Solution of Equations and Systems of Equations

Preface to the First EditionPreface to the Second Edition1. Divided Differences Divided Differences for Distinct Arguments Symmetry Integral Representation Mean Value Formulas Divided Differences with Repeated Arguments A Formula for Confluent Divided Differences Newton's Interpolation Formula General Interpolation Problem Polynomial Interpolation The Remainder for a General Interpolating Function Triangular Schemes for Computing Divided Differences2. Inverse Interpolation. Derivatives of the Inverse Function. One Interpolation Point The Concept of Inverse Interpolation Darboux's Theorem on Values of f(x) Derivatives of the Inverse Function One Interpolation Point A Development of a Zero of f(x)3. Method of False Position (regula Falsi) Definition of the Regula Falsi Use of Inverse Interpolation Geometric Interpretation (Fourier's Conditions) Iteration with Successive Adjacent Points Homer Units and Efficiency Index The Rounding-Off Rule Locating the Zero with the Regula Falsi Examples of Computation by the Regula Falsi4. Iteration A Convergence Criterion for an Iteration Points of Attraction and Repulsion Improving the Convergence5. Further Discussion of Iterations. Multiple Zeros Iterations by Monotonic Iterating Functions Multiple Zeros Connection of the Regula Falsi with the Theory of Iteration6. Newton-Raphson Method The Idea of the Newton-Raphson Method The Use of Inverse Interpolation Comparison of Regula Falsi and Newton-Raphson Method7. Fundamental Existence Theorems for Newton-Raphson Iteration Error Estimates a Priori and a Posteriori Fundamental Existence Theorems8. An Analog of the Newton-Raphson Method for Multiple Roots9. Fourier Bounds for Newton-Raphson Iteration10. Dandelin Bounds for Newton-Raphson Iteration11. Three Interpolation Points Interpolation by Linear Fractions Two Coincident Interpolation Points Error Estimates Use in Iteration Procedure12. Linear Difference Equations Inhomogeneous and Homogeneous Difference Equations General Solution of the Homogeneous Equation Lemma on Division of Power Series Asymptotic Behavior of Solutions of (12.1) Asymptotic Behavior of Errors in the Regula Falsi Iteration A Theorem on Roots of Certain Equations13. n Distinct Points of Interpolation Error Estimates Iteration with n Distinct Points of Interpolation Discussion of the Roots of Some Special Equations14. n + 1 Coincident Points of Interpolation and Taylor Development of the Root Statement of the Problem A Theorem on Inverse Functions and Conformal Mapping Theorem on the Error of the Taylor Approximation to the Root Discussion of the Conditions of the Theorem15. The Square Root Iteration16. Further Discussion of Square Root Iteration17. A General Theorem on Zeros of interpolating Polynomials18. Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Simple Roots19. Norms of Vectors and Matrices20. Two Theorems on Convergence of Products of Matrices21. A Theorem on Divergence of Products of Matrices22. Characterization of Points of Attraction and Repulsion for Iterations with Several Variables An Example23. Further Discussion of Norms Matrices △q(A) Triangle Inequality Bilinear and Quadratic Forms of Symmetric Matrices Estimate of △p(ABC) Variation of △p(A-l) Length of Arc in the