Solution of Equations and Systems of Equations【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1．Introduction．Remainder Terms of Interpolation Formulas1

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 Point7

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．Iteration26

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 Roots32

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 Method40

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 Iteration43

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 Roots51

9．Fourier Bounds for Newton-Raphson Iterations55

10．Dandelin Bounds for Newton-Raphson Iterations60

11．Three Interpolation Points67

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 Equations73

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 Interpolation84

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 Root94

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 Matrices103

16．Two Theorems on the Convergence of Products of Matrices110

17．A Theorem on the Divergence of Products of Matrices113

18．Characterization of Points of Attraction and Repulsion for Iterations with Several Variables118

Appendices123

Appendix A．Continuity of the Roots of Algebraic Equations125

Appendix B．Relative Continuity of the Roots of Algebraic Equations130

Appendix C．An Explicit Formula for the nth Derivative of the Inverse Function141

Appendix D．Analog of the Regula Falsi for Two Equations with Two Unknowns146

Appendix E．Steffensen＇s Improved Iteration Rule148

Appendix F．The Newton-Raphson Algorithm for Quadratic Polynomials154

Appendix G．Some Modifications and an Improvement of the Newton-Raphson Method158

Appendix H．Rounding Off in Inverse Interpolation164

Appendix I．Accelerating Iterations with Superlinear Convergence175

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 Matrix192

BIBLIOGRAPHICAL NOTES195

INDEX201