声明
ACKNOWLEDGMENTS
ABSTRACT
摘要
CONTENTS
LIST OF FIGURES
LIST OF TABLES
CHAPTER 1 INTRODUCTION
1．1 Literature reviews
1．2 Outlines of the Dissertation
CHAPTER 2 MATHEMATICAL PRELIMINARY CONCEPTS
2．1 Introduction to polynomial roots and their perturbations
2．2 Reviews of some mathematical terms
2．3 Sources of nonlinear equations
2．4 Formulating problems and steps in mathematical methods
2．4．1 Construction of iterative methods
2．5 The performance analyses of iterative algorithms
2．6 Complexity and generic choice of an algorithm
CHAPTER 3 REVIEWS OF NUMERICAL ERRORS AND THEIR SOURCES
3．1．Types and sources of errors
3．1．1 Basic introduction of errors
3．1．2 Round off errors
3．1．3 Truncation errors (in function approximations)
3．1．4 Errors in function evaluations(computations of functions)
3．1．5 Arithmetic errors
3．1．6 Errors due to numerical algorithm for a given numerical method
3．2 Errors，Stability and conditioning of a problem
3．2．1 Reviews of frequently used Vector norms
3．2．2 Conditioning of mathematical problem
3．2．3 Conditioning of root finding for real valued scalar equations
3．3 Errors in floating point operations
CHAPTER 4 THE CONCEPT OF MODELlNG AND ERRORS
4．1 Introduction to modeling
4．1．1 Modeling-Definition
4．2 Things of interests in modeling
4．3 Interpolation models and iterative methods
4．3．1 Newton’s forward finite difference interpolation
4．3．2 Lagrange’s Interpolation
4．3．3 Generalized Newton’s Formula (Newton’s Divided Difference Formula，NDDF)
4．3．4 Lagrange’s inverse-Interpolation
4．3．5 Errors in the interpolation models
4．4 Stability of numerical methods in model solutions
4．4．1 The two main stopping criteria for iterative methods
4．4．2 A priori and a posteriori errors analyses
4．4．3 Modeling and measures to reduce computational errors
4．5 Example on source of nonlinear (models) equations
4．5．1 Stability analysis of numerical ODE as a source of nonlinear equations
CHAPTER 5 SCALAR POLYNOMIALS AND ILL-CONDITIONING
5．1 Introduction to the solution of scalar equations
5．2 Basic concepts on roots of polynomial equations
5．3 Tests for existence of roots or zeros and root locations
5．3．1 Location principle for root of a polynomial and its derivative
5．3．2 Bound of Real roots
5．3．3 Rational root test
5．4 Graphical-Method [Geometric Approach]
5．5 Further discussions on polynomial roots
5．5．1 Center of mass and the convex hull of roots
5．6 Roots of a polynomial and its derivatives (Bounds and localizations)
5．6．1 Localization of the (complex) roots of the derivatives
5．6．2 Lehmer’s method
5．6．3 Common roots of two polynomials
5．6．4 Separation of Roots
5．6．5 The relations of coefficients and roots and degree of a polynomial
5．6．6 Division algorithm and Sturm sequence
5．6．7 Factoring a polynomial (remainder theorem and factor theorem)
5．6．8 Solving cubic and quartic polynomials，Cardano’s formula
5．7 The Horner’s method and polynomial representations
5．7．1 Horner’s nested multiplication algorithm for polynomials
5．7．2 Synthetic Division algorithm (SYNDV)
5．8 perturbation methods for solving roots of algebraic equations
5．9 Software root solvers (SRS)
5．9．1 Matlab root solvers (MRS)
5．9．2 Maple root finders
5．9．3 Mathematica root solvers
5．10 The Bernstein polynomials and applications
5．11 Sensitivity of a polynomial root to the perturbation in its coefficients (One main focus of the thesis)
CHAPTER 6 ITERATIVE ALGORITHMS FOR SCALAR NONLINEAR EQUATIONS
6．1 Induction
6．1．1 Some existing iterative methods
6．1．2 Bisection Method (BM)
6．1．3 Regula Falsi Method (RFM)
6．1．4 Newton’s Method
6．1．5 Fixed-Point Iteration Method (FPM)
6．1．6 The secant method (SM)
6．1．7 Steffensen’s Method and other derivative free methods
6．1．8 Iterative Methods using Quadratic Interpolation
6．2 Methods with higher order derivatives
6．3 Classical Methods
6．4 Algorithms using perturbation theory and Taylor’s series
6．5 Algorithms using Taylor’s Approximation (Extensions of Newton’s formula)
6．6 Iterative method for multiple roots
6．6．1 Generalized Newton’s method(modified Newton’s Method)
6．7 Variational methods for multiple roots
6．8 Simultaneous root finders
6．9 Rate and order of convergence
CHAPTER 7 FURTHER ANALYSES AND SUGGESTIONS OF NEW ITERATIVE METHODS
7．1 New iterative methods for simple roots using function construction
7．2 Iterative algorithms using derivative estimations in Taylor’s third order interpolation
7．2．1 A statement of the higher derivative estimations
7．2．2 Iterative methods applying derivative estimations
7．2．3 Convergence analysis
7．2．4 Test equations and numerical results
7．2．5 Summary
7．3 Construction of iterative methods for multiple roots
7．3．1 Newton method for multiple roots and Newton-correction
7．3．2 Construction of third order methods for multiple roots
7．4 Convergence analysis
7．5 Test equations and numerical results
7．6 Summary (Concluding remarks)
7．7 Application of Root finding
7．7．1 Application of root finding in the stability analysis of numetical ODE (FE，EB，RK2，RK3，RK4)
7．7．2．Application in model solutions
7．8 perturbation effect of a polynomial on the results of an iterative algorithm
7．9 C++ implementations
7．10 Numerical results
7．11 Research summary
REFERENCES
