New progress in real and complex polynomial root-finding

Abstract

a b s t r a c tMatrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out to be competitive with the most popular root-finders because of its excellence in exploiting matrix structure. To advance the iteration we preprocess the matrix and incorporate Newton's linearization, repeated squaring, homotopy continuation techniques, and some heuristics. The resulting algorithms accelerate the known numerical root-finders for univariate polynomial and secular equations, and are particularly well suited for the acceleration by using parallel processing. Furthermore, even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots.

Local Convergence of Solvers with Eighth Order Having Weak Conditions

“…If the zero has been determined with sufficient accuracy, the polynomial is deflated and the algorithm is applied again on the deflated polynomial. In this way, we can determine all zeros simultaneously and also have theoretical importance [8] for the details of methods can be seen in [9][10][11][12][13][14][15][16][17][18][19]. Therefore, we have extended amount of iterative solvers to solve problems like expression (1).…”

Characterization and Enumeration of Good Punctured Polynomials over Finite Fields

“…From the fundamental theorem of algebra, every polynomial over the rational numbers Q (or over the real numbers R) has a root in C. However, it is not guaranteed that a polynomial has a root in Q or in R. Therefore, for a given polynomial over Q (resp., R), it is of natural interest to determine whether it has a root in Q (resp., R). In general, determining whether a given polynomial has a root in a nonalgebraically closed field is an interesting problem and has been extensively studied (see, e.g., [1][2][3][4]).…”

The Polynomial Pivots as Initial Values for a New Root-Finding Iterative Method

“…In addition, McNamee [6] has proposed an indicator to measure the efficiency of an iterative method and has applied it to the most common approaches. Other relevant reviews on algorithms to search zeros have been presented by Pan [11][12][13][14] and by Pan and Zheng [14].…”