Third Edition of Solution of Equations and Systems of Equations

Content: 
Edited by
Page iii

Copyright page
Page iv

Preface to the Third Edition
Pages xiii-xv

List of Notations and Abbreviations
Pages xvii-xx

1A Divided Differences
Pages 1-7

1B Confluent Case. Interpolation
Pages 8-17

2 Inverse Interpolation. Derivatives of the Inverse Function. One Interpolation Point
Pages 18-26

3 Method of False Position (Réegula Falsi)
Pages 27-37

4 Iteration
Pages 38-46

5 Further Discussion of Iterations Multiple Zeros
Pages 47-52

6 The Newton—Raphson Method
Pages 53-55

7 Fundamental Existence Theorems in the Newton-Raphson Iteration
Pages 56-60

8 An Analog of the Newton-Raphson Method for Multiple Roots
Pages 61-68

9 Fourier Bounds for the Newton-Raphson Iteration
Pages 69-72

10 Dandelin Bounds for the Newton-Raphson Iteration
Pages 73-78

11 Three Interpolation Points
Pages 79-83

12 Linear Difference Equations
Pages 84-93

13 n Distinct Points of Interpolation
Pages 94-102

14 n+1 Coincident Interpolation Points and Taylor Development of the Root
Pages 103-109

15 The Square Root Iteration
Pages 110-118

16 Further Discussion of Square Root Iteration
Pages 119-126

17 A General Theorem on Zeros of Interpolating Polynomials
Pages 127-130

18 Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Simple Roots
Pages 131-134

19 Norms of Vectors and Matrices
Pages 135-142

20 Two Theorems on Convergence of Products of Matrices
Pages 143-145

21 A Theorem on Divergence of Products of Matrices
Pages 146-149

22 Characterization of Points of Attraction and Repulsion for Iterations with Several Variables
Pages 150-154

23 Euclidean Norms
Pages 155-158

24 Minkowski Norms, Δp (A), Δp,p′ (A)
Pages 159-165

25 Method of Steepest Descent. Convergence of the Procedure
Pages 166-172

26 Method of Steepest Descent. Weakly Linear Convergence of the ζμ
Pages 173-177

27 Method of Steepest Descent. Linear Convergence of the ζμ
Pages 178-185

28 Convergent Procedures for Polynomial Equations
Pages 186-193

29 J-Test and J-Routine
Pages 194-199

30 q-Acceleration. The Practice of the Procedure
Pages 200-206

31 Normed Linear Spaces
Pages 207-212

32 Metric Spaces
Pages 214-218

33 Operators in Normed Linear Spaces
Pages 219-223

34 Inverse Operators
Pages 224-228

35 Operators Mapping a Linear Interval
Pages 229-235

36 The Directional Derivatives and Gradients of Operators
Pages 236-240

37 Central Existence Theorem
Pages 241-246

38 Newton-Raphson Iteration in Banach Spaces. Statement of the Theorems
Pages 247-252

39 Proof of Theorems 38.1–38.3
Pages 253-259

40 Complements to the Newton-Raphson Method
Pages 260-264

41 Central Existence Theorem for Finite Systems of Equations
Pages 265-269

42 Newton-Raphson Iteration for Finite Systems of Equations
Pages 270-274

A Continuity of the Roots of Algebraic Equations
Pages 276-280

B Relative Continuity of the Roots of Algebraic Equations
Pages 281-289

C An Explicit Formula for the nth Derivative of the Inverse Function
Pages 290-293

D Analog of the Regula Falsi for Two Equations with Two Unknowns
Pages 294-295

E Steffensen\'s Improved Iteration Rule
Pages 296-300

F The Newton-Raphson Algorithm for Quadratic Polynomials
Pages 301-305

G Some Modifications and Improvements of the Newton-Raphson Method
Pages 306-309

H Rounding Off in Inverse Interpolation
Pages 310-319

I Accelerating Iterations with Supralinear Convergence
Pages 320-323

J Roots of f(z) = 0 in Terms of the Coefficients of the Development of 1/f(z)
Pages 324-333

K Continuity of the Fundamental Roots as Functions of the Elements of the Matirx
Pages 334-335

L The Determinantal Formulas for Divided Differences
Pages 336-338

M Remainder Terms in Interpolation Formulas
Pages 339-342

N Generalization of Schröder\'s Series to the Case of Multiple Roots
Pages 343-352

O Laguerre Iterations
Pages 353-362

P Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Multiple Zeros
Pages 363-371

Q Feedback Techniques for Error Estimates
Pages 372-377

R Reduced Polynomial Equations
Pages 378-382

S Discussion of the q-Acceleration
Pages 383-388

T Remainder in the Taylor Formula for Analytic Functions
Pages 389-390

U Equality Conditions for the Newton-Raphson Iteration
Pages 391-398

Bibliographical Notes
Pages 399-407

Index
Pages 409-412