Numerical Analysis with Applications in Mechanics and EngineeringISBN: 9781118077504
646 pages
June 2013, WileyIEEE Press

A muchneeded guide on how to use numerical methods to solve practical engineering problems
Bridging the gap between mathematics and engineering, Numerical Analysis with Applications in Mechanics and Engineering arms readers with powerful tools for solving realworld problems in mechanics, physics, and civil and mechanical engineering. Unlike most books on numerical analysis, this outstanding work links theory and application, explains the mathematics in simple engineering terms, and clearly demonstrates how to use numerical methods to obtain solutions and interpret results.
Each chapter is devoted to a unique analytical methodology, including a detailed theoretical presentation and emphasis on practical computation. Ample numerical examples and applications round out the discussion, illustrating how to work out specific problems of mechanics, physics, or engineering. Readers will learn the core purpose of each technique, develop handson problemsolving skills, and get a complete picture of the studied phenomenon. Coverage includes:
 How to deal with errors in numerical analysis
 Approaches for solving problems in linear and nonlinear systems
 Methods of interpolation and approximation of functions
 Formulas and calculations for numerical differentiation and integration
 Integration of ordinary and partial differential equations
 Optimization methods and solutions for programming problems
Numerical Analysis with Applications in Mechanics and Engineering is a oneofakind guide for engineers using mathematical models and methods, as well as for physicists and mathematicians interested in engineering problems.
1 Errors in Numerical Analysis 1
1.1 Enter Data Errors, 1
1.2 Approximation Errors, 2
1.3 RoundOff Errors, 3
1.4 Propagation of Errors, 3
1.4.1 Addition, 3
1.4.2 Multiplication, 5
1.4.3 Inversion of a Number, 7
1.4.4 Division of Two Numbers, 7
1.4.5 Raising to a Negative Entire Power, 7
1.4.6 Taking the Root of pth Order, 7
1.4.7 Subtraction, 8
1.4.8 Computation of Functions, 8
1.5 Applications, 8
Further Reading, 14
2 Solution of Equations 17
2.1 The Bipartition (Bisection) Method, 17
2.2 The Chord (Secant) Method, 20
2.3 The Tangent Method (Newton), 26
2.4 The Contraction Method, 37
2.5 The Newton–Kantorovich Method, 42
2.6 Numerical Examples, 46
2.7 Applications, 49
Further Reading, 52
3 Solution of Algebraic Equations 55
3.1 Determination of Limits of the Roots of Polynomials, 55
3.2 Separation of Roots, 60
3.3 Lagrange’s Method, 69
3.4 The Lobachevski–Graeffe Method, 72
3.4.1 The Case of Distinct Real Roots, 72
3.4.2 The Case of a Pair of Complex Conjugate Roots, 74
3.4.3 The Case of Two Pairs of Complex Conjugate Roots, 75
3.5 The Bernoulli Method, 76
3.6 The Bierge–Vi`ete Method, 79
3.7 Lin Methods, 79
3.8 Numerical Examples, 82
3.9 Applications, 94
Further Reading, 109
4 Linear Algebra 111
4.1 Calculation of Determinants, 111
4.1.1 Use of Definition, 111
4.1.2 Use of Equivalent Matrices, 112
4.2 Calculation of the Rank, 113
4.3 Norm of a Matrix, 114
4.4 Inversion of Matrices, 123
4.4.1 Direct Inversion, 123
4.4.2 The Gauss–Jordan Method, 124
4.4.3 The Determination of the Inverse Matrix by its Partition, 125
4.4.4 Schur’s Method of Inversion of Matrices, 127
4.4.5 The Iterative Method (Schulz), 128
4.4.6 Inversion by Means of the Characteristic Polynomial, 131
4.4.7 The Frame–Fadeev Method, 131
4.5 Solution of Linear Algebraic Systems of Equations, 132
4.5.1 Cramer’s Rule, 132
4.5.2 Gauss’s Method, 133
4.5.3 The Gauss–Jordan Method, 134
4.5.4 The LU Factorization, 135
4.5.5 The Schur Method of Solving Systems of Linear Equations, 137
4.5.6 The Iteration Method (Jacobi), 142
4.5.7 The Gauss–Seidel Method, 147
4.5.8 The Relaxation Method, 149
4.5.9 The Monte Carlo Method, 150
4.5.10 Infinite Systems of Linear Equations, 152
4.6 Determination of Eigenvalues and Eigenvectors, 153
4.6.1 Introduction, 153
4.6.2 Krylov’s Method, 155
4.6.3 Danilevski’s Method, 157
4.6.4 The Direct Power Method, 160
4.6.5 The Inverse Power Method, 165
4.6.6 The Displacement Method, 166
4.6.7 Leverrier’s Method, 166
4.6.8 The L–R (Left–Right) Method, 166
4.6.9 The Rotation Method, 168
4.7 QR Decomposition, 169
4.8 The Singular Value Decomposition (SVD), 172
4.9 Use of the Least Squares Method in Solving the Linear Overdetermined Systems, 174
4.10 The PseudoInverse of a Matrix, 177
4.11 Solving of the Underdetermined Linear Systems, 178
4.12 Numerical Examples, 178
4.13 Applications, 211
Further Reading, 269
5 Solution of Systems of Nonlinear Equations 273
5.1 The Iteration Method (Jacobi), 273
5.2 Newton’s Method, 275
5.3 The Modified Newton’s Method, 276
5.4 The Newton–Raphson Method, 277
5.5 The Gradient Method, 277
5.6 The Method of Entire Series, 280
5.7 Numerical Example, 281
5.8 Applications, 287
Further Reading, 304
6 Interpolation and Approximation of Functions 307
6.1 Lagrange’s Interpolation Polynomial, 307
6.2 Taylor Polynomials, 311
6.3 Finite Differences: Generalized Power, 312
6.4 Newton’s Interpolation Polynomials, 317
6.5 Central Differences: Gauss’s Formulae, Stirling’s Formula, Bessel’s Formula, Everett’s Formulae, 322
6.6 Divided Differences, 327
6.7 NewtonType Formula with Divided Differences, 331
6.8 Inverse Interpolation, 332
6.9 Determination of the Roots of an Equation by Inverse Interpolation, 333
6.10 Interpolation by Spline Functions, 335
6.11 Hermite’s Interpolation, 339
6.12 Chebyshev’s Polynomials, 340
6.13 Mini–Max Approximation of Functions, 344
6.14 Almost Mini–Max Approximation of Functions, 345
6.15 Approximation of Functions by Trigonometric Functions (Fourier), 346
6.16 Approximation of Functions by the Least Squares, 352
6.17 Other Methods of Interpolation, 354
6.17.1 Interpolation with Rational Functions, 354
6.17.2 The Method of Least Squares with Rational Functions, 355
6.17.3 Interpolation with Exponentials, 355
6.18 Numerical Examples, 356
6.19 Applications, 363
Further Reading, 374
7 Numerical Differentiation and Integration 377
7.1 Introduction, 377
7.2 Numerical Differentiation by Means of an Expansion into a Taylor Series, 377
7.3 Numerical Differentiation by Means of Interpolation Polynomials, 380
7.4 Introduction to Numerical Integration, 382
7.5 The Newton–Cˆotes Quadrature Formulae, 384
7.6 The Trapezoid Formula, 386
7.7 Simpson’s Formula, 389
7.8 Euler’s and Gregory’s Formulae, 393
7.9 Romberg’s Formula, 396
7.10 Chebyshev’s Quadrature Formulae, 398
7.11 Legendre’s Polynomials, 400
7.12 Gauss’s Quadrature Formulae, 405
7.13 Orthogonal Polynomials, 406
7.13.1 Legendre Polynomials, 407
7.13.2 Chebyshev Polynomials, 407
7.13.3 Jacobi Polynomials, 408
7.13.4 Hermite Polynomials, 408
7.13.5 Laguerre Polynomials, 409
7.13.6 General Properties of the Orthogonal Polynomials, 410
7.14 Quadrature Formulae of Gauss Type Obtained by Orthogonal Polynomials, 412
7.14.1 Gauss–Jacobi Quadrature Formulae, 413
7.14.2 Gauss–Hermite Quadrature Formulae, 414
7.14.3 Gauss–Laguerre Quadrature Formulae, 415
7.15 Other Quadrature Formulae, 417
7.15.1 Gauss Formulae with Imposed Points, 417
7.15.2 Gauss Formulae in which the Derivatives of the Function Also Appear, 418
7.16 Calculation of Improper Integrals, 420
7.17 Kantorovich’s Method, 422
7.18 The Monte Carlo Method for Calculation of Definite Integrals, 423
7.18.1 The OneDimensional Case, 423
7.18.2 The Multidimensional Case, 425
7.19 Numerical Examples, 427
7.20 Applications, 435
Further Reading, 447
8 Integration of Ordinary Differential Equations and of Systems of Ordinary Differential Equations 451
8.1 State of the Problem, 451
8.2 Euler’s Method, 454
8.3 Taylor Method, 457
8.4 The Runge–Kutta Methods, 458
8.5 Multistep Methods, 462
8.6 Adams’s Method, 463
8.7 The Adams–Bashforth Methods, 465
8.8 The Adams–Moulton Methods, 467
8.9 Predictor–Corrector Methods, 469
8.9.1 Euler’s Predictor–Corrector Method, 469
8.9.2 Adams’s Predictor–Corrector Methods, 469
8.9.3 Milne’s FourthOrder Predictor–Corrector Method, 470
8.9.4 Hamming’s Predictor–Corrector Method, 470
8.10 The Linear Equivalence Method (LEM), 471
8.11 Considerations about the Errors, 473
8.12 Numerical Example, 474
8.13 Applications, 480
Further Reading, 525
9 Integration of Partial Differential Equations and of Systems of Partial Differential Equations 529
9.1 Introduction, 529
9.2 Partial Differential Equations of First Order, 529
9.2.1 Numerical Integration by Means of Explicit Schemata, 531
9.2.2 Numerical Integration by Means of Implicit Schemata, 533
9.3 Partial Differential Equations of Second Order, 534
9.4 Partial Differential Equations of Second Order of Elliptic Type, 534
9.5 Partial Differential Equations of Second Order of Parabolic Type, 538
9.6 Partial Differential Equations of Second Order of Hyperbolic Type, 543
9.7 Point Matching Method, 546
9.8 Variational Methods, 547
9.8.1 Ritz’s Method, 549
9.8.2 Galerkin’s Method, 551
9.8.3 Method of the Least Squares, 553
9.9 Numerical Examples, 554
9.10 Applications, 562
Further Reading, 575
10 Optimizations 577
10.1 Introduction, 577
10.2 Minimization Along a Direction, 578
10.2.1 Localization of the Minimum, 579
10.2.2 Determination of the Minimum, 580
10.3 Conjugate Directions, 583
10.4 Powell’s Algorithm, 585
10.5 Methods of Gradient Type, 585
10.5.1 The Gradient Method, 585
10.5.2 The Conjugate Gradient Method, 587
10.5.3 Solution of Systems of Linear Equations by Means of Methods of Gradient Type, 589
10.6 Methods of Newton Type, 590
10.6.1 Newton’s Method, 590
10.6.2 QuasiNewton Method, 592
10.7 Linear Programming: The Simplex Algorithm, 593
10.7.1 Introduction, 593
10.7.2 Formulation of the Problem of Linear Programming, 595
10.7.3 Geometrical Interpretation, 597
10.7.4 The Primal Simplex Algorithm, 597
10.7.5 The Dual Simplex Algorithm, 599
10.8 Convex Programming, 600
10.9 Numerical Methods for Problems of Convex Programming, 602
10.9.1 Method of Conditional Gradient, 602
10.9.2 Method of Gradient’s Projection, 602
10.9.3 Method of Possible Directions, 603
10.9.4 Method of Penalizing Functions, 603
10.10 Quadratic Programming, 603
10.11 Dynamic Programming, 605
10.12 Pontryagin’s Principle of Maximum, 607
10.13 Problems of Extremum, 609
10.14 Numerical Examples, 611
10.15 Applications, 623
Further Reading, 626
Index 629
PETRE TEODORESCU, PhD, is a Professor in the Faculty of Mathematics and Computer Science at the University of Bucharest in Romania and the author of 250 papers and twentyeight books.
NICOLAEDORU STÂNESCU, PhD, is a Professor in the Faculty of Mechanics and Technology at the University of Piteşti in Romania and the author of 200 papers and ten books.
NICOLAE PANDREA, PhD, is a Professor in the Faculty of Mechanics and Technology at the University of Piteşti in Romania and the author of 250 papers and six books.