Numerical Methods for Engineers, Second Edition

by Griffiths; D. Vaughan

• ISBN13: 9781584884019
• ISBN10: 1584884010
• Edition: 2nd
• Format: Hardcover
• Copyright: 2006-06-22
• Publisher: Chapman & Hall/

Summary

Although pseudocodes, Mathematica®, and MATLAB® 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).

Table of Contents

1 Introduction and Programming Preliminaries	1	(14)
1.1 Introduction	1	(1)
1.2 Running programs	1	(1)
1.3 Hardware	2	(1)
1.4 External Fortran subprogram libraries	2	(2)
1.5 A simple Fortran program	4	(3)
1.6 Some simple Fortran constructs	7	(1)
1.7 Intrinsic functions	7	(1)
1.8 User-supplied functions and subroutines	8	(1)
1.9 Errors and accuracy	9	(3)
1.9.1 Roundoff	9	(1)
1.9.2 Truncation	10	(1)
1.9.3 Cancellation	11	(1)
1.9.4 Intrinsic and library-supplied precision routines	11	(1)
1.10 Graphical output	12	(1)
1.11 Conclusions	13	(2)
2 Linear Algebraic Equations	15	(74)
2.1 Introduction	15	(1)
2.2 Gaussian elimination	15	(5)
2.2.1 Observations on the elimination process	20	(1)
2.3 Equation solution using factorization	20	(5)
2.3.1 Observations on the solution process by factorization	25	(1)
2.4 Equations with a symmetrical coefficient matrix	25	(8)
2.4.1 Quadratic form and positive definiteness	30	(1)
2.4.2 Cholesky's method	31	(2)
2.5 Banded equations	33	(2)
2.6 Compact storage for variable bandwidths	35	(3)
2.7 Pivoting	38	(4)
2.7.1 Ill-conditioning	41	(1)
2.8 Equations with prescribed solutions	42	(4)
2.9 Iterative methods	46	(15)
2.9.1 The iterative process	46	(6)
2.9.2 Very sparse systems	52	(1)
2.9.3 The Gauss-Seidel method	52	(5)
2.9.4 Successive overrelaxation	57	(4)
2.10 Gradient methods	61	(7)
2.10.1 The method of 'steepest descent'	61	(3)
2.10.2 The method of 'conjugate gradients'	64	(4)
2.10.3 Convergence of iterative methods	68	(1)
2.11 Unsymmetrical systems	68	(4)
2.12 Preconditioning	72	(9)
2.13 Comparison of direct and iterative methods	81	(1)
2.14 Exercises	82	(7)
3 Nonlinear Equations	89	(42)
3.1 Introduction	89	(2)
3.2 Iterative substitution	91	(3)
3.3 Multiple roots and other difficulties	94	(3)
3.4 Interpolation methods	97	(6)
3.4.1 Bisection method	97	(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