Integer Arithmetic Algorithms for Polynomial Real Zero Determination

This thesis discusses two sets of fully specified algorithms which, given a univariate polynomial with integer coefficients (with possible multiple roots) and a positive rational error bound, uses infinite-precision arithmetic and Sturm's Theorem to compute intervals containing the real roots 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 roots in any interval.The first set of algorithms uses infinite-precision integer arithmetic to compute the sequence of polynomials required by Sturm's Theorem and also to do all the required polynomial evaluations. The second set uses congruence arithmetic to compute the images of the sequence of polynomials over the finite fields GF(p i ) for several primes p i and then uses the natural homomorphisms from the integers onto the finite fields GF(p i ) and the Chinese Remainder Theorem to do the evaluations. The primary advantage of the use of congruence arithmetic in the second set of algorithms is that the computing time to multiply any two elements of GF(p i ) is a constant whereas the time to multiply any two integers is proportional to the square of their lengths.The computing times of both sets of algorithms are analyzed in detail using the techniques of computing time analysis developed in the past few years by G. E. Collins and D. Knuth. It is shown that if P is a primitive, positive, univariate polynomial over the integers of positive degree, m, whose coefficients are bounded in magnitude by d, and which has L real roots, and ε is some positive rational number, 0 < ε ≤ 1, then the computing time to approximate the roots of P to within is:[EQUATION]using either set of algorithms.The empirical results of applying the two sets of algorithms to random polynomials and the first ten Legendre and Chebyshev polynomials are reported. Neither set of algorithms proved to be faster for all the polynomials tested.

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.

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.

Starting from Laguerre's method and using Newton's and Halley's corrections for a multiple zero, new simultaneous methods of Laguerre's type for finding multiple (real or complex) zeros of polynomials are constructed. The convergence order of the proposed methods is five and six, respectively. By applying the Gauss–Seidel approach, these methods are further accelerated. The lower bounds of the R-order of convergence of the improved (single-step) methods are derived. Faster convergence of all proposed methods is attained with negligible number of additional operations, which provides a high computational efficiency of these methods. A detailed convergence analysis and numerical results are given.

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.

In this work, we propose a new method to find monic irreducible polynomials with integer coefficients, only real roots, and span less than 4. The main idea is to reduce the search of such polynomials to the solution of Integer Linear Programming problems. In this frame, the coefficients of the polynomials we are looking for are the integer unknowns. We give inequality constraints specified by the properties that the polynomials should have, such as the typical distribution of their roots. These properties can be inferred from those of polynomials already treated in the literature on this topic.

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=Qα is a simple algebraic extension of the rational numbers. We revisit two approaches for the problem. In the first approach, using resultant computations, we perform a reduction to a polynomial with integer coefficients and we deduce a bound of OB(N10) for isolating the real roots of Bα, where N is an upper bound on all the quantities (degree and bitsize) of the input polynomials. In the second approach we isolate the real roots working directly on the polynomial of the input. We compute improved separation bounds for the roots and we prove that they are optimal, under mild assumptions. For isolating the real roots we consider a modified Sturm algorithm, and a modified version of Descartes' algorithm introduced by Sagraloff. For the former we prove a complexity bound of OB(N8) and for the latter a bound of OB(N7). We implemented the algorithms in C as part of the core library of Mathematica and we illustrate their efficiency over various data sets. Finally, we present complexity results for the general case of the first approach, where the coefficients belong to multiple extensions.

Our results concern the Macdonald q, t-Kostka coefficients K λµ(q, t). More precisely we work here with the expressions \(\tilde K_{\lambda \mu } (q,t) = t^{n(\mu )} K_{\lambda \mu } (q,1/t)\) where n(µ) denotes the sum of the legs of the cells of µ. The K λµ (q, t) have been conjectured by Macdonald to be polynomials in q, t with positive integer coefficients. We prove here the Macdonald conjecture for arbitrary µ when λ is an augmented hook. Our proof is based on explicit formulas yielding \(\tilde K_{\lambda \mu } (q,t)\) as a symmetric polynomial plethystically evaluated at \(\tilde K_{(n - 1,1),\mu } (q,t)\).More precisely, it can be shown that for each γ ⊢ k there is a unique symmetric polynomial kγ (x; q, t), of degree ≤ k in x, such that for any λ of the form λ = (n − k, γ), we have \(\tilde K_{(n - 1,1),\mu } (q,t) = k_\lambda \left[ {\tilde K_{(n - 1,1),\mu } (q,t);q,t} \right]\). The proof of existence of the polynomials kγ (x; q, t) is algorithmic. There are now two separate algorithms yielding kγ (x; q, t). We present here the chronologically first one. The second one is presented in a paper by Garsia-Tesler [12] where it yields that kγ (x;q, t) is a polynomial with integer coefficients in x, q, t, 1/q, 1/t. Although the first algorithm is not as good with denominators, we present it here since, combined with some recent work of Lapointe-Vinet, it yields what may be the simplest proof that the K λµ (q, t) are polynomials with integer coefficients.

We provide irreducibility criteria for multivariate polynomials with coefficients in an arbitrary field that extend a classical result of Pólya for polynomials with integer coefficients. In particular, we provide irreducibility conditions for polynomials of the form f(X)(Y − f 1(X))…(Y − f n (X)) + g(X), with f, f 1, ⋅, f n , g univariate polynomials over an arbitrary field.

This paper gives an algorithm for finding the irreducible factors of any multivariate polynomial with integer coefficients. The algorithm begins by making substitutions for all but one of the variable. This univariate polynomial is then factored by a known method, which uses an algorithm of Berlekamp for factoring univariate polynomials over finite fields. After this factorization is done, the multivariate factors are recovered from the univariate ones by a kind of Hensel algorithm. A number of ideas are given which greatly speed the computation in some special cases.

The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [11] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is to put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.

Utilising a polar-plot approach, a theorem by Mitra and Hurtig on positive real (p.r.) functions with noninterlaced real distinct poles and zeros is shown to be a special case of a generalised theorem on p.r. functions with noninterlaced real multiple poles and zeros.

Very recent work introduces an asymptotically fast subdivision algorithm, denoted ANewDsc, for isolating the real roots of a univariate real polynomial. The method combines Descartes' Rule of Signs to test intervals for the existence of roots, Newton iteration to speed up convergence against clusters of roots, and approximate computation to decrease the required precision. It achieves record bounds on the worst-case complexity for the considered problem, matching the complexity of Pan's method for computing all complex roots and improving upon the complexity of other subdivision methods by several magnitudes. In the article at hand, we report on an implementation of ANewDsc on top of the RS root isolator. RS is a highly efficient realization of the classical Descartes method and currently serves as the default real root solver in Maple. We describe crucial design changes within ANewDsc and RS that led to a high-performance implementation without harming the theoretical complexity of the underlying algorithm. With an excerpt of our extensive collection of benchmarks, available online at http://anewdsc.mpi-inf.mpg.de/, we illustrate that the theoretical gain in performance of ANewDsc over other subdivision methods also transfers into practice. These experiments also show that our new implementation outperforms both RS and mature competitors by magnitudes for notoriously hard instances with clustered roots. For all other instances, we avoid almost any overhead by integrating additional optimizations and heuristics.

As a zero set of some multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. This paper presents an algorithm for isolating real roots of the zero-dimensional piecewise algebraic variety which is based on interval evaluation and the interval zeros of univariate interval polynomials in Bernstein form. An example is provided to show the proposed algorithm is effective.