Numerical Methods for Engineers, Second Edition

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ISBN: 9781584884019 | 1584884010
Cover: Hardcover
Copyright: 6/22/2006

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Although pseudocodes, MathematicaÂ®, and MATLABÂ® illustrate how algorithms work, designers of engineering systems write the vast majority of large computer programs in the Fortran language. Using Fortran 95 to solve a range of practical engineering problems, Numerical Methods for Engineers, Second Edition provides an introduction to numerical methods, incorporating theory with concrete computing exercises and programmed examples of the techniques presented. Covering a wide range of numerical applications that have immediate relevancy for engineers, the book describes forty-nine programs in Fortran 95. Many of the programs discussed use a sub-program library called nm_lib that holds twenty-three subroutines and functions. In addition, there is a precision module that controls the precision of calculations. Well-respected in their field, the authors discuss a variety of numerical topics related to engineering. Some of the chapter features include'¦ The numerical solution of sets of linear algebraic equations Roots of single nonlinear equations and sets of nonlinear equations Numerical quadrature, or numerical evaluation of integrals An introduction to the solution of partial differential equations using finite difference and finite element approaches Describing concise programs that are constructed using sub-programs wherever possible, this book presents many different contexts of numerical analysis, forming an excellent introduction to more comprehensive subroutine libraries such as the numerical algorithm group (NAG).

1 Introduction and Programming Preliminaries

(14)

1.1 Introduction

(1)

1.2 Running programs

(1)

1.3 Hardware

(1)

1.4 External Fortran subprogram libraries

(2)

1.5 A simple Fortran program

(3)

1.6 Some simple Fortran constructs

(1)

1.7 Intrinsic functions

(1)

1.8 User-supplied functions and subroutines

(1)

1.9 Errors and accuracy

(3)

1.9.1 Roundoff

(1)

1.9.2 Truncation

(1)

1.9.3 Cancellation

(1)

1.9.4 Intrinsic and library-supplied precision routines

(1)

1.10 Graphical output

(1)

1.11 Conclusions

(2)

2 Linear Algebraic Equations

(74)

2.1 Introduction

(1)

2.2 Gaussian elimination

(5)

2.2.1 Observations on the elimination process

(1)

2.3 Equation solution using factorization

(5)

2.3.1 Observations on the solution process by factorization

(1)

2.4 Equations with a symmetrical coefficient matrix

(8)

2.4.1 Quadratic form and positive definiteness

(1)

2.4.2 Cholesky's method

(2)

2.5 Banded equations

(2)

2.6 Compact storage for variable bandwidths

(3)

2.7 Pivoting

(4)

2.7.1 Ill-conditioning

(1)

2.8 Equations with prescribed solutions

(4)

2.9 Iterative methods

(15)

2.9.1 The iterative process

(6)

2.9.2 Very sparse systems

(1)

2.9.3 The Gauss-Seidel method

(5)

2.9.4 Successive overrelaxation

(4)

2.10 Gradient methods

(7)

2.10.1 The method of 'steepest descent'

(3)

2.10.2 The method of 'conjugate gradients'

(4)

2.10.3 Convergence of iterative methods

(1)

2.11 Unsymmetrical systems

(4)

2.12 Preconditioning

(9)

2.13 Comparison of direct and iterative methods

(1)

2.14 Exercises

(7)

3 Nonlinear Equations

(42)

3.1 Introduction

(2)

3.2 Iterative substitution

(3)

3.3 Multiple roots and other difficulties

(3)

3.4 Interpolation methods

(6)

3.4.1 Bisection method

(3)

3.4.2 False position method

100

(3)

3.5 Extrapolation methods

103

(9)

3.5.1 Newton-Raphson method

104

(3)

3.5.2 A modified Newton-Raphson method

107

(5)

3.6 Acceleration of convergence

112

(1)

3.7 Systems of nonlinear equations

112

(13)

3.7.1 Iterative substitution for systems

113

(3)

3.7.2 Newton-Raphson for systems

116

(5)

3.7.3 Modified Newton-Raphson method for systems

121

(4)

3.8 Exercises

125

(6)

4 Eigenvalue Equations

131

(62)

4.1 Introduction

131

(5)

4.1.1 Orthogonality and normalization of eigenvectors

132

(2)

4.1.2 Properties of eigenvalues and eigenvectors

134

(2)

4.1.3 Solution methods for eigenvalue equations

136

(1)

4.2 Vector iteration

136

(12)

4.2.1 Shifted vector iteration

140

(3)

4.2.2 Shifted inverse iteration

143

(5)

4.3 Intermediate eigenvalues by deflation

148

(2)

4.4 The generalized eigenvalue problem [K]{x} = λ[M] {x}

150

(8)

4.4.1 Conversion of generalized problem to symmetrical standard form

154

(4)

4.5 Transformation methods

158

(22)

4.5.1 Comments on Jacobi diagonalization

167

(1)

4.5.2 Householder's transformation to tridiagonal form

167

(4)

4.5.3 Lanczos transformation to tridiagonal form

171

(5)

4.5.4 LR transformation for eigenvalues of tridiagonal matrices

176

(4)

4.6 Characteristic polynomial methods

180

(8)

4.6.1 Evaluating determinants of tridiagonal matrices

180

(1)

4.6.2 The Sturm sequence property

181

(6)

4.6.3 General symmetrical matrices, e.g., band matrices

187

(1)

4.7 Exercises

188

(5)

5 Interpolation and Curve Fitting

193

(52)

5.1 Introduction

193

(1)

5.2 Interpolating polynomials

193

(14)

5.2.1 Lagrangian polynomials

194

(4)

5.2.2 Difference methods

198

(1)

5.2.3 Difference methods with equal intervals

199

(8)

5.3 Interpolation using cubic spline functions

207

(7)

5.4 Numerical differentiation

214

(12)

5.4.1 Interpolating polynomial method

215

(4)

5.4.2 Taylor series method

219

(7)

5.5 Curve fitting

226

(11)

5.5.1 Least squares

226

(3)

5.5.2 Linearization of data

229

(8)

5.6 Exercises

237

(8)

6 Numerical Integration

245

(72)

6.1 Introduction

245

(2)

6.2 Newton-Cotes rules

247

(18)

6.2.1 Introduction

247

(1)

6.2.2 Rectangle rule, (n = 1)

247

(1)

6.2.3 Trapezoid rule, (n = 2)

248

(2)

6.2.4 Simpson's rule, (n = 3)

250

(2)

6.2.5 Higher order Newton-Cotes rules (n > 3)

252

(1)

6.2.6 Accuracy of Newton-Cotes rules

253

(1)

6.2.7 Summary of Newton-Cotes rules

254

(1)

6.2.8 Repeated Newton-Cotes rules

255

(9)

6.2.9 Remarks on Newton-Cotes rules

264

(1)

6.3 Gauss-Legendre rules

265

(13)

6.3.1 Introduction

265

(1)

6.3.2 Midpoint rule, (n = 1)

265

(2)

6.3.3 Two-point Gauss-Legendre rule, (n = 2)

267

(3)

6.3.4 Three-point Gauss-Legendre rule, (n = 3)

270

(1)

6.3.5 Changing the limits of integration

271

(6)

6.3.6 Accuracy of Gauss-Legendre rules

277

(1)

6.4 Adaptive integration rules

278

(6)

6.5 Special integration rules

284

(8)

6.5.1 Gauss-Chebyshev rules

288

(1)

6.5.2 Fixed weighting coefficients

289

(1)

6.5.3 Hybrid rules

290

(1)

6.5.4 Sampling points outside the range of integration

290

(2)

6.6 Multiple integrals

292

(15)

6.6.1 Introduction

292

(7)

6.6.2 Integration over a general quadrilateral area

299

(8)

6.7 Exercises

307

(10)

7 Numerical Solution of Ordinary Differential Equations

317

(76)

7.1 Introduction

317

(1)

7.2 Definitions and types of ODE

317

(2)

7.3 Initial value problems

319

(43)

7.3.1 One-step methods

321

(9)

7.3.2 Reduction of high order equations

330

(2)

7.3.3 Solution of simultaneous first order equations

332

(11)

7.3.4 theta-methods for linear equations

343

(6)

7.3.5 Predictor-corrector methods

349

(10)

7.3.6 Stiff equations

359

(1)

7.3.7 Error propagation and numerical stability

360

(1)

7.3.8 Concluding remarks on initial value problems

361

(1)

7.4 Boundary value problems

362

(24)

7.4.1 Finite difference methods

362

(6)

7.4.2 Shooting methods

368

(8)

7.4.3 Weighted residual methods

376

(10)

7.5 Exercises

386

(7)

8 Introduction to Partial Differential Equations

393

(52)

8.1 Introduction

393

(1)

8.2 Definitions and types of PDE

393

(1)

8.3 First order equations

394

(5)

8.4 Second order equations

399

(2)

8.5 Finite difference method

401

(29)

8.5.1 Elliptic systems

404

(13)

8.5.2 Parabolic systems

417

(10)

8.5.3 Hyperbolic systems

427

(3)

8.6 Finite element method

430

(4)

8.7 Exercises

434

(11)

A Descriptions of Library Subprograms

445

(2)

B Fortran 95 Listings of Library Subprograms

447

(22)

C References and Additional Reading

469

(6)

Index

475

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Numerical Methods for Engineers, Second Edition

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