Solution Of Cubic Equation Research Articles

On the Solution of Quartic and Cubic Equations

A method of solving a quartic equation, which does not require extracting the roots of complex numbers is explained in details. In the process, the solution of a cubic equation has also been presented, with the same degree of simplicity.

Numerical and analytical study on the pull-in instability of micro-structure under electrostatic loading

The one-mode analysis method on the pull-in instability of micro-structure under electrostatic loading is presented. Taylor series are used to expand the electrostatic loading term in the one-mode analysis method, which makes analytical solution available. The one-mode analysis is the combination of Galerkin method and Cardan solution of cubic equation. The one-mode analysis offers a direct computation method on the pull-in voltage and displacement. In low axial loading range, it shows little difference with the established multi-mode analysis on predicting the pull-in voltages for three different structures (cantilever, clamped–clamped beams and the plate with four edges simply-supported) studied here. For numerical multi-mode analysis, we also show that using the structural symmetry to select the symmetric mode can greatly reduce both the computation effort and the numerical fluctuation.

Solution of Cubic Equations by Iteration Methods on a Pocket Calculator

Sometimes it is necessary to solve cubic or higher-order equations in the investigation of chemical equilibrium systems. Solving these kinds of equations may cause difficulties for chemistry students. Methods for solving cubic equations by inexpensive pocket-size programmable calculators are presented. Using these methods, students are able to solve cubic equations arising from analysis of the chemical equilibrium states without the need for a PC. In addition, by practicing such programs, students will learn to write more useful programs on their calculators.

Numerical methods for the solution of a system of Eikonal equations with Dirichlet boundary conditions

In this Note, we discuss the numerical solution of a system of Eikonal equations with Dirichlet boundary conditions. Since the problem under consideration has infinitely many solutions, we look for those solutions which are nonnegative and of maximal (or nearly maximal) L 1-norm. The computational methodology combines penalty, biharmonic regularization, operator splitting, and finite element approximations. Its practical implementation requires essentially the solution of cubic equations in one variable and of discrete linear elliptic problems of the Poisson and Helmholtz type. As expected, when the spatial domain is a square whose sides are parallel to the coordinate axes, and when the Dirichlet data vanishes at the boundary, the computed solutions show a fractal behavior near the boundary, and particularly, close to the corners. To cite this article: B. Dacorogna et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

A unified approach to teaching quadratic and cubic equations

A simple method for teaching the algebraic solution of cubic equations is presented. The approach is via ‘completion of the cube’. It is found that this method is readily accepted by students already familiar with completion of the square as a method for quadratic equations. Many people are surprised at how simple the algebraic solution of cubic equations turns out to be. It is not normally necessary to resort to numerical techniques for these equations.

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.

Solutions of cubic equations in quadratic fields

Let K be any quadratic field with O-K its ring of integers. We study the solutions of cubic equations, which represent elliptic curves defined over Q, in quadratic fields and prove some interesting results regarding the solutions by using elementary tools. As an application we consider the Diophantine equation r + s + t = rst = 1 in O-K. This Diophantine equation gives an elliptic curve defined over Q with finite Mordell-Weil group. Using our study of the solutions of cubic equations in quadratic fields we present a simple proof of the fact that except for the ring of integers of Q(i) and Q(root 2), this Diophantine equation is not solvable in the ring of integers of any other quadratic fields, which is already proved in [4].

Asymptotic Evaluation of Integrals Related to Time-Dependent Fields Near Caustics

Integrals which arise in the theory of time-dependent wave fields near caustics in nondispersive problems are much simpler than their time-harmonic counterparts because they involve delta functions rather than exponentials. Fields near smooth caustics, for instance, involve solutions of cubic equations, which can be expressed in elementary terms, instead of the Airy functions of the better known time-harmonic theory. The Pearcey functions, which arise in the theory of the cusped caustic, are similarly replaced by expressions involving solutions of quartic equations, etc. As is well known, each type of caustic is related to a particular type of catastrophe. The theory presented here deals conveniently with fields reducible to single integrals since the delta function and the integral will annihilate each other, leaving closed-form expressions; fields involving multiple integrals will be reduced somewhat in complexity, but not in general to closed form, and are not treated here.

A simplification of Cardano's Solution of cubic equation

A simplification of Cardano's Solution of cubic equation

Comment on ‘laminar film condensation in a tube with upward vapour flow’ by Seban and Hodgson

In this paper analytical solution is presented for condensation when the friction factor augmentation factor (F) [1] is negligible. For the cases where F is significant an alternative route involving the solution of cubic equation in Δ is shown. It is demonstrated that the results satisfactorily represent the numerically computed values of Seban and Hodgson.

Table for the Solution of Cubic Equations

Table for the Solution of Cubic Equations

Khayyam's Solution of Cubic Equations

Khayyam's Solution of Cubic Equations

Khayyam's Solution of Cubic Equations

Khayyam's Solution of Cubic Equations

H. E. Salzer, C. H. Richards and I. Arsham, Table for the solution of Cubic Equations (McGraw-Hill Book Company, New York, 1958), 161 pp., 50s.

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Some theoretical considerations in analytical chemistry : A method of calculating asymmetrical titration curves avoiding the use of cubic equations

A method is offered for the exact calculation of data for unsymmetrical titration curves in which the solution of cubic equations is avoided. Instead of calculating ion concentrations from the volume of titrant added in the conventional way, the calculation is reversed, and titrant volumes are calculated from supplied values of the ion which is not in excess. The necessary precision is attained more easily by calculating the difference between the titrant volume and the equivalence point volume, so that the use of seven figure logarithms is unnecessary.

Table for the Solution of Cubic Equations.

Table for the Solution of Cubic Equations.

REVIEWS

Journal Article REVIEWS Get access Table for the Solution of Cubic Equations. By H. E. SALZER et al. U.S.A. and London: McGraw-Hill Book Co. Inc. 1958. Pp. xv + 160. 58s. D. E. BARTON D. E. BARTON Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 46, Issue 3-4, December 1959, Page 499, https://doi.org/10.1093/biomet/46.3-4.499 Published: 01 December 1959

The solution of cubic equations by iteration

L'autore tratta la convergenza del metodo iterativo diLin nel caso di un'equazione cubica. Stabilisce una relazione fra la convergenza e la grandezza delle parti reali delle radici, e dimostra che una radice puo essere sempre trovata in questo modo.

A Supplementary Note on Solutions of Cubic Equations on a Slide Rule

This department will present comments on papers previously published in the MATHEMATICS MAGAZINE, lists of new books, and book reviews. In order that errors may be corrected, results extended, and interesting aspects further illuminated, comments on published papers in all departments are invited. Communications intended for this department should be sent in duplicate to H. V. Craig, Department of Applied Mathematics, Universityof Texas, Austin 12, Texas.

Historically Speaking,— Omar Khayyam's solution of cubic equations

Omar Khayyam was the first to solve geometrically, so far as positive roots are concerned, every type of cubic equation.