Roots Of Cubic Equation Research Articles

On the Relationship Between the Roots of Cubic Equations of State and New Perspectives of the Vapor–Liquid Equilibrium Calculation

On the Relationship Between the Roots of Cubic Equations of State and New Perspectives of the Vapor–Liquid Equilibrium Calculation

Thermodynamic analysis of formulations to discriminate multiple roots of cubic equations of state in process models

Several discrimination criteria have been proposed to select the correct root in the presence of multiple real roots to a cubic equation of state. Herein, a criterion which exploits the well-known method of Cardan to solve cubic equations is presented. The criterion is straightforward, continuous and covers different root scenarios. As a reference, minimization of Gibbs free energy is formulated for const. (P, T). From its KKT conditions, it follows that, in case a phase vanishes, relaxation of cubic equality is an alternative to relaxation of root discrimination as proposed by Kamath et al. (2010). The proposed criterion and those from literature satisfy at most a necessary condition, while minimization of Gibbs free energy obviously is a necessary and sufficient condition for thermodynamic stability of the selected root. In certain cases, formulations from literature are found to exclude valid solutions, whereas the proposed criterion overcomes this deficiency.

Local Descriptions of Roots of Cubic Equations over P-adic Fields

One of the most frequently asked question in the p-adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $${\mathbb {Z}}_p^{*}, {\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}, {\mathbb {Z}}_p, {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p^{*}, {\mathbb {Q}}_p{\setminus }({\mathbb {Z}}_p{\setminus }{\mathbb {Z}}_p^{*}), {\mathbb {Q}}_p{\setminus }{\mathbb {Z}}_p, {\mathbb {Q}}_p $$ or not. However, this question was open even for lower-degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the p-adic fields for $$p>3$$ .

A simple approach to solving cubic equations

Finding the roots of cubic equations has been the focus of research by many mathematicians. Omar Khayyam, the 11th century Iranian mathematician, astronomer, philosopher and poet, discovered a geometrical method for solving cubic equations by intersecting conic sections [1]. In more recent times, various methods have been presented to find the roots of cubic equations. Some methods require complex number calculations, a number of techniques use graphical methods to find the roots [e.g. 2, 3] and some other techniques use trigonometric functions [e.g. 4]. The method presented in this paper does not use graphical techniques as in [2] and [3], does not involve complex number calculations, and does not require using trigonometric functions. By using this fairly simple method, the roots of cubic equations can be found in a short time without using complicated formulas.

Normals to a parabola

Given a parabola in the standard form corresponding to three points on the parabola, such that the normals at these three points concur at a point the equation of the circumscribing circle through the three points provides a tremendous opportunity to illustrate ‘The Art of Algebraic Manipulations’. The equation of the circle exists as a separate entity depending on the values of the parameters The graphical illustrations are based on Scientific WorkPlace 5.5 and they provide interesting interpretations of the roots of cubic equations. Geometry, algebra and the theory of cubic equations are amply illustrated in this article.

Relations between roots and coefficients of cubic equations with one root negative the reciprocal of another

Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x 3 + ax 2 + bx + c = 0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied.

RANKING STOCKS USING THE FUZZY MULTIPLE CRITERIA DECISION MAKING APPROACH

This work applies the Fuzzy Multiple Criteria Decision Making (FMCDM) approach to assist in making stock investment decisions. The proposed approach applies to quantitative data and can accommodate qualitative information which is normally difficult to be integrated by traditional finance and accounting methods. A major-sub-criteria hierarchy is established to reduce the possibility of over-weighing some dependent criteria existing in a single-level structure. The ratings of stocks versus qualitative sub-criteria and the weights of major and sub-criteria are assessed in linguistic terms represented by fuzzy numbers. Each sub-criterion is in a benefit, cost, or balanced nature. New standardization methods for cost-nature and balanced-nature criteria are presented. The algorithms of membership functions of the final aggregation are derived from the roots of cubic equations of multiplications of triple fuzzy numbers. Since these algorithms are clearly developed, the investor can easily calculate the fuzzy aggregation values. The defuzzified final aggregation judges the performance of alternative stocks. Moreover, the ratio of the market price to performance (PP) is suggested to filter the over and under-pricing of alternative stocks. A set of buying and selling rules are recommended based on the performance and PP ratio. Finally, an empirical example of evaluating a set of TSE-listed stocks tests the proposed approach.

Fallibility of analytic roots of cubic equations of state in low temperature region

It is showed by several examples that, in low temperature region, the analytical solutions of cubic equation of state (EOS) lead to irrational results, while the iterative solutions of cubic EOS by using the Newton–Raphson method produce valid results. Errors caused by the limitation of significant figures of the computer languages are revealed, and a magnification of errors is defined which is a main factor bringing out the irrational results of the analytical solution of cubic EOS.

On formulae for roots of cubic equation

On formulae for roots of cubic equation

Sharaf al-Dīn al- [formula omitted]ūsī on the number of positive roots of cubic equations

In the second part of his Algebra, Sharaf al-Dīn al- T ̇ ūsī (12th-century) correctly determines the number of positive roots of cubic equations in terms of the coefficients. R. Rashed has recently published an edition of the Algebra [al- T ̇ ūsī 1985], and he has discussed al- T ̇ ūsī's work in connection with 17th century and more recent mathematical methods (see also [Rashed 1974]). In this paper we summarize and analyze the work of al- T ̇ ūsī using ancient and medieval mathematical methods. We show that al- T ̇ ūsī probably found his results by means of manipulations of squares and rectangles on the basis of Book II of Euclid's Elements. We also discuss al- T ̇ ūsī's geometrical proof of an algorithm for the numerical approximation of the smallest positive root of x 3 + c = ax 2. We argue that al- T ̇ ūsī discovered some of the fundamental ideas in his Algebra when he was searching for geometrical proofs of such algorithms.

A Method for Finding the Real Roots of Cubic Equations by Using the Slide Rule

(4) R 1=r1-A/3, R2 = r2-A/3 and R3 =r3A/3. Thus, our task is to first find the real roots of equation (2) by using the slide rule, and then use equations (4) to find the roots of equation (I). We shall solve equation (2) by considering the following cases. CASE 1. When q is a positive number, c, and p is a negative number, -d, in the special cubic equation (2). For this case, we may write equation (2) as

The Graphical Interpretation of the Complex Roots of Cubic Equations

The Graphical Interpretation of the Complex Roots of Cubic Equations

XVIII.—On the Approximation to the Roots of Cubic Equations by help of Recurring Chain-Fractions

In the twenty-ninth volume of the Society's Transactions, at page 59, Lord Brouncker's process for finding the ratio of two quantities (commonly known as the method of continued fractions) is extended to the comparison of three or more magnitudes. It is there shown that recurrence, which was believed to belong exclusively to equations of the second degree, extends to those of higher orders, and examples of this extension are given in determining the proportions of the heptagon and enneagon.

1. Approximation to the Roots of Cubic Equations by help of Recurring Chain Fractions

In the progression of fractions here used, each new term is got by corresponding multiples of the three preceding terms, thus:this mode of formation being applied separately to the numerators and denominators of the fractions.