Algorithms for exact polynomial root calculation

This paper discusses a set of algorithms which given a univariate polynomial with integer coefficients (with possible multiple zeros) and a positive rational error bound, uses infinite-precision integer arithmetic and Sturm's Theorem to compute intervals containing the real zeros of the polynomial and whose lengths are less than the given error bound. The algorithms also provide a simple means of determining the number of real zeros in any interval. Theoretical computing time bounds are developed for the algorithms and some empirical results are reported.

This paper presents a general analysis of all the quartic equations with real coefficients and multiple roots; this analysis revealed some unknown formulae to solve each kind of these equations and some precisions about the relation between these ones and the Resolvent Cubic; for example, it is well-known that any quartic equation has multiple roots whenever its Resolvent Cubic also has multiple roots; however, this analysis reveals that any non-biquadratic quartic equation and its Resolvent Cubic always have the same number of multiple roots; additionally, the four roots of any quartic equation with multiple roots are real whenever some specific forms of its Resolvent Cubic have three non-negative real roots. This analysis also proves that any method to solve third-degree equations is unnecessary to solve quartic equations with multiple roots, despite the existence of the Resolvent Cubic; finally, here is developed a generalized variation of the Ferrari Method and the Descartes Method, which help to avoid complex arithmetic operations during the resolution of any quartic equation with real coefficients, even though this equation has non-real roots; and a new, more simplified form of the discriminant of the quartic equations is also featured here.

We present an efficient exponentiation algorithm in a finite field GF(q n ) using a Gauss period of type (n,1). Though the Gauss period α of type (n,1) in GF(q n ) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of α which is a primitive element in GF(q n ). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(q n ), where a Gauss period of type (n,k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(q n )×.

A two-stage algorithm is proposed to compute switching angles for selective harmonic elimination or selective harmonic control. In the first offline computing stage, the multivariate nonlinear equations are equivalently converted to a group of univariate linear equations by using the theories of symmetric polynomials and Gröbner bases, which are then stored into the target microcontrollers (MCUs) and solved online. In the second online computing stage, the linear equations are solved, and their solutions are used to construct a polynomial whose real roots are actually the final solutions; then, an algebraic-numerical hybrid method is developed to solve the constructed polynomial, which uses the Sturm's theorem to bracket each real root into an individual interval, and then, the bisection method is used to find all the exact real roots. Compared to the traditional numerical methods, the executing efficiency is increased significantly. This method is implemented on several commonly used MCUs and shows very good real-time performance. Experiments verify the correctness of the solved switching angles and demonstrate the feasibility of applying this method in the real control system.

This paper presents a formally verified decision procedure for determinining the satisfiability of a system of univariate polynomial relations over the real line. The procedure combines a root counting function, based on Sturm's theorem, with an interval subdivision algorithm. Given a system of polynomial relations over the same variable, the decision procedure progressively subdivides the real interval into smaller intervals. The subdivision continues until the satisfiability of the system can be determined on each subinterval using Sturm's theorem on a subset of the system's polynomials. The decision procedure has been formally verified in the Prototype Verification System (PVS). In PVS, the decision procedure is specified as a computable boolean function on a deep embedding of polynomial relations. This function is used to define a proof producing strategy for automatically proving existential and universal statements on polynomial systems. The soundness of the strategy solely depends on the internal logic of PVS.

Congruence arithmetic algorithms for polynomial real zero determination

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = Q(α1,..., αe) is an algebraic extension of the rational numbers. Our bounds are single exponential in e and match the ones presented in [34] for the case e = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is OB(N4e+4), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes' rule of sign. We achieve a Boolean complexity of OB [equation], respectively. We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets.

Finite field arithmetic logic is central in the implementation of Reed-Solomon coders and in some cryptographic algorithms. There is a need for good multiplication and inversion algorithms that can be easily realized on VLSI chips. Massey and Omura recently developed a new multiplication algorithm for Galois fields based on a normal basis representation. In this paper, a pipeline structure is developed to realize the Massey-Omura multiplier in the finite field GF(2m). With the simple squaring property of the normal basis representation used together with this multiplier, a pipeline architecture is developed for computing inverse elements in GF(2m). The designs developed for the Massey-Omura multiplier and the computation of inverse elements are regular, simple, expandable, and therefore, naturally suitable for VLSI implementation.

The paper described a novel method of computing the trace of an element from a given finite field GF(pn). By applying the Newton Formula, this method shows a simplex and linear structure of traces of different elements, and the usual procedure of computing the products and sums of elements of GF(pn) is not required. The paper also studied The structure of the minimal polynomial of an element from GF(pn). A new fast way for computing the minimal polynomial of an element in GF(pn) is derived through combining the structure with the trace computation.

Historically, work on multiple roots for the internal rate of return (PW = 0) equation focuses on tests for the root's uniqueness and interpreting the meaningfulness of computed roots. Instead, this article starts with problems that consist of P, A, and F cash flows. Then for all possible values of these cash flows, this article develops the boundary conditions for no, single, and multiple real roots of either positive or negative sign. The results of characterizing these roots are applied to two examples as a foundation for conclusions about those multiple root problems and hypotheses about other problems with multiple roots. The first example is an important class of resource development projects that may require significant expenditures for environmental remediation at the conclusion of operation. The second example of buying versus leasing a home is often used to demonstrate the value of economic analysis to students.

Sturm's theorem finds much use in linear passive network synthesis. It forms a concluding part of the test for positive real functions, to determine the existence of any positive roots and their multiplicity. By applying the Sturm's theorem to a polynomial, its multiple roots are placed in evidence. The character and multiplicity of roots. of the polynomial under test is determined by inspection or by a simple test, or, if necessary, by repeated use of Sturm's test itself. While dealing with highorder polynomials, repeated use of Sturm's test is necessary, and a computer desirable, since Sturm's test becomes very complex. For a digital computer, the process should be based on a suitable algorithm capable of handling a general case. In this paper, an attempt has been made to present such an algorithm. A flow chart has been developed for this algorithm, process computerized and program tested on a digital computer.

Cellular automata (CA) have been accepted as a good evolutionary computational model for the simulation of complex physical systems. They have been used for various applications, such as parallel processing computations and number theory. In this paper, we studied the applications of cellular automata for the modular multiplications; we proposed two new architectures of multipliers based on cellular automata over finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ). Since they have regularity, modularity and concurrency, they are suitable for VLSI implementation. The proposed architectures can be easily implemented into the hardware design of crypto-coprocessors.

Tools for analyzing the intersection curve between two quadrics through projection and lifting

CHAPTER ELEVEN - ROOTS OF POLYNOMIALS

CHAPTER TEN - ROOTS OF POLYNOMIALS