Numerical Methods

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9780534407612: Numerical Methods
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This text emphasizes the intelligent application of approximation techniques to the type of problems that commonly occur in engineering and the physical sciences. Students learn why the numerical methods work, what type of errors to expect, and when an application might lead to difficulties. The authors also provide information about the availability of high-quality software for numerical approximation routines. The techniques are essentially the same as those covered in the authors' top-selling Numerical Analysis text, but in this text, full mathematical justifications are provided only if they are concise and add to the understanding of the methods. The emphasis is placed on describing each technique from an implementation standpoint, and on convincing the student that the method is reasonable both mathematically and computationally.

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About the Author:

J. Douglas Faires is a Professor of Mathematics at Youngstown State University. His research interests include analysis, numerical analysis, and mathematics history. Dr. Faires has won many awards, including Outstanding College-University Teacher of Mathematics, Ohio Section of MAA (1996) and Youngstown State University, Distinguished Professor for Teaching (1995-1996).

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1. MATHEMATICAL PRELIMINARIES AND ERROR ANALYSIS. Introduction. Review of Calculus. Round-off Error and Computer Arithmetic. Errors in Scientific Computation. Computer Software. 2. SOLUTIONS OF EQUATIONS OF ONE VARIABLE. Introduction. The Bisection Method. The Secant Method. Newton's Method. Error Analysis and Accelerating Convergence. Muller's Method. Survey of Methods and Software. 3. INTERPOLATION AND POLYNOMIAL APPROXIMATION. Introduction. Lagrange Polynomials. Divided Differences. Hermite Interpolation. Spline Interpolation. Parametric Curves. Survey of Methods and Software. 4. NUMERICAL INTEGRATION AND DIFFERENTIATION. Introduction. Basic Quadrature Rules. Composite Quadrature Rules. Romberg Integration. Gaussian Quadrature. Adaptive Quadrature. Multiple Integrals. Improper Integrals. Numerical Differentiation. Survey of Methods and Software. 5. NUMERICAL SOLUTION OF INITIAL-VALUE PROBLEMS. Introduction. Taylor Methods. Runge-Kutta Methods. Predictor-Corrector Methods. Extrapolation Methods. Adaptive Techniques. Methods for Systems of Equations. Stiff Differentials Equations. Survey of Methods and Software. 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS. Introduction. Gaussian Elimination. Pivoting Strategies. Linear Algebra and Matrix Inversion. Matrix Factorization. Techniques for Special Matrices. Survey of Methods and Software. 7. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. Introduction. Convergence of Vectors. Eigenvalues and Eigenvectors. The Jacobi and Gauss-Seidel Methods. The SOR Method. Error Bounds and Iterative Refinement. Survey of Methods and Software. 8. APPROXIMATION THEORY. Introduction. Discrete Least Squares Approximation. Continuous Least Squares Approximation. Chebyshev Polynomials. Rational Function Approximation. Trigonometric Polynomial Approximation. Fast Fourier Transforms. Survey of Methods and Software. 9. APPROXIMATING EIGENVALUES. Introduction. Isolating Eigenvalues. The Power Method. Householder's Method. The QR Method. Survey of Methods and Software. 10. SOLUTIONS OF SYSTEMS OF NONLINEAR EQUATIONS. Introduction. Newton's Methods for Systems. Quasi-Newton Methods. The Steepest Descent Method. Survey of Methods and Software. 11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS. Introduction. The Linear Shooting Method. Linear Finite Difference Methods. The Nonlinear Shooting Method. Nonlinear Finite-Difference Methods. Variational Techniques. Survey of Methods and Software. 12. NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS. Introduction. Finite-Difference Methods for Elliptic Problems. Finite-Difference Methods for Parabolic Problems. Finite-Difference Methods for Hyperbolic Problems. Introduction to the Finite-Element Method. Survey of Methods and Software.

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