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This book sets out to state computationally verifiable initial conditions for predicting the immediate appearance of the guaranteed and fast convergence of iterative root finding methods. Attention is paid to iterative methods for simultaneous determination of polynomial zeros in the spirit of Smale's point estimation theory, introduced in 1986. Some basic concepts and Smale's theory for Newton's method, together with its modifications and higher-order methods, are presented in the first two chapters. The remaining chapters contain the recent author's results on initial conditions guaranteing convergence of a wide class of iterative methods for solving algebraic equations. These conditions are of practical interest since they depend only on available data, the information of a function whose zeros are sought and initial approximations. The convergence approach presented can be applied in designing a package for the simultaneous approximation of polynomial zeros.

Fix-point estimation --- Equations, Roots of --- Mathematical Statistics --- Applied Mathematics --- Mathematics --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Fix-point estimation. --- Equations, Roots of. --- Roots of equations --- Point estimation --- Mathematics. --- Numerical analysis. --- Numerical Analysis. --- Mathematical analysis --- Math --- Science --- Estimation theory

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Numerical Methods for Roots of Polynomials - Part II along with Part I (9780444527295) covers most of the traditional methods for polynomial root-finding such as interpolation and methods due to Graeffe, Laguerre, and Jenkins and Traub. It includes many other methods and topics as well and has a chapter devoted to certain modern virtually optimal methods. Additionally, there are pointers to robust and efficient programs. This book is invaluable to anyone doing research in polynomial roots, or teaching a graduate course on that topic. First comprehensive treatment of Root-

Equations, Roots of. --- Polynomials -- Mathematical models. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Polynomials --- Mathematical models. --- Mathematical models --- Roots of equations

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On the Higher-Order Sheffer Orthogonal Polynomial Sequences sheds light on the existence/non-existence of B-Type 1 orthogonal polynomials. This book presents a template for analyzing potential orthogonal polynomial sequences including additional higher-order Sheffer classes. This text not only shows that there are no OPS for the special case the B-Type 1 class, but that there are no orthogonal polynomial sequences for the general B-Type 1 class as well. Moreover, it is quite provocative how the seemingly subtle transition from the B-Type 0 class to the B-Type 1 class leads to a drastically more difficult characterization problem. Despite this issue, a procedure is established that yields a definite answer to our current characterization problem, which can also be extended to various other characterization problems as well. Accessible to undergraduate students in the mathematical sciences and related fields, This book functions as an important reference work regarding the Sheffer sequences. The author takes advantage of Mathematica 7 to display unique detailed code and increase the reader's understanding of the implementation of Mathematica 7 and facilitate further experimentation. In addition, this book provides an excellent example of how packages like Mathematica 7 can be used to derive rigorous mathematical results.

Equations, Roots of. --- Number theory -- Congresses. --- Polynomials -- Congresses. --- Polynomials -- Mathematical models. --- Orthogonal polynomials --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Orthogonal polynomials. --- Sequences (Mathematics) --- Mathematical sequences --- Numerical sequences --- Mathematics. --- Matrix theory. --- Algebra. --- Computer mathematics. --- Linear and Multilinear Algebras, Matrix Theory. --- Computational Science and Engineering. --- Fourier analysis --- Functions, Orthogonal --- Polynomials --- Computer science. --- Informatics --- Science --- Computer mathematics --- Electronic data processing --- Mathematical analysis