingncontinuousderivativeson X 4
C∞(X) Setofallfunctionshavingderivativesofallorderson X 4
R Setofrealnumbers 11
0.3¯ Adecimalinwhichthenumeral3repeatsindefinitely 11
fl(y) Floating-pointformoftherealnumber y 19
O(·) Orderofconvergence 35
(cid:19) (cid:20) Floorfunction,(cid:19)x(cid:20),thegreatestintegerlessthanorequaltox 42
(cid:21) (cid:22) Ceilingfunction,(cid:21)x(cid:22),thesmallestintegergreaterthanorequaltox 42
sgn(x) Signofthenumberx:1ifx >0,−1ifx <0 50
(cid:4) Forwarddifference 84
(cid:10)z¯ (cid:11) Complexconjugateofthecomplexnumberz 91
n Thekthbinomialcoefficientofordern 118
k
f[·] Divideddifferenceofthefunction f 119
∇ Backwarddifference 123
Rn Setoforderedn-tuplesofrealnumbers 251
τ Localtruncationerrorattheithstep 266
i
→ Equationreplacement 345
↔ Equationinterchange 345
(a ) Matrixwitha astheentryintheithrowand jthcolumn 347
ij ij
x ColumnvectororelementofRn 347
[A,b] Augmentedmatrix 348
O Amatrixwithallzeroentries 371
δ Kroneckerdelta:1ifi = j,0ifi (cid:13)= j 372
ij
I n×nidentitymatrix 372
n
A−1 Inversematrixofthematrix A 374
At Transposematrixofthematrix A 378
M Minorofamatrix 383
ij
detA Determinantofthematrix A 384
0 Vectorwithallzeroentries 386
||x|| Arbitrarynormofthevectorx 418
||x|| Thel normofthevectorx 419
2 2
||x||∞ Thel∞ normofthevectorx 419
||A|| Arbitrarynormofthematrix A 424
||A|| Thel normofthematrix A 425
2 2
||A||∞ Thel∞ normofthematrix A 425
ρ(A) Thespectralradiusofthematrix A 433
K(A) Theconditionnumberofthematrix A 455
"x,y# Innerproductofthen-dimensionalvectorsxandy 464
(cid:22) Setofallpolynomialsofdegreenorless 498
n
(cid:22)5 Setofallmonicpolynomialsofdegreen 507
n
T Setofalltrigonometricpolynomialsofdegreenorless 524
n
C Setofcomplexnumbers 554
F FunctionmappingRn intoRn 599
A(x) MatrixwhoseentriesarefunctionsformRn intoR 608
J(x) Jacobianmatrix 610
∇g Gradientofthefunctiong 625
Numerical Analysis
EIGHTH EDITION
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Numerical Analysis
EIGHTH EDITION
Richard L. Burden
YoungstownStateUniversity
J. Douglas Faires
YoungstownStateUniversity
Australia • Canada • Mexico • Singapore • Spain
UnitedKingdom • UnitedStates
NumericalAnalysis,eighthedition
RichardL.Burden,J.DouglasFaires
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Contents
1 Mathematical Preliminaries and Error Analysis 1
1.1 ReviewofCalculus 2
1.2 Round-offErrorsandComputerArithmetic 17
1.3 AlgorithmsandConvergence 30
1.4 NumericalSoftware 38
2 Solutions of Equations in One Variable 45
2.1 TheBisectionMethod 46
2.2 Fixed-PointIteration 53
2.3 Newton’sMethod 63
2.4 ErrorAnalysisforIterativeMethods 75
2.5 AcceleratingConvergence 83
2.6 ZerosofPolynomialsandMu¨ller’sMethod 87
2.7 SurveyofMethodsandSoftware 97
3 Interpolation and Polynomial Approximation 101
3.1 InterpolationandtheLagrangePolynomial 104
3.2 DividedDifferences 119
3.3 HermiteInterpolation 130
3.4 CubicSplineInterpolation 137
3.5 ParametricCurves 157
3.6 SurveyofMethodsandSoftware 164
4 Numerical Differentiation and Integration 167
4.1 NumericalDifferentiation 168
4.2 Richardson’sExtrapolation 179
4.3 ElementsofNumericalIntegration 187
4.4 CompositeNumericalIntegration 196
v
vi Contents
4.5 RombergIntegration 207
4.6 AdaptiveQuadratureMethods 212
4.7 GaussianQuadrature 220
4.8 MultipleIntegrals 226
4.9 ImproperIntegrals 241
4.10 SurveyofMethodsandSoftware 246
5 Initial-Value Problems for Ordinary Differential
Equations 249
5.1 TheElementaryTheoryofInitial-ValueProblems 250
5.2 Euler’sMethod 256
5.3 Higher-OrderTaylorMethods 266
5.4 Runge–KuttaMethods 273
5.5 ErrorControlandtheTheRunge–Kutta–FehlbergMethod 283
5.6 MultistepMethods 291
5.7 VariableStep-SizeMultistepMethods 302
5.8 ExtrapolationMethods 308
5.9 Higher-OrderEquationsandSystemsofDifferentialEquations 313
5.10 Stability 325
5.11 StiffDifferentialEquations 335
5.12 SurveyofMethodsandSoftware 342
6 Direct Methods for Solving Linear Systems 345
6.1 LinearSystemsofEquations 346
6.2 PivotingStrategies 360
6.3 LinearAlgebraandMatrixInversion 370
6.4 TheDeterminantofaMatrix 383
6.5 MatrixFactorization 388
6.6 SpecialTypesofMatrices 398
6.7 SurveyofMethodsandSoftware 413
7 Iterative Techniques in Matrix Algebra 417
7.1 NormsofVectorsandMatrices 418
7.2 EigenvaluesandEigenvectors 429
7.3 IterativeTechniquesforSolvingLinearSystems 436
7.4 ErrorBoundsandIterativeRefinement 454
7.5 TheConjugateGradientMethod 464
7.6 SurveyofMethodsandSoftware 479
Contents vii
8 Approximation Theory 481
8.1 DiscreteLeastSquaresApproximation 482
8.2 OrthogonalPolynomialsandLeastSquaresApproximation 494
8.3 ChebyshevPolynomialsandEconomizationofPowerSeries 503
8.4 RationalFunctionApproximation 512
8.5 TrigonometricPolynomialApproximation 523
8.6 FastFourierTransforms 532
8.7 SurveyofMethodsandSoftware 544
9 Approximating Eigenvalues 547
9.1 LinearAlgebraandEigenvalues 548
9.2 ThePowerMethod 557
9.3 Householder’sMethod 574
9.4 TheQRAlgorithm 582
9.5 SurveyofMethodsandSoftware 594
10 Numerical Solutions of Nonlinear Systems
of Equations 597
10.1 FixedPointsforFunctionsofSeveralVariables 598
10.2 Newton’sMethod 607
10.3 Quasi-NewtonMethods 617
10.4 SteepestDescentTechniques 624
10.5 HomotopyandContinuationMethods 631
10.6 SurveyofMethodsandSoftware 639
11 Boundary-Value Problems for Ordinary
Differential Equations 641
11.1 TheLinearShootingMethod 642
11.2 TheShootingMethodforNonlinearProblems 649
11.3 Finite-DifferenceMethodsforLinearProblems 656
11.4 Finite-DifferenceMethodsforNonlinearProblems 662
11.5 TheRayleigh–RitzMethod 668
11.6 SurveyofMethodsandSoftware 684