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- 出版年份：1960
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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