Handbook of Numerical Methods for the Solution of Algebraic and Transcendental Equations

ForewordAuthor's PrefaceIntroductionChapter I - Initial Information About Polynomials and Transcendental Functions 1. Root of a Function 2. Basic Properties of Polynomials 3. Divisibility of Polynomials 3.1. Exact Division 3.2. Bezu's Theorem 3.3. Reduction of the Order of an Algebraic Equation 3.4. Euclid's Algorithm. Extraction of Multiple Roots 4. Changing the Argument of a Polynomial 4.1. The Process of Successive Division 4.2. Calculation of the Values of Derivatives 5. Polynomials with Real and Complex Coefficients 5.1. Complex Conjugate Numbers 5.2. Replacing an Equation with Complex Coefficients by an Equation with Real Coefficients 5.3. Replacement of an Equation with Complex Coefficients by Simultaneous Equations 5.4. The Method of A.P.Domoryad 5.5. The Conjugate Property of Complex Roots 5.6. Reduction of the Order of an Equation in the Case of a Complex Root 6. The Calculational Schemes for the Multiplication and Division of Polynomials 6.1. A Calculational Scheme for the Multiplication of Polynomials 6.2. The Strip Method 6.3. A Scheme for Division of One Polynomial by Another 6.4. The Strip Method (for Division) 7. Horner's Scheme and its Application. Calculation of the Value of a Polynomial 7.1. Calculational Scheme for a Transformation of the Form y = x — a of the Argument 7.2. Calculation of the Value of a Polynomial with Real Coefficients for a Complex Value of the Argument x = a + iβ 7.3. Calculation of the Derivative for a Complex Value of the Argument 8. The Number of Roots and the Limits for the Roots of a Polynomial 8.1. The Number of Roots 8.2. Determination of the Limits for Roots by Maclaurin's Theorem 8.3. Limits for Complex RootsChapter II - Operations with Approximate Numbers 1. An Approximate Number. Absolute and Relative Errors 1.1. The Writing of Approximate Numbers 1.2. Abbreviated Writing of Approximate Numbers 2. Rounding Off 3. Operations with Approximate Numbers 3.1. The Error of a Sum 3.2. The Error of a Product 3.3. Summation of Errors 4. The Error in Calculating the Value of a Polynomial 5. The Solution of Approximate Equations 5.1. Unconditional, Conditional and Complete Errors 5.2. Calculation of the Unconditional Error 6. Plan for the Solution of an Approximate Equation 6.1. Calculation of the Conditional Error 6.2. Example 7. Reduction of Accuracy when the Order of an Algebraic is LoweredChapter III - Methods for Approximate Determination of Roots 1. The Graphical Method 1.1. Solution of Equations 1.2. Some Peculiarities of Algebraic Equations 2. Approximate Determination of the Roots of a Polynomial by Means of Viet's Formulæ 2.1. Viet's Formulæ 2.2. Calculation of the Larger Roots 2.3. Calculation of the Smaller Roots 3. The Method of N.I.Lobachevskii 3.1. The Basic Idea of the Method 3.2. Transformation of a Polynomial 3.3. The Calculational Scheme for Transformation of a Polynomial 3.4. Calculation of the Moduli of Roots 3.5. Calculation of Roots whose Moduli are Known 3.6. A Pair of Unknown Roots 3.7. Two Pairs of Complex Roots with Different Moduli 3.8. The Brodetskii-Smil Method. The Basic Idea 3.9. The Brodetskii-Smil Method. The Calculational Scheme 3.10. Best's Formulæ 3.11. Loss of Accuracy in Lobachevskii's Method 3.12. Example 3.13. The Solution of Transcendental Equations 4. I.Bernoulli's Method 4.1. The Calculational Scheme for the Case of an Algebraic Equation 4.2. A Property of the Sequence {μk} 4.3. Calculation of the Roots 4.4. Calculation of the Second Largest Roots 4.5. Loss of Accuracy in Calculation of the First Root 4.6. Rapidity of Convergence 4.7. A General Observation 4.8. Example 4.9. The Solution of Transcendental Equations 5. The Method of Iteration 5.1. The Essence of the Method 5.2. The Conditions of Convergence 5.3. Rapidity of Convergence 5.4. Loss of Accuracy 5.5. Example 6. Lin's Method 6.1. The Essence of the Method 6.2. The Calculational Scheme 6.3. The Conditions and the Rapidity of Convergence 7. The Method of N.V.Paluver 7.1. The Basic Idea of the Method 7.2. The Calculational Scheme 8. Comparison of the MethodsChapter IV - Methods of Making More Accurate Roots Already Found 1. The Method of Linear Interpolation 1.1. Calculations in the Method of Linear Interpolation 1.2. Convergence of the Process 1.3. Example 2. Newton's Method 2.1. Calculations 2.2. On the Accuracy of the Calculations 2.2. Examples 2.4. Newton's Method Generalized 3. Berstoi's Method 3.1. The Essence of the Method 3.2. The Calculational Scheme 3.3. Example 3.4. Application of the Method of Calculating Complex Roots whose Moduli are Known 4. The Method of A.Y.Belostotskii 4.1. Essence of the Method 4.2. The Calculational Scheme 4.3. Example 5. Iteration with Quadratic Convergence 5.1. The Basic Idea of the Method 5.2. The First Method of Calculating K(x) 5.3. The Second Method of Calculating K(x) 5.4. Making Real Roots More Accurate 5.5. Making Quadratic Divisors More Accurate 6. Methods of Improving Convergence 6.1. Convergence Like that of a Geometrical Progression 6.2. Oscillatory Convergence 6.3. Quadratic Convergence 6.4. Application of Methods of Improving Convergence 6.5. Examples 6.6. Improvement of Convergence in Lin's Method 7. Comparison of the MethodsChapter V - Solution of Equations of Low Orders and Extraction of Roots 1. Quadratic Equations 1.1. General Formulæ 1.2. Calculation of the Roots by Means of a Slide Rule 2. Equations of Third Order 2.1. Solution by Means of Tables 2.2. Cardano's Formula 2.3. Calculation of Roots by Means of a Slide Rule 3. Equations of Fourth Order 4. Equations of Fifth Order 5. Extraction of Roots 5.1. Newton's Method 5.2. The Generalization of Newton's Method 5.3. The Method of Iteration 5.4. Iteration with Quadratic ConvergenceChapter VI - Solution of Simultaneous Equations 1. The Method of Elimination 2. The Graphical Method 3. The Method of Iteration 3.1. The Calculational Scheme 3.2. Conditions of Convergence 3.3. Zeidel's Method 3.4. Rapidity of Convergence 3.5. Example 4. Newton's Method 4.1. The Essence of the Method 4.2. Example 4.3. Generalized Methods 4.4. An Observation 5· The Method of Steepest Descent 6. Comparison of Methods of Solution of Simultaneous Non-Linear Equations 7. Methods of Solution of Simultaneous Linear EquationsAppendix - A Table for the Solution of Cubic EquationsReferencesIndex