Computing Methods

PrefaceIntroduction § 1. Object of Computer Mathematics § 2. Computer Mathematics, Its Method 1. Functional Metric Spaces 2. Functions Determined in Functional Spaces 3. The Method of Computer Mathematics § 3. Means of Computation § 4. Computing Methods as a Branch of Computer Mathematics. Synopsis of CourseChapter 1 Operation on Approximate Quantities § 1. Classification Of Errors 1. Sources of Error in the Results of Computation 2. The Problems Arising When Working with Approximate Quantities 3. Rules for Rounding Numbers 4. Classification of Errors § 2. The Irremovable Error 1. The Absolute and Relative Errors of a Number 2. Number of Significant Digits 3. The Irremovable Error in the Value of Functions for Approximate Values of the Argument. Errors in the Results of Arithmetic Operations § 3. Errors in Rounding § 4. Total Error § 5. The Concept of Statistical Methods for Evaluating Errors § 6. R.M.S. Errors 1. Systematic And Random Errors 2. R.M.S. Errors 3. Treatment of the Results Using the Method of Least Squares 4. The R.M.S. Error of a Function 5. The R.M.S. Error of a Uniformly Distributed Quantity Exercises References Chapter 2 The Theory of Interpolation and Certain Application § 1. Statement of the Problem 1. Linear Sets. Linearly Independent Systems Of Elements 2. The Problem of Interpolation 3. Construction of an Interpolating Function 4. The Chebyshev System 5. Fundamental Problems in the Theory of Interpolation § 2. The Lagrange Interpolation Polynomial 1. The Construction of Lagrange Interpolation Polynomials 2. The Lagrange Interpolation Polynomial for Equidistant Nodes 3. The Aitken Interpolation Scheme (Neville's Modification) § 3. Errors in The Lagrange Interpolation Formula 1. The Residual Term in the Lagrange Formula and its Estimation 2. Choosing the Points of Interpolation 3. The Irremovable Error in the Lagrange Formula § 4. The Residual Term of the General Interpolation Formula § 5. The Newton Interpolation Formula for Unequal Intervals 1. Divided Differences and Their Properties 2. Derivation of the Newton Formula for Unequal Intervals 3. The Residual Term in the Newton Formula § 6. Newton Interpolation Formula for Equal Intervals 1. Finite Differences and Their Properties 2. Derivation of Newton Interpolation Formula 3. The Residual Terms Of The Newton Interpolation Formula § 7. Interpolation Formula Using Central Differences 1. The Interpolation Formula of Gauss, Stirling, Bessel and Everett 2. The Residual Terms of Interpolation Formula Using Central Differences § 8. Other Approaches to the Derivation of Interpolation Formula for Equal Intervals 1. The Fraser Lozenge Diagram 2. The Operational Method of Deriving Interpolation Formula § 9. The Convergence of the Interpolation Process § 10. Interpolation of Periodic Functions § 11. The General Problem of Interpolation by Algebraic Polynomials 1. The Hermite Interpolation Polynomial 2. The General Form of the Hermite Interpolation Polynomial 3. The Residual Term in the Hermite Interpolation Formula 4. Divided Differences with Recurring Values of the Argument 5. The Generalized Newton Divided Difference interpolation Formula § 12. Interpolation Functions of Many Independent Variables 1. The Difficulties Involved in Interpolating Functions of Many Variables 2. Generalization of the Newton Interpolation Formula in the Case of Functions of Many Variables 3. Other Methods of Producing Interpolation Polynomials for Functions of Many Variables § 13. Interpolating a Function of a Complex Variable § 14. The Use of Interpolation in Compiling Tables § 15. Inverse Interpolation Exercises References Chapter 3 Numerical Differentiation and Integration § 1. Numerical Differentiation § 2. Formula for Numerical Differentiation 1. Formula for Non-Equidistant Nodes 2. Formula For Equidistant Nodes 3. Formula Without Differences 4. The Method of Undetermined Coefficients 5. The Expression of Differences by Derivatives § 3. Numerical Integration § 4. Newton-Coates Formula 1. Derivation 2. Residual Terms of the Formula 3. The Trapezoid Formula and the Simpson Formula § 5. Gauss Formula for Numerical Integration 1. Construction of the Formula. The Abscissae of the Gauss Formula 2. The Residual Term of the Gauss Formula 3. Coefficients of the Gauss Formula 4. The Hermite Numerical Integration Formula 5. The Markov Numerical Integration Formula § 6. Chebyshev Numerical Integration Formula 1. Producing the Formula 2. The Residual Term of the Chebyshev Formula § 7. Convergence in Quadrature Processes § 8. The Euler Formula 1. Bernoulli Polynomials and Numbers 2. The Euler Formula and Examples § 9. Formula Containing Differences of the Integrand 1. The Gregory Formula 2. The Laplace Formula and Other Formula § 10. A Few Remarks About Numerical Integration Formula 1. The "Runge" Method of Estimating the Error Innumerical Integration 2. Note on the Calculation of Integrals with a Variable Upper Limit § 11. Improper Integrals 1. The Method of Isolating Singularities 2. Special Methods § 12. Estimation of Multiple Integrals 1. The Method of Reapplying Quadrature Formula 2. Substitution of an Interpolation Polynomial for the Integrand 3. Lyusternik and Ditkin's Method 4. The Monte Carlo Method Exercises References Chapter 4 Approximations § 1. The Best Approximation in Linearly Normalized Spaces 1. Linearly Normalized Space 2. Best Approximations 3. The Existence of Best Approximations 4. The Uniqueness of Best Approximations § 2. The Best Approximation of Continuous Functions by Generalized Polynomials 1. The Best Approximation in Space C 2. The "Khaar" Theorem 3. The Chebyshev Theorem § 3. Algebraic Polynomials for the Best Approximation 1. The Weierstrass Theorem 2. The Theorem Concerning the Order of Approximation with Bernstein Polynomials § 4. Trigonometrical Polynomials of Best Approximation § 5. Certain Theorems Concerning the Order of the Best Approximation for Continuous Functions § 6. Devising Approximate Algebraic Polynomials for the Best Approximation 1. Preliminary Remarks 2. The First Method of Approximating the Polynomial for the Best Approximation 3. The Second Method of Approximating the Best Polynomial Approximation Exercises References Chapter 5 Least Square Approximations § 1. Hubert Space § 2. Ortho-Normal Sets in Hilbert Space. Fourier Series § 3. Approximations in Hilbert Space 1. Constructing the Best Approximation § 4. Least Square Approximation of Functions by Algebraic Polynomials 1. Orthogonal Sets of Polynomials 2. Recurrence Relations for Orthogonal Polynomials 3. The Christoffel-Darboux Identity 4. Properties of Orthogonal Polynomials § 5. Certain Special Cases of Orthogonal Sets of Polynomials 1. Jacobi Polynomials 2. Legendre Polynomials 3. Chebyshev Polynomials (Kinds 1 and 2) 4. Laguerre and Hermite Polynomials § 6. The Convergence of Series of Orthogonal Sets of Polynomials § 7. Least Square Approximations of Functions by Trigonometrical Polynomials § 8. Least Square Approximations of Tabulated Functions § 9. Least Square Approximations by Algebraic Polynomials 1. Orthogonal Sets of Polynomials in a Set of Equidistant Points § 10. Smoothing The Results of Observation by the Least Squares Method § 11. Empirical Formula by the Least Squares Method. The Solution of Linear Algebraic Equations by the Least Squares Method § 12. Least Square Approximations for Tabulated Functions by Trigonometrical Polynomials § 13. The Runge Method of Calculating the Coefficients A0, Aft, Bft If N = 4p Exercises References Index