Root Isolation Research Articles

Exact algorithms for the implementation of cauchy's rule

Cauchy's little known rule for computing a lower (or upper) bound on the values of the positive roots of a polynomial equation has proven to be of great importance; namely it constitutes an indispensable and crucial part of the fastest method existing for the isolation of the real roots of an equation, a method which was recently developed by the author of this article. In this paper efficient, exact (infinite precision) algorithms, along with their computing time analysis, are presented for the implementation of this important rule.

Vincent's forgotten theorem, its extension and application

Vincent's theorem of 1836, which was only recently discovered by the author of this article, is of extreme importance because it constitutes the basis of the fastest method existing for the isolation of the real roots of a polynomial equation (using exact integer arithmetic). In this paper this forgotten theorem is presented both in its original form and in an extended version, and is followed by a general discussion of its application.

The fastest exact algorithms for the isolation of the real roots of a polynomial equation

This paper discusses a set of algorithms which, given a polynomial equation with integer coefficients and without any multiple roots, uses exact (infinite precision) integer arithmetic and the Vincent-Uspensky-Akritas theorem to compute intervals containing the real roots of the polynomial equation. Theoretical computing time bounds are developed for these algorithms which are proven to be the fastest existing; this fact is also verified by the empirical results which are included in this article.

An improved method for the isolation and the quantitative measurement of crop roots.

The isolation of crop roots from soil samples, making use of the dispersing action of CO2 evolved by the reaction of added oxalic acid on carbonates in the soil was described. The roots were photographed in a thin layer of water against a matt black background, resulting in sharp negative pictures that can be used by a Quantimet image analyser to measure length or other properties embodied in the photographs. The methods were applied in a study of the rooting characteristics of potato crops in a rotation experiment. (Abstract retrieved from CAB Abstracts by CABI’s permission)

An implementation of Vincent's theorem

A new method is presented for the isolation of the real roots of a polynomial equation; it is based on Vincent's forgotten theorem of 1836 and has been implemented using exact (infinite precision) integer arithmetic algorithms. A theoretical analysis of the computing time of this method is given along with some empirical results.

On the forgotten theorem of Mr. Vincent

A little known theorem concerning the isolation of roots of polynomial equations, published in 1836 by a mathematician known only as Mr. Vincent, is discussed. Mr. Vincent's method is of historical and practical interest because it requires fewer computations than Sturm's method. The advantages afforded by this theorem are particularly relevant to software systems for computerized algebra. Certain computational results which offer an empirical comparison of the two methods are also presented. Ci'examine´est un theóre`me peu connu relatifa`l'isolement des racines dese´quations polynoˆmes, publie´en 1836 par un mathe´maticien connu simplement sous le nom de Monsieur Vincent. La me´thode de Vincent est d'un inte´reˆt historique aussi bien que pratique puisqu'elle exige moins de calculs que celle de Sturm. Les avantages admis par ce the´ore`me sont particulie`rment applicablesa`des programmes de manipulations alge`briques. Certains re´sultats qui offrent une comparaison empirique des deux me´thodes sonte´galement pre´sente´s. Diskutiert wird ein wenig bekannter Lehrsatz, der die Isolation von Wurzeln polynomischer Gleichungen betrifft und 1836 von einem nur als Herr Vincent bekannten Mathematiker vero¨ffentlicht wurde. Herrn Vincents Methode hat historisches und praktisches Interesse, weil sie weniger Berechnungen als Sturms Methode verlangt. Die Vorteile, die dieser Lehrsatz hat, sind fu¨r Software Systeme fu¨r Computeralgebra besonders wichtig. Eigeschlossen sind auch gewisse Berechnungsergebnisse, die einen empirischen Vergleich der beiden Methoden darbieten.

A short note on a new method for polynomial real root isolation

The purpose of this short note is to make known to our community some of the results of my Ph.D. Dissertation [1]. More details can also be found elsewhere [2], [3].

Real root isolation for algebraic polynomials

Several algorithms are known to separate the real zeros of a polynomial. In his thesis Heindel [He70] showed, that the computing time of his algorithm using Sturm sequences is polynomially bounded in the length of the coefficients. The polynomials are assumed to have integral coefficients. In his Diplomarbeit [Lü76] Ludicke gave a modified Sturm algorithm for real algebraic polynomials with a polynomially bounded, but very high computing time. He described and analyzed the algorithm, but did not implement it. In the present paper we extend the Collins/Loos - algorithm [CL76] from integral to real algebraic coefficients and gain empirical computing times.

A heuristic for root isolation

AbstractA heuristic for isolating the real roots of an equation involving a continuous function over a specified region is presented. Root isolation strategies are well known for polynomial and rational functions; however, there have been no really effective root isolation strategies for transcendental functions. For a transcendental function and a given interval, there has been no way to decide the existence of roots within the interval unless the function changed sign at the endpoints or unless evaluations were performed within the interval. This paper presents a new decision procedure using only interval endpoints. The methodology developed does not guarantee to detect real roots in an interval; however, the technique will effectively indicate real roots in a given interval unless the behaviour of the function is quite erratic.

ALFALFA DISEASE IN SOME ALBERTA SOILS

Attempts were made to evaluate the potential biological factor causing poor growth (stunting) of alfalfa (Medicago sativa L.) on some soils in Central Alberta. Paratylenchus projectus which associates with stunted alfalfa was extracted from an affected soil. Alfalfa seedlings grown in sterilized sand culture assemblies were inoculated with P. projectus alone and in combination with dilutions from sterilized and non-sterilized affected and normal soils. Other studies included the effects of various sterilized and non-sterilized soil dilutions from an affected soil on plant growth and incidence of alfalfa stunting symptoms. Results indicate that P. projectus is not the biological factor causing alfalfa stunting. Typical stunting symptoms were observed on plants receiving non-sterilized soil dilutions from an affected soil, and the consistency of development of irregular reddish-brown root lesions on stunted plant roots confirmed the presence of other biological factor(s) causing the disease. It was reasonably well established that the root rot disease was caused by a microorganism(s). Further studies regarding root lesioning, isolation and identification of the microorganisms are being carried on in this laboratory.

A Root Isolation Method for Testing Root-Active Chemicals

A method for isolating roots of a growing plant was devised. The technique allows treatment of an isolated portion of the root system without disturbance or injury to the untreated root mass. A specially designed planter-box was used to induce natural separation of the root system in 3 to 4 weeks. Weeping willow (Salix babylonica L.) seedlings were used to test the performance of dichlobenil (2,6-dichlorobenzonitrile), paraquat (1,1′-dimethyl-4,4′-bipyridinium ion), sodium hydroxide (NaOH), and copper sulfate (CuSO4 • 5H2O) in reducing the viability of treated roots. A tetrazolium chloride assay was used to determine the difference in viability of the treated roots. Copper sulfate and paraquat-treated plants produced symptoms of systemic injury.

Root Isolation through Curve Analysis

Root Isolation through Curve Analysis

The physiology of Indian Nodule Bacteria

1. The physiological reactions of the three types of Indian nodule bacteria (C. indicus, D. biflorus andPs. tetragonolobus) have been studied and ascertained, as is indicated in the tables. 2. In the isolation of the nodule organism fromCajanus indicus roots we found with certain characters of constancy a contaminant or rather a concomitant, which has not the characteristics ofB. radiobacter, the only organism likely to be met with. 3. TheCajanus indicus root nodule organism is placed by various authors Lohnis and Leonard, in the Cowpea group, the members of which have got an alkaline reaction on milk medium and also on nearly all sugars. The organism ofC. indicus isolated by us differs in this respect from the members of the Cowpea group in having acid reaction on litmus milk, and mixed reactions on sugars, acidic and alkaline. 4. The use of gelatin stabs, commonly followed in the study of gelatin reactions, may account for much of the irregularity of reaction and disagreement observed in the conclusions of different authors. Our procedure of the gelatin slants gives clear results, sometimes in half the time, and great advantage will be derived from its general adoption. 5. Garman and Didlake’s method proved successful as regards the culturing of host plants, but not as regards the nodule formation. Pending the study of a modification of the same, we devised an easy and reliable method of pot cultures which can be depended upon to give clear results in our exacting conditions of temperature and moisture.