# Numerical Analysis with Applications in Mechanics and Engineering

## Books A much-needed 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 real-world 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 hands-on problem-solving 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 one-of-a-kind guide for engineers using mathematical models and methods, as well as for physicists and mathematicians interested in engineering problems.

Preface xi

1 Errors in Numerical Analysis 1

1.1 Enter Data Errors, 1

1.2 Approximation Errors, 2

1.3 Round-Off Errors, 3

1.4 Propagation of Errors, 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

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

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

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

4.5 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 Pseudo-Inverse of a Matrix, 177

4.11 Solving of the Underdetermined Linear Systems, 178

4.12 Numerical Examples, 178

4.13 Applications, 211

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

5.6 The Method of Entire Series, 280

5.7 Numerical Example, 281

5.8 Applications, 287

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 Newton-Type 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

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

7.11 Legendre’s Polynomials, 400

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

7.15.1 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 One-Dimensional Case, 423

7.18.2 The Multidimensional Case, 425

7.19 Numerical Examples, 427

7.20 Applications, 435

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

8.9 Predictor–Corrector Methods, 469

8.9.1 Euler’s Predictor–Corrector Method, 469

8.9.3 Milne’s Fourth-Order 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

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

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

10.5.2 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 Quasi-Newton 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

10.11 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

Index 629

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