# Numerical Methods for Roots of Polynomials

Produk Detail:
• Author : J.M. McNamee
• Publisher : Newnes
• Pages : 728 pages
• ISBN : 008093143X
• Rating : /5 from reviews

## Numerical Methods for Roots of Polynomials

• Author : J.M. McNamee,Victor Pan
• Publisher : Newnes
• Release : 19 July 2013

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

## Numerical Methods for Roots of Polynomials

• Author : J.M. McNamee
• Publisher : Elsevier
• Release : 17 August 2007

Numerical Methods for Roots of Polynomials - Part I (along with volume 2 covers most of the traditional methods for polynomial root-finding such as Newton’s, as well as numerous variations on them invented in the last few decades. Perhaps more importantly it covers recent developments such as Vincent’s method, simultaneous iterations, and matrix methods. There is an extensive chapter on evaluation of polynomials, including parallel methods and errors. There are pointers to robust and efficient programs. In short, it

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

First we consider the Jenkins–Traub 3-stage algorithm. In stage 1 we defineIn the second stage the factor is replaced by for fixed , and in the third stage by where is re-computed at each iteration. Then a root. A slightly different algorithm is given for real polynomials. Another class of methods uses minimization, i.e. we try to find such that is a minimum, where . At this minimum we must have , i.e. . Several authors search along the coordinate axes or

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

The zeros of a polynomial can be readily recovered from its linear factors. The linear factors can be approximated by first splitting a polynomial numerically into the product of its two nonconstant factors and then recursively splitting every computed nonlinear factor in similar fashion. For both the worst and average case inputs the resulting algorithms solve the polynomial factorization and root-finding problems within fixed sufficiently small error bounds by using nearly optimal arithmetic and Boolean time, that is using nearly

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

This chapter treats several topics, starting with Bernoulli’s method. This method iteratively solves a linear difference equation whose coefficients are the same as those of the polynomial. The ratios of successive iterates tends to the root of largest magnitude. Special versions are used for complex and/or multiple roots. The iteration may be accelerated, and Aitken’s variation finds all the roots simultaneously. The Quotient-Difference algorithm uses two sequences(with a similar one for ). Then, if the roots are

## Numerical Methods for Roots of Polynomials

• Author : J.M. McNamee,Victor Pan
• Publisher : Elsevier Science
• Release : 11 September 2013

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

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

We consider proofs that every polynomial has one zero (and hence n) in the complex plane. This was proved by Gauss in 1799, although a flaw in his proof was pointed out and fixed by Ostrowski in 1920, whereas other scientists had previously made unsuccessful attempts. We give details of Gauss’ fourth (trigonometric) proof, and also more modern proofs, such as several based on integration, or on minimization. We also treat the proofs that polynomials of degree 5 or more cannot in general

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

We deal here with low-degree polynomials, mostly closed-form solutions. We describe early and modern solutions of the quadratic, and potential errors in these. Again we give the early history of the cubic, and details of Cardan’s solution and Vieta’s trigonometric approach. We consider the discriminant, which decides what type of roots the cubic has. Then we describe several ways (both old and new) of solving the quartic, most of which involve first solving a “resolvent” cubic. The quintic

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

We discuss the secant method:where are initial guesses. In the Regula Falsi variation we start with initial guesses and such that ; after an iteration similar to the above we replace either a or b by the new value depending on which of or has the same sign as . Often one of the points gets “stuck,” and several variants such as the Illinois or Pegasus methods and variations are used to “unstick” it. We discuss convergence and efficiency of most

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

We discuss Graeffes’s method and variations. Graeffe iteratively computes a sequence of polynomialsso that the roots of are those of raised to the power . Then the roots of can be expressed in terms of the coefficients of . Special treatment is given to complex and/or multiple modulus roots. A method of Lehmer’s finds the argument as well as the modulus of the roots, while other authors show how to reduce the danger of overflow. Variants such as the

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

In considering the stability of mechanical systems we are led to the characteristic equation . Continuous-time systems are stable if all the roots of this equation are in the left half-plane (Hurwitz stability), while discrete-time systems require all (Schur stability). Hurwitz stability has been treated by the Cauchy index and Sturm sequences, leading to various determinantal criteria and Routh’s array, and several other methods. We also have to consider the question of robust stability, i.e. whethera system remains stable

## Initial Approximations and Root Finding Methods

• Author : Nikolay V. Kyurkchiev
• Publisher : Wiley-VCH
• Release : 27 October 1998

Polynomials as mathematical objects have been studied extensively for a long time, and the knowledge collected about them is enormous. Polynomials appear in various fields of applied mathematics and engineering, from mathematics of finance up to signal theory or robust control. The calculation of the roots of a polynomial is a basic problems of numerical mathematics. In this book, an update on iterative methods of calculating simultaneously all roots of a polynomial is given: a survey on basic facts, a

## Numerical Methods for Roots of Polynomials Part II

• Author : J.M. McNamee,V.Y. Pan
• Publisher : Elsevier Inc. Chapters
• Release : 19 July 2013

Whereas Newton’s method involves only the first derivative, methods discussed in this chapter involve the second or higher. The “classical” methods of this type (such as Halley’s, Euler’s, Hansen and Patrick’s, Ostrowski’s, Cauchy’s and Chebyshev’s) are all third order with three evaluations, so are slightly more efficient than Newton’s method. Convergence of some of these methods is discussed, as well as composite variations (some of which have fairly high efficiency). We describe