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Numerical Mathematics and Computing,9780534201128

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Numerical Mathematics and Computing

by E. Ward Cheney; David R. Kincaid; Ward Cheney

Edition:

3rd

ISBN13:

9780534201128

ISBN10:

0534201121

Format:

Paperback

Pub. Date:

1/13/1994

Publisher(s):

Brooks Cole

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Table of Contents

1 Introduction

1

(38)

Preliminary Remarks

2

(2)

Nested Multiplication

2

(1)

Rounding and Chopping

3

(1)

1.1 Programming Suggestions

4

(14)

Programming and Coding Advice

4

(2)

Case Studies

6

(4)

Programming Experiment

10

(8)

1.2 Review of Taylor Series

18

(21)

Taylor Series

18

(3)

Taylor's Theorem in Terms of (x - c)

21

(2)

Mean-Value Theorem

23

(1)

Taylor's Theorem in Terms of h

24

(2)

Alternating Series

26

(13)

2 Number Representation and Errors

39

(43)

2.1 Representation of Numbers in Different Bases

39

(10)

Base Beta Numbers

40

(1)

Conversion of Integer Part

41

(2)

Conversion of Fractional Part

43

(2)

Base Conversion 10 (XXX) 8 (XXX) 2

45

(2)

Base 16

47

(1)

More Examples

47

(2)

2.2 Floating-Point Representation

49

(16)

Normalized Floating-Point Representation

50

(2)

Hypothetical Marc-32 Computer

52

(3)

Computer Errors in Representing Numbers

55

(2)

Notation fl(x) and Inverse Error Analysis

57

(8)

2.3 Loss of Significance

65

(17)

Significant Digits

65

(1)

Computer-Caused Loss of Significance

66

(2)

Theorem on Loss of Precision

68

(1)

Avoiding Loss of Significance in Subtraction

69

(3)

Range Reduction

72

(10)

3 Locating Roots of Equations

82

(37)

3.1 Bisection Method

84

(9)

Bisection Algorithm and Pseudocode

84

(2)

Use of Pseudocode

86

(2)

Convergence Analysis

88

(5)

3.2 Newton's Method

93

(16)

Interpretations of Newton's Method

93

(2)

Pseudocode

95

(1)

Illustration

95

(2)

Convergence Analysis

97

(12)

3.3 Secant Method

109

(10)

Interpretations of Secant Method

109

(1)

Secant Algorithm

110

(1)

Convergence Analysis

111

(3)

Summary

114

(5)

4 Interpolation and Numerical Differentiation

119

(46)

4.1 Polynomial Interpolation

120

(22)

Existence of Interpolating Polynomial

120

(2)

Newton Interpolating Polynomial

122

(1)

Nested Form

123

(2)

Calculating Coefficients a(i) Using Divided Differences

125

(5)

Algorithms and Pseudocode

130

(3)

Inverse Interpolation

133

(1)

Lagrange Interpolating Polynomial

134

(8)

4.2 Errors in Polynomial Interpolation

142

(9)

Examples

143

(1)

Theorems on Interpolation Errors

144

(7)

4.3 Estimating Derivatives and Richardson Extrapolation

151

(14)

First-Derivative Formulas via Taylor Series

152

(2)

Richardson Extrapolation

154

(3)

First-Derivative Formulas via Interpolation Polynomials

157

(3)

Second-Derivative Formulas via Taylor Series

160

(5)

5 Numerical Integration

165

(52)

5.1 Definite Integral

165

(9)

Definite and Indefinite Integrals

165

(2)

Lower and Upper Sums

167

(2)

Riemann-Integrable Functions

169

(1)

Examples and Pseudocode

170

(4)

5.2 Trapezoid Rule

174

(14)

Description

174

(2)

Uniform Spacing

176

(1)

Error Analysis

177

(3)

Applying the Error Formula

180

(2)

Recursive Trapezoid Formula for 2^n Equal Subintervals

182

(6)

5.3 Romberg Algorithm

188

(11)

Description

188

(1)

Pseudocode

189

(1)

Derivation

190

(3)

General Extrapolation

193

(6)

5.4 An Adaptive Simpson's Scheme

199

(7)

Simpson's Rule

199

(2)

Adaptive Simpson's Algorithm

201

(2)

Pseudocode

203

(1)

Example Using Adaptive Simpson Procedure

204

(2)

5.5 Gaussian Quadrature Formulas

206

(11)

Description

206

(2)

Gaussian Nodes and Weights

208

(2)

Legendre Polynomials

210

(7)

6 Systems of Linear Equations

217

(61)

6.1 Naive Gaussian Elimination

218

(12)

Numerical Example

218

(2)

Algorithm

220

(2)

Pseudocode

222

(3)

Testing the Pseudocode

225

(1)

Residual and Error Vectors

226

(4)

6.2 Gaussian Elimination with Scaled Partial Pivoting

230

(19)

Examples Where Naive Gaussian Elimination Fails

230

(2)

Gaussian Elimination with Scaled Partial Pivoting

232

(2)

Numerical Example

234

(2)

Pseudocode

236

(3)

Long Operation Count

239

(2)

Examples of Software Packages

241

(8)

6.3 Tridiagonal and Banded Systems

249

(9)

Tridiagonal Systems

250

(2)

Diagonal Dominance

252

(1)

Pentadiagonal Systems

252

(6)

6.4 LU Factorization

258

(20)

Numerical Example

259

(3)

Formal Derivation

262

(4)

Solving Linear Systems Using LU Factorization

266

(2)

Computing A^(-1)

268

(1)

Example of a Software Package

268

(10)

7 Approximation by Spline Functions

278

(47)

7.1 First-Degree and Second-Degree Splines

278

(10)

First-Degree Spline

278

(3)

Second-Degree Splines

281

(1)

Quadratic Spline Q(x)

282

(1)

Subbotin Quadratic Spline

283

(5)

7.2 Natural Cubic Splines

288

(20)

Introduction

288

(2)

Natural Cubic Spline

290

(1)

Algorithm for Natural Cubic Spline

291

(5)

Pseudocode for Natural Cubic Splines

296

(2)

Using Pseudocode for Interpolating and Curve Fitting

298

(3)

Smoothness Property

301

(7)

7.3 B Splines

308

(8)

7.4 Interpolation and Approximation by B Splines

316

(9)

Pseudocode and a Curve-Fitting Example

318

(2)

Schoenberg's Process

320

(1)

Pseudocode

321

(4)

8 Ordinary Differential Equations

325

(34)

Initial-Value Problem: Analytical vs. Numerical Solution

325

(2)

8.1 Taylor Series Methods

327

(8)

Euler's Method Pseudocode

328

(1)

Taylor Series Method of Higher Order

329

(2)

Types of Errors

331

(4)

8.2 Runge-Kutta Methods

335

(12)

Taylor Series for f(x, y)

336

(1)

Runge-Kutta Method of Order 2

337

(2)

Pseudocode

339

(8)

8.3 Stability and Adaptive Runge-Kutta Methods

347

(12)

Stability Analysis

347

(2)

An Adaptive Runge-Kutta-Fehlberg Method

349

(4)

Solving Differential Equations and Integration

353

(6)

9 Systems of Ordinary Differential Equations

359

(22)

9.1 Methods for First-Order Systems

359

(10)

Uncoupled and Coupled Systems

360

(1)

Taylor Series Method

361

(1)

Simplification

362

(2)

Taylor Series Method: Vector Notation

364

(1)

Runge-Kutta Method

365

(4)

9.2 Higher-Order Equations and Systems

369

(6)

Higher-Order Differential Equations

370

(1)

Systems of Differential Equations

371

(4)

9.3 Adams-Moulton Methods

375

(6)

A Predictor-Corrector Scheme

375

(1)

Pseudocode

376

(4)

An Adaptive Scheme

380

(1)

10 Smoothing of Data and the Method of Least Squares

381

(29)

10.1 The Method of Least Squares

381

(8)

Linear Least Squares

381

(3)

Linear Example

384

(1)

Nonpolynomial Example

384

(1)

Basis Functions (g0, g1,...,gn)

385

(4)

10.2 Orthogonal Systems and Chebyshev Polynomials

389

(13)

Orthonormal Basis Functions (g0, g1,...,gn)

389

(3)

Outline of Algorithm

392

(2)

Smoothing Data: Polynomial Regression

394

(8)

10.3 Other Examples of the Least Squares Principle

402

(8)

Weight Function w(x)

403

(2)

Nonlinear Example

405

(1)

Linear/Nonlinear Example

406

(4)

11 Monte Carlo Methods and Simulation

410

(30)

11.1 Random Numbers

411

(11)

Random-Number Algorithms/Generators

411

(3)

Examples

414

(2)

Uses of Pseudocode random

416

(6)

11.2 Estimation of Areas and Volumes by Monte Carlo Techniques

422

(8)

Numerical Integration

422

(1)

Example and Pseudocode

423

(2)

Computing Volumes

425

(1)

Ice Cream Cone Example

426

(4)

11.3 Simulation

430

(10)

Loaded Die Problem

430

(1)

Birthday Problem

431

(1)

Buffon's Needle Problem

432

(1)

Two Dice Problem

433

(1)

Neutron Shielding

434

(6)

12 Boundary Value Problems for Ordinary Differential Equations

440

(18)

12.1 Shooting Method

441

(6)

Shooting Method Algorithm

442

(2)

Modifications and Refinements

444

(3)

12.2 A Discretization Method

447

(11)

Finite Difference Approximations

447

(1)

The Linear Case

448

(1)

Pseudocode and Numerical Example

449

(2)

Shooting Method in Linear Case

451

(1)

Pseudocode and Numerical Example

452

(6)

13 Partial Differential Equations

458

(30)

Some Partial Differential Equations from Applied Problems

458

(2)

13.1 Parabolic Problems

460

(8)

Heat Equation Model Problem

460

(1)

Finite-Difference Method

460

(2)

Pseudocode for Explicit Method

462

(1)

Crank-Nicolson Method

463

(1)

Pseudocode for Crank-Nicolson Method

464

(1)

Alternative Version of Crank-Nicolson Method

465

(3)

13.2 Hyperbolic Problems

468

(7)

Wave Equation Model Problem

468

(1)

Analytic Solution

469

(2)

Numerical Solution

471

(1)

Pseudocode

472

(3)

13.3 Elliptic Problems

475

(13)

Helmholtz Equation Model Problem

475

(1)

Finite-Difference Method

476

(4)

Gauss-Seidel Iterative Method

480

(1)

Numerical Example and Computer Program

481

(7)

14 Minimization of Multivariate Functions

488

(28)

Unconstrained and Constrained Minimization Problems

489

(1)

14.1 One-Variable Case

489

(13)

Unimodal F

490

(1)

The Fibonacci Search Algorithm

491

(3)

The Golden Section Search Algorithm

494

(2)

Taylor Series Algorithm

496

(6)

14.2 Multivariate Case

502

(14)

Taylor Series for F: Gradient Vector and Hessian Matrix

502

(2)

Alternative Form of Taylor Series

504

(2)

Steepest Descent Procedure

506

(1)

Contour Diagrams

507

(1)

More Advanced Algorithms

507

(2)

Minimum, Maximum, and Saddle Points

509

(1)

Positive Definite Matrix

510

(6)

15 Linear Programming

516

(25)

15.1 Standard Forms and Duality

516

(12)

First Primal Form

516

(1)

Numerical Example

517

(2)

Transforming Problems into First Primal Form

519

(1)

Dual Problem

520

(2)

Second Primal Form

522

(6)

15.2 Simplex Method

528

(5)

Vertices in K and Linearly Independent Columns of A

529

(2)

Simplex Method

531

(2)

15.3 Approximate Solution of Inconsistent Linear Systems

533

(8)

L(1) Problem

533

(3)

L(Infinity) Problem

536

(5)

APPENDIX A Linear Algebra Concepts and Notation

541

Answers for Selected Problems

553

(4)

Bibliography

567

(6)

Index

573

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