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We study the following problem: given a regular symmetric form (linear functional) v, find all the regular symmetric forms u which satisfy the equation x 3 u = λ x v , λ ∈ C − { 0 }. We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. Moreover, in the case where v is a semi-classical form of class s, we show that u is semi-classical and its class is analyzed in term of the class of v. An example is highlighted. [ABSTRACT FROM AUTHOR]

It is known that the immediate basins of attraction of zeros of a complex polynomial f(z) with simple zeros under the iteration of family of iterations, i.e., Schröder's formula of the second kind (König's formula or the Basic Family) with the convergence degree m ≥ 2, converge to the Voronoi diagram of the zeros as m → ∞ . In a previous work, we consider a region, i.e., the intersection of the regions surrounded by the circles of Apollonius for zeros and show that for each zero, this region included in the immediate basin of attraction of the zero converges to the Voronoi cell of the zero as m → ∞ . Here, for a polynomial f(z) with multiple zeros, by extending our previous work, we show how the immediate basins of attraction of the family to the rational function f (z) / f ′ (z) , whose zeros are simple, converge to the Voronoi diagram. Honorato has shown a similar result, but our approach is more direct and quantitative. Numerical examples illustrate the relationship between the regions above, the immediate basins of attraction and the Voronoi cells, and show that the region above monotonically enlarges to the Voronoi cell. [ABSTRACT FROM AUTHOR]

Resultants are important special functions used to describe nonlinear phenomena. The resultant $$R_{r_1 \ldots r_n }$$ determines a consistency condition for a system of n homogeneous polynomials of degrees r, ..., r in n variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new iteration scheme is proposed to solve the roots of an algebraic equation f(x)=0. Given an initial guess, x 0, the roots of the equation can be obtained using the following iteration scheme:This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods. [Copyright &y& Elsevier]

We obtain an integral formula for a solution to a general algebraic equation. In this formula the integrand is an elementary function and integration is carried out over an interval. The advantage of this formula over the well-known Mellin formula is that the integral has a broader convergence domain. This circumstance makes it possible to describe the monodromy of a solution for trinomial equations. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new algorithm based on Adomian method to solve algebraic equations. [Copyright &y& Elsevier]

We further develop our recent direct method for the optimized effective potential (OEP) in density functional theory (DFT) [Yang and Wu, Phys. Rev. Lett.89, 143002 (2002)]. First, we show that the stationary condition in our optimization approach leads to a proper nonlinear algebraic equation for the OEP in a finite basis set, which differs from other finite basis set approaches. Then by constructing an approximate second derivative matrix of the energy functional in conjunction with the use of the Newton method, we significantly accelerate the convergence of the iterative optimization for OEP. Enhancement of the method is made in using the Tikhonov regularization method for the inversion of the second derivative matrix when it is singular or nearly singular and the direct inversion in the iterative space. It is shown that under a fixed stepsize condition, the optimization approach is equivalent to the self-consistent solution to the nonlinear algebraic equation for OEP. Because the approximate second derivatives are easy to compute and the iteration numbers are small now, the computation costs of OEP become comparable to that of regular DFT calculations as shown by calculations of some molecules, small and larger ones. We show how to find balanced results between energies and potentials when choosing a basis set for potentials. [ABSTRACT FROM AUTHOR]