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1．Introduction．Remainder Terms of Interpolation Formulas 1

Lemma on Functions with Several Roots(§§ 2-4) 1

Theorem on Quotients of Functions with Common Roots(§§ 5-6) 2

Interpolating Functions(§ 7) 3

Remainder Term in General Interpolation(§ 8) 4

Hermite Interpolating Polynomial(§§ 9-10) 5

2．Inverse Interpolation．Derivatives of the Inverse Function One Interpolation Point 7

The Concept of Inverse Interpolation(§§ 1-2) 7

Darboux＇s Theorem on Values of f＇(x)(§ 3) 8

Derivatives of the Inverse Function(§§ 4-8) 9

One Interpolation Point(§ 9) 11

3．Method of False Position(Regula Falsi) 13

Definition of the Regula Falsi(§§ 1-2) 13

Use of Inverse Interpolation(§§ 3-5) 14

Geometric Interpretation(Fourier＇s Conditions)(§§ 6-7) 16

Iteration with Successive Adjacent Points(§§ 8-10) 17

Horner Units and Efficiency Index(§ 11) 19

The Roundihg-Off Rule(§§ 12-14) 20

Locating the Root with the Regula Falsi(§§ 15-16) 22

Examples of Computation by the Regula Falsi(§ 17) 23

4．Iteration 26

A Convergence Criterion for an Iteration(§ 1) 26

Points of Attraction and Repulsion(§§ 2-5) 26

Improving the Convergence(§§ 6-8) 28

5．Further Discussion of Iterations．Multiple Roots 32

Iterations by Monotonic Iterating Functions(§§ 1-5) 32

Multiple Roots(§§ 6-9) 34

Connection of the Regula Falsi with the Theory of Iteration(§ 10) 38

6．Newton-Raphson Method 40

The Idea of the Newton-Raphson Method(§ 1) 40

The Use of Inverse Interpolation(§§ 2-3) 41

Comparison of Regula Falsi and Newton-Raphson(§ 4) 42

7．Fundamental Existence Theorems in the Newton-Raphson Iteration 43

Error Estimates a Priori and a Posteriori(§ 1) 43

Fundamental Existence Theorems(§§ 2-11) 43

8．An Analog of the Newton-Raphson Method for Multiple Roots 51

9．Fourier Bounds for Newton-Raphson Iterations 55

10．Dandelin Bounds for Newton-Raphson Iterations 60

11．Three Interpolation Points 67

Interpolation by Linear Fractions(§ 1) 67

Two Coincident Interpolation Points(§§ 2-3) 68

Error Estimates(§§ 4-5) 69

Use in Iteration Procedure(§§ 6-7) 71

12．Linear Difference Equations 73

Inhomogeneous and Homogeneous Difference Equations(§§ 1-3) 73

General Solution of the Homogeneous Equation(§§ 4-5) 74

Lemmata on Division of Power Series(§§ 6-7) 75

Asymptotic Behavior of Solutions(§§ 8-10) 77

Asymptotic Behavior of Errors in the Regula Falsi Iteration(§§ 11-12) 80

A Theorem on Roots of Certain Equations(§ 13) 82

13．n Distinct Points of Interpolation 84

Error Estimates(§§ 1-2) 84

Iteration with n Distinct Points of Interpolation(§§ 3-7) 86

Discussion of the Roots of the Characteristic Polynomial(§§ 8-18) 87

14．n+1 Coincident Points of Interpolation and the Taylor Development of the Root 94

Statement of the Problem(§ 1) 94

A Theorem on Inverse Functions and Conformal Mapping(§§ 2-6) 94

Theorem on the Error of the Taylor Approximation to the Root(§§ 7-8) 98

Discussion of the Conditions of the Theorem(§§ 9-11) 99

15．Norms of Vectors and Matrices 103

16．Two Theorems on the Convergence of Products of Matrices 110

17．A Theorem on the Divergence of Products of Matrices 113

18．Characterization of Points of Attraction and Repulsion for Iterations with Several Variables 118

Appendices 123

Appendix A．Continuity of the Roots of Algebraic Equations 125

Appendix B．Relative Continuity of the Roots of Algebraic Equations 130

Appendix C．An Explicit Formula for the nth Derivative of the Inverse Function 141

Appendix D．Analog of the Regula Falsi for Two Equations with Two Unknowns 146

Appendix E．Steffensen＇s Improved Iteration Rule 148

Appendix F．The Newton-Raphson Algorithm for Quadratic Polynomials 154

Appendix G．Some Modifications and an Improvement of the Newton-Raphson Method 158

Appendix H．Rounding Off in Inverse Interpolation 164

Appendix I．Accelerating Iterations with Superlinear Convergence 175

Appendix J．Roots of f(z)=0 from the Coefficients of the Development of l/f(z) 180

Appendix K．Continuity of the Fundamental Roots as Functions of the Elements of the Matrix 192

BIBLIOGRAPHICAL NOTES 195

INDEX 201