Numerical Analysis

Summary: 1. MATHEMATICAL PRELIMINARIES Review of Calculus / Round-off Errors and Computer Arithmetic / Algorithms and Convergence / Numerical Software 2. SOLUTIONS OF EQUATIONS IN ONE VARIABLE The Bisection Method / Fixed-Point Iteration / The Newton's Method / Error Analysis for Iterative Methods / Accelerating Convergence / Zeros of Polynomials and Muller's Method / Survey of Methods and Software 3. INTERPOLATION AND POLYNO...MIAL APPROXIMATION Interpolation and the LaGrange Polynomial / Divided Differences / Hermite Interpolation / Cubic Spline Interpolation / Parametric Curves / Survey of Methods and Software 4. NUMERICAL DIFFERENTIATION AND INTEGRATION Numerical Differentiation / Richardson's Extrapolation / Elements of Numerical Integration / Composite Numerical Integration / Romberg Integration / Adaptive Quadrature Methods / Gaussian Quadrature / Multiple Integrals / Improper Integrals / Survey of Methods and Software 5. INITIAL-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS The Elementary Theory of Initial-Value Problems / Euler's Method / Higher-Order Taylor Methods / Runge-Kutta Methods / Error Control and the Runge-Kutta-Fehlberg Method / Multi-Step Methods / Variable Step-Size Multi-Step Methods / Extrapolation Methods / Higher-Order Equations and Systems of Differential Equations / Stability / Stiff Differential Equations / Survey of Methods and Software 6. DIRECT METHODS FOR SOLVING LINEAR SYSTEMS Linear Systems of Equations / Pivoting Strategies / Linear Algebra and Matrix Inversion / The Determinant of a Matrix / Matrix Factorization / Special Types of Matrices / Survey of Methods and Software 7. ITERATIVE TECHNIQUES IN MATRIX ALGEBRA Norms of Vectors and Matrices / Eigenvalues and Eigenvectors / Iterative Techniques for Solving Linear Systems / Error Bounds and Iterative Refinement / The Conjugate Gradient Method / Survey of Methods and Software 8. APPROXIMATION THEORY Discrete Least Squares Approximation / Orthogonal Polynomials and Least Squares Approximation / Chebyshev Polynomials and Economization of Power Series / Rational Function Approximation / Trigonometric Polynomial Approximation / Fast Fourier Transforms / Survey of Methods and Software 9. APPROXIMATING EIGENVALUES Linear Algebra and Eigenvalues / The Power Method / Householder's Method / The QR Algorithm / Survey of Methods and Software 10. NUMERICAL SOLUTIONS OF NONLINEAR SYSTEMS OF EQUATIONS Fixed Points for Functions of Several Variables / Newton's Method / Quasi-Newton Methods / Steepest Descent Techniques / Homotopy and Continuation Methods / Survey of Methods and Software 11. BOUNDARY-VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS The Linear Shooting Method / The Shooting Method for Nonlinear Problems / Finite-Difference Methods for Linear Problems / Finite-Difference Methods for Nonlinear Problems / The Rayleigh-Ritz Method / Survey of Methods and Software 12. NUMERICAL SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS Elliptic Partial-Differential Equations / Parabolic Partial-Differential Equations / Hyperbolic Partial-Differential Equations / An Introduction to the Finite-Element Method / Survey of Methods and Software / BIBLIOGRAPHY / ANSWERS TO SELECTED EXERCISES / INDEX [read more]