دانلود کتاب A first course in numerical analysis

The 2006 Abel symposium is focusing on contemporary research involving interaction between computer science, computational science and mathematics. In recent years, computation has been affecting pure mathematics in fundamental ways. Conversely, ideas and methods of pure mathematics are becoming increasingly important within computational and applied mathematics. At the core of computer science is the study of computability and complexity for discrete mathematical structures. Studying the foundations of computational mathematics raises similar questions concerning continuous mathematical structures. There are several reasons for these developments. The exponential growth of computing power is bringing computational methods into ever new application areas. Equally important is the advance of software and programming languages, which to an increasing degree allows the representation of abstract mathematical structures in program code. Symbolic computing is bringing algorithms from mathematical analysis into the hands of pure and applied mathematicians, and the combination of symbolic and numerical techniques is becoming increasingly important both in computational science and in areas of pure mathematics Introduction and Preliminaries -- What Is Numerical Analysis? -- Sources of Error -- Error Definitions and Related Matters -- Significant Digits -- Error in Functional Evaluation -- Norms -- Roundoff Error -- The Probabilistic Approach to Roundoff: A Particular Example -- Computer Arithmetic -- Fixed-Point Arithmetic -- Floating-Point Numbers -- Floating-Point Arithmetic -- Overflow and Underflow -- Single- and Double-Precision Arithmetic -- Error Analysis -- Backward Error Analysis -- Condition and Stability -- Approximation and Algorithms -- Approximation -- Classes of Approximating Functions -- Types of Approximations -- The Case for Polynomial Approximation -- Numerical Algorithms -- Functionals and Error Analysis -- The Method of Undetermined Coefficients -- Interpolation -- Lagrangian Interpolation -- Interpolation at Equal Intervals -- Lagrangian Interpolation at Equal Intervals -- Finite Differences -- The Use of Interpolation Formulas -- Iterated Interpolation -- Inverse Interpolation -- Hermite Interpolation -- Spline Interpolation -- Other Methods of Interpolation; Extrapolation -- Numerical Differentiation, Numerical Quadrature, and Summation -- Numerical Differentiation of Data -- Numerical Differentation of Functions -- Numerical Quadrature: The General Problem -- Numerical Integration of Data -- Gaussian Quadrature -- Weight Functions -- Orthogonal Polynomials and Gaussian Quadrature -- Gaussian Quadrature over Infinite Intervals -- Particular Gaussian Quadrature Formulas -- Gauss-Jacobi Quadrature -- Gauss-Chebyshev Quadrature -- Singular Integrals -- Composite Quadrature Formulas -- Newton-Cotes Quadrature Formulas -- Composite Newton-Cotes Formulas -- Romberg Integration -- Adaptive Integration -- Choosing a Quadrature Formula -- Summation -- The Euler-Maclaurin Sum Formula -- Summation of Rational Functions; Factorial Functions -- The Euler Transformation -- The Numerical Solution of Ordinary Differential Equations -- Statement of the Problem -- Numerical Integration Methods -- The Method of Undetermined Coefficients -- Truncation Error in Numerical Integration Methods -- Stability of Numerical Integration Methods -- Convergence and Stability -- Propagated-Error Bounds and Estimates -- Predictor-Corrector Methods -- Convergence of the Iterations -- Predictors and Correctors -- Error Estimation -- Stability -- Starting the Solution and Changing the Interval -- Analytic Methods -- A Numerical Method -- Changing the Interval -- Using Predictor-Corrector Methods -- Variable-Order-Variable-Step Methods -- Some Illustrative Examples -- Runge-Kutta Methods -- Errors in Runge-Kutta Methods -- Second-Order Methods -- Third-Order Methods -- Fourth-Order Methods -- Higher-Order Methods -- Practical Error Estimation -- Step-Size Strategy -- Stability -- Comparison of Runge-Kutta and Predictor-Corrector Methods -- Other Numerical Integration Methods -- Methods Based on Higher Derivatives -- Extrapolation Methods -- Stiff Equations -- Functional Approximation: Least-Squares Techniques -- The Principle of Least Squares -- Polynomial Least-Squares Approximations -- Solution of the Normal Equations -- Choosing the Degree of the Polynomial -- Orthogonal-Polynomial Approximations -- An Example of the Generation of Least-Squares Approximations -- The Fourier Approximation -- The Fast Fourier Transform -- Least-Squares Approximations and Trigonometric Interpolation -- Functional Approximation: Minimum Maximum Error Techniques -- General Remarks -- Rational Functions, Polynomials, and Continued Fractions -- Pade Approximations -- An Example -- Chebyshev Polynomials -- Chebyshev Expansions -- Economization of Rational Functions -- Economization of Power Series -- Generalization to Rational Functions -- Chebyshev's Theorem on Minimax Approximations -- Constructing Minimax Approximations -- The Second Algorithm of Remes -- The Differential Correction Algorithm -- The Solution of Nonlinear Equations -- Functional Iteration -- Computational Efficiency -- The Secant Method -- One-Point Iteration Formulas -- Multipoint Iteration Formulas -- Iteration Formulas Using General Inverse Interpolation -- Derivative Estimated Iteration Formulas -- Functional Iteration at a Multiple Root -- Some Computational Aspects of Functional Iteration -- The [delta superscript 2] Process -- Systems of Nonlinear Equations -- The Zeros of Polynomials: The Problem -- Sturm Sequences -- Classical Methods -- Bairstow's Method -- Graeffe's Root-Squaring Method -- Bernoulli's Method -- Laguerre's Method -- The Jenkins-Traub Method -- A Newton-based Method -- The Effect of Coefficient Errors on the Roots; Ill-conditioned Polynomials -- The Solution of Simultaneous Linear Equations -- The Basic Theorem and the Problem -- General Remarks -- Direct Methods -- Gaussian Elimination -- Compact Forms of Gaussian Elimination -- The Doolittle, Crout, and Cholesky Algorithms -- Pivoting and Equilibration -- Error Analysis -- Roundoff-Error Analysis -- Iterative Refinement -- Matrix Iterative Methods -- Stationary Iterative Processes and Related Matters -- The Jacobi Iteration -- The Gauss-Seidel Method -- Roundoff Error in Iterative Methods -- Acceleration of Stationary Iterative Processes -- Matrix Inversion -- Overdetermined Systems of Linear Equations -- The Simplex Method for Solving Linear Programming Problems -- Miscellaneous Topics -- The Calculation of Elgenvalues and Eigenvectors of Matrices -- Basic Relationships -- Basic Theorems -- The Characteristic Equation -- The Location of, and Bounds on, the Eigenvalues -- Canonical Forms -- The Largest Eigenvalue in Magnitude by the Power Method -- Acceleration of Convergence -- The Inverse Power Method -- The Eigenvalues and Eigenvectors of Symmetric Matrices -- The Jacobi Method -- Givens' Method -- Householder's Method -- Methods for Nonsymmetric Matrices -- Lanczos' Method -- Supertriangularization -- Jacobi-Type Methods -- The LR and QR Algorithms -- The Simple QR Algorithm -- The Double QR Algorithm -- Errors in Computed Eigenvalues and Eigenvectors