Numerical Analysis with Applications in Mechanics and Engineering

Cover Image Copyright Year: 2013
Author(s): Petre P. Teodorescu; Nicolae-Doru Stanescu; Nicolae Pandrea
Book Type: Wiley-IEEE Press
Content Type : Books
Topics: Components, Circuits, Devices & Systems ;  Computing & Processing ;  Engineered Materials, Dielectrics & Plasmas ;  Power, Energy, & Industry Applications ;  Robotics & Control Systems
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      Frontmatter

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      The prelims comprise:
      Half Title
      IEEE Press Editorial Board 2013
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      Contents
      Preface View full abstract»

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      Errors in Numerical Analysis

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      This chapter deals with the most encountered errors in numerical analysis, that is, enter data errors, approximation errors, round-off errors, and propagation of errors. Propagation of errors during addition, multiplication, inversion of a number, division of two numbers, subtraction and computation of functions is discussed in detail. View full abstract»

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      Solution of Equations

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      This chapter deals with several methods of approximate solutions of equations, that is, the bipartition method, the chord (secant) method, the tangent method (Newton), the contraction method, and the Newton-Kantorovich method. These are followed by applications. View full abstract»

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      Solution of Algebraic Equations

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      This chapter deals with the determination of limits of the roots of polynomials, including their separation. Three methods are considered, namely, Lagrange's method, the Lobachevski-Graeffe method, and Bernoulli's method. In addition, the chapter talks about Bierge-Viete method and Lin methods. These are followed by applications. View full abstract»

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      Linear Algebra

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      This chapter on linear algebra first discusses calculation of determinants and rank, norm of a matrix, and inversion of matrices. Next, it deals with solution of linear algebraic systems of equations. Then, it gives definitions of QR decomposition and the singular value decomposition (SVD). The use of the least squares method in solving the linear overdetermined systems, the pseudo-inverse of a matrix, and solving of the underdetermined linear systems are considered. These are followed by applications. View full abstract»

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      Solution of Systems of Nonlinear Equations

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      This chapter deals with the solution of systems of nonlinear equations. The methods of solution discussed are the iteration method (Jacobi), Newton's method, the modified Newton's method, the Newton-Raphson method, and the gradient method. These are followed by applications. View full abstract»

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      Interpolation and Approximation of Functions

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      This chapter deals with the interpolation and approximation of functions. The interpolation includes Lagrange's interpolation polynomial, Taylor polynomials, Newton's interpolation polynomials, inverse interpolation, interpolation by Spline functions, and Hermite's interpolation. The mini-max approximation of functions, almost mini-max approximation of functions, approximation of functions by trigonometric functions (Fourier) and the least squares are considered. These are followed by applications. View full abstract»

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      Numerical Differentiation and Integration

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      This chapter gives introduction to numerical differentiation by means of an expansion into a Taylor series and interpolation polynomials, and numerical integration. The numerical integration formulas include the Newton-CÿTes quadrature formulae, the trapezoid formula, Simpson's formula, Euler's and Gregory's formulae, Romberg's formula, and Chebyshev's quadrature formulae. In addition, the chapter considers quadrature formulae of Gauss type obtained by orthogonal polynomials, calculation of improper integrals, Kantorovich's method, and the Monte Carlo method for calculation of definite integrals. These are followed by applications. View full abstract»

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      Integration of Ordinary Differential Equations and of Systems of Ordinary Differential Equations

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      This chapter presents the numerical methods for the integration of ordinary differential equations and of systems of differential equations. It presents Euler's method, Taylor's method, the Runge-Kutta methods, the multistep methods, and the predictor-corrector methods. The chapter ends with some applications. View full abstract»

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      Integration of Partial Differential Equations and of Systems of Partial Differential Equations

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      Many problems of science and technique lead to partial differential equations. The mathematical theories of such equations, especially of the nonlinear ones, are very intricate, such that their numerical study becomes inevitable. To classify the partial differential equations, this chapter uses various criteria, that is, considering the order of the derivatives (first order, second order, or nth order), the linearity character (linear, quasilinear, or nonlinear equations), the influence of the integration domain at a point (equations of elliptic, parabolic, or hyperbolic type), and the types of limit conditions (Dirichlet, Neumann, or mixed problems). It presents the numerical methods for the integration of partial differential equations and of systems of partial differential equations. The numerical methods are point matching method, Ritz's method, Galerkin's method, and method of the least squares. Finally, the chapter gives numerical examples and some applications. View full abstract»

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      Optimizations

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      A method of optimization solves the problem of determination of the minimum (maximum) of an objective (purpose) function U. In general, in case of optimization problems, the global minimum is of interest. Such a point of global minimum is found between the points of local minimum; it can be unique or multiple. This chapter first considers minimization along a direction. The Powell algorithm gives a procedure to determine n conjugate directions without using the matrix 2U(x). Next, the chapter discusses the methods of gradient type, which are characterized by the use of the gradient of the function to be optimized, U(x), and the methods of Newton type using the Hessian matrix 2U(x). Other covered topics are linear programming, numerical methods for problems of convex programming, quadratic programming, dynamic programming, and Pontryagin's principle of maximum. These are followed by numerical examples and some applications. View full abstract»

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      Index

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      No abstract. View full abstract»