Numerical Analysis with Applications in Mechanics and Engineering

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 twenty-eight books. NICOLAE-DORU STĂNESCU, 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.

• Preface xi1 Errors in Numerical Analysis 11.1 Enter Data Errors 11.2 Approximation Errors 21.3 Round-Off Errors 31.4 Propagation of Errors 31.4.1 Addition 31.4.2 Multiplication 51.4.3 Inversion of a Number 71.4.4 Division of Two Numbers 71.4.5 Raising to a Negative Entire Power 71.4.6 Taking the Root of pth Order 71.4.7 Subtraction 81.4.8 Computation of Functions 81.5 Applications 8Further Reading 142 Solution of Equations 172.1 The Bipartition (Bisection) Method 172.2 The Chord (Secant) Method 202.3 The Tangent Method (Newton) 262.4 The Contraction Method 372.5 The Newton–Kantorovich Method 422.6 Numerical Examples 462.7 Applications 49Further Reading 523 Solution of Algebraic Equations 553.1 Determination of Limits of the Roots of Polynomials 553.2 Separation of Roots 603.3 Lagrange’s Method 693.4 The Lobachevski–Graeffe Method 723.4.1 The Case of Distinct Real Roots 723.4.2 The Case of a Pair of Complex Conjugate Roots 743.4.3 The Case of Two Pairs of Complex Conjugate Roots 753.5 The Bernoulli Method 763.6 The Bierge–Viète Method 793.7 Lin Methods 793.8 Numerical Examples 823.9 Applications 94Further Reading 1094 Linear Algebra 1114.1 Calculation of Determinants 1114.1.1 Use of Definition 1114.1.2 Use of Equivalent Matrices 1124.2 Calculation of the Rank 1134.3 Norm of a Matrix 1144.4 Inversion of Matrices 1234.4.1 Direct Inversion 1234.4.2 The Gauss–Jordan Method 1244.4.3 The Determination of the Inverse Matrix by its Partition 1254.4.4 Schur’s Method of Inversion of Matrices 1274.4.5 The Iterative Method (Schulz) 1284.4.6 Inversion by Means of the Characteristic Polynomial 1314.4.7 The Frame–Fadeev Method 1314.5 Solution of Linear Algebraic Systems of Equations 1324.5.1 Cramer’s Rule 1324.5.2 Gauss’s Method 1334.5.3 The Gauss–Jordan Method 1344.5.4 The LU Factorization 1354.5.5 The Schur Method of Solving Systems of Linear Equations 1374.5.6 The Iteration Method (Jacobi) 1424.5.7 The Gauss–Seidel Method 1474.5.8 The Relaxation Method 1494.5.9 The Monte Carlo Method 1504.5.10 Infinite Systems of Linear Equations 1524.6 Determination of Eigenvalues and Eigenvectors 1534.6.1 Introduction 1534.6.2 Krylov’s Method 1554.6.3 Danilevski’s Method 1574.6.4 The Direct Power Method 1604.6.5 The Inverse Power Method 1654.6.6 The Displacement Method 1664.6.7 Leverrier’s Method 1664.6.8 The L–R (Left–Right) Method 1664.6.9 The Rotation Method 1684.7 QR Decomposition 1694.8 The Singular Value Decomposition (SVD) 1724.9 Use of the Least Squares Method in Solving the Linear Overdetermined Systems 1744.10 The Pseudo-Inverse of a Matrix 1774.11 Solving of the Underdetermined Linear Systems 1784.12 Numerical Examples 1784.13 Applications 211Further Reading 2695 Solution of Systems of Nonlinear Equations 2735.1 The Iteration Method (Jacobi) 2735.2 Newton’s Method 2755.3 The Modified Newton’s Method 2765.4 The Newton–Raphson Method 2775.5 The Gradient Method 2775.6 The Method of Entire Series 2805.7 Numerical Example 2815.8 Applications 287Further Reading 3046 Interpolation and Approximation of Functions 3076.1 Lagrange’s Interpolation Polynomial 3076.2 Taylor Polynomials 3116.3 Finite Differences: Generalized Power 3126.4 Newton’s Interpolation Polynomials 3176.5 Central Differences: Gauss’s Formulae Stirling’s Formula Bessel’s Formula Everett’s Formulae 3226.6 Divided Differences 3276.7 Newton-Type Formula with Divided Differences 3316.8 Inverse Interpolation 3326.9 Determination of the Roots of an Equation by Inverse Interpolation 3336.10 Interpolation by Spline Functions 3356.11 Hermite’s Interpolation 3396.12 Chebyshev’s Polynomials 3406.13 Mini–Max Approximation of Functions 3446.14 Almost Mini–Max Approximation of Functions 3456.15 Approximation of Functions by Trigonometric Functions (Fourier) 3466.16 Approximation of Functions by the Least Squares 3526.17 Other Methods of Interpolation 3546.17.1 Interpolation with Rational Functions 3546.17.2 The Method of Least Squares with Rational Functions 3556.17.3 Interpolation with Exponentials 3556.18 Numerical Examples 3566.19 Applications 363Further Reading 3747 Numerical Differentiation and Integration 3777.1 Introduction 3777.2 Numerical Differentiation by Means of an Expansion into a Taylor Series 3777.3 Numerical Differentiation by Means of Interpolation Polynomials 3807.4 Introduction to Numerical Integration 3827.5 The Newton–Côtes Quadrature Formulae 3847.6 The Trapezoid Formula 3867.7 Simpson’s Formula 3897.8 Euler’s and Gregory’s Formulae 3937.9 Romberg’s Formula 3967.10 Chebyshev’s Quadrature Formulae 3987.11 Legendre’s Polynomials 4007.12 Gauss’s Quadrature Formulae 4057.13 Orthogonal Polynomials 4067.13.1 Legendre Polynomials 4077.13.2 Chebyshev Polynomials 4077.13.3 Jacobi Polynomials 4087.13.4 Hermite Polynomials 4087.13.5 Laguerre Polynomials 4097.13.6 General Properties of the Orthogonal Polynomials 4107.14 Quadrature Formulae of Gauss Type Obtained by Orthogonal Polynomials 4127.14.1 Gauss–Jacobi Quadrature Formulae 4137.14.2 Gauss–Hermite Quadrature Formulae 4147.14.3 Gauss–Laguerre Quadrature Formulae 4157.15 Other Quadrature Formulae 4177.15.1 Gauss Formulae with Imposed Points 4177.15.2 Gauss Formulae in which the Derivatives of the Function Also Appear 4187.16 Calculation of Improper Integrals 4207.17 Kantorovich’s Method 4227.18 The Monte Carlo Method for Calculation of Definite Integrals 4237.18.1 The One-Dimensional Case 4237.18.2 The Multidimensional Case 4257.19 Numerical Examples 4277.20 Applications 435Further Reading 4478 Integration of Ordinary Differential Equations and of Systems of Ordinary Differential Equations 4518.1 State of the Problem 4518.2 Euler’s Method 4548.3 Taylor Method 4578.4 The Runge–Kutta Methods 4588.5 Multistep Methods 4628.6 Adams’s Method 4638.7 The Adams–Bashforth Methods 4658.8 The Adams–Moulton Methods 4678.9 Predictor–Corrector Methods 4698.9.1 Euler’s Predictor–Corrector Method 4698.9.2 Adams’s Predictor–Corrector Methods 4698.9.3 Milne’s Fourth-Order Predictor–Corrector Method 4708.9.4 Hamming’s Predictor–Corrector Method 4708.10 The Linear Equivalence Method (LEM) 4718.11 Considerations about the Errors 4738.12 Numerical Example 4748.13 Applications 480Further Reading 5259 Integration of Partial Differential Equations and of Systems of Partial Differential Equations 5299.1 Introduction 5299.2 Partial Differential Equations of First Order 5299.2.1 Numerical Integration by Means of Explicit Schemata 5319.2.2 Numerical Integration by Means of Implicit Schemata 5339.3 Partial Differential Equations of Second Order 5349.4 Partial Differential Equations of Second Order of Elliptic Type 5349.5 Partial Differential Equations of Second Order of Parabolic Type 5389.6 Partial Differential Equations of Second Order of Hyperbolic Type 5439.7 Point Matching Method 5469.8 Variational Methods 5479.8.1 Ritz’s Method 5499.8.2 Galerkin’s Method 5519.8.3 Method of the Least Squares 5539.9 Numerical Examples 5549.10 Applications 562Further Reading 57510 Optimizations 57710.1 Introduction 57710.2 Minimization Along a Direction 57810.2.1 Localization of the Minimum 57910.2.2 Determination of the Minimum 58010.3 Conjugate Directions 58310.4 Powell’s Algorithm 58510.5 Methods of Gradient Type 58510.5.1 The Gradient Method 58510.5.2 The Conjugate Gradient Method 58710.5.3 Solution of Systems of Linear Equations by Means of Methods of Gradient Type 58910.6 Methods of Newton Type 59010.6.1 Newton’s Method 59010.6.2 Quasi-Newton Method 59210.7 Linear Programming: The Simplex Algorithm 59310.7.1 Introduction 59310.7.2 Formulation of the Problem of Linear Programming 59510.7.3 Geometrical Interpretation 59710.7.4 The Primal Simplex Algorithm 59710.7.5 The Dual Simplex Algorithm 59910.8 Convex Programming 60010.9 Numerical Methods for Problems of Convex Programming 60210.9.1 Method of Conditional Gradient 60210.9.2 Method of Gradient’s Projection 60210.9.3 Method of Possible Directions 60310.9.4 Method of Penalizing Functions 60310.10 Quadratic Programming 60310.11 Dynamic Programming 60510.12 Pontryagin’s Principle of Maximum 60710.13 Problems of Extremum 60910.14 Numerical Examples 61110.15 Applications 623Further Reading 626Index 629