XII.—Further Numerical Studies in Algebraic Equations and Matrices

1932 ◽  
Vol 51 ◽  
pp. 80-90 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Algebraic Equation ◽  
Symmetric Functions ◽  
Algebraic Equations ◽  
Numerical Studies ◽  
The Given

In a former paper on the same subject the writer pointed out that the sequence used by D. Bernoulli for approximating to the greatest root of an algebraic equation could be further utilised in such a way as to give all the roots. It is suggested in the present paper that there is really no need to compute a first Bernoullian sequence at all, but that by the theory of dual symmetric functions the coefficients in the given equation may be used with equal convenience. In a practical respect this simplifies the technique of root-evaluation.

scholarly journals XIX. On the differential equations which determine the form of the roots of algebraic equations

1864 ◽  
Vol 154 ◽  
pp. 733-755 ◽  
Keyword(s):  
Differential Equation ◽  
Positive Integer ◽  
Differential Equations ◽  
Algebraic Equation ◽  
General Theorem ◽  
Algebraic Equations ◽  
The Given ◽  
Lagrange's Theorem

1. Mr. Harley has shown that any root of the equation y n — ny +(n—1 x =0 satisfies the differential equation y ‒ (D‒2 n ‒1/ n ) (D‒3 n ‒2/ n ) . . (D‒ n 2 ‒ n + 1/ n )/D(D‒1) . . (D‒ n + 1) e ( n ‒1) θ y =0, . . . (1) in which e θ = x , and D= d / dθ provided that n be a positive integer greater than 2. This result, demonstrated for particular values of , and raised by induction into a general theorem, was subsequently established rigorously by Mr. Cayley by means of Lagrange’s theorem. For the case of n =2, the differential equation was found by Mr. Harley to be y ‒D‒3/2/D e θ y =1/2 e θ ............(2) Solving these differential equations for the particular cases of n =2 and n =3, Mr. Harley arrived at the actual expression of the roots of the given algebraic equation for these cases. That all algebraic equations up to the fifth degree can be reduced to the above trinomial form, is well known.

scholarly journals Two matrix methods for solution of nonlinear and linear Lane-Emden type equations with mixed condition by operational matrix

2015 ◽  
Vol 4 (3) ◽  
pp. 420 ◽  
Author(s):  
Behrooz Basirat ◽  
Mohammad Amin Shahdadi
Keyword(s):  
Bernstein Polynomials ◽  
Operational Matrix ◽  
Numerical Procedure ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Matrix Methods ◽  
Numerical Examples ◽  
Linear Algebraic Equations ◽  
The Given

<p>The aim of this article is to present an efficient numerical procedure for solving Lane-Emden type equations. We present two practical matrix method for solving Lane-Emden type equations with mixed conditions by Bernstein polynomials operational matrices (BPOMs) on interval [<em>a; b</em>]. This methods transforms Lane-Emden type equations and the given conditions into matrix equation which corresponds to a system of linear algebraic equations. We also give some numerical examples to demonstrate the efficiency and validity of the operational matrices for solving Lane-Emden type equations (LEEs).</p>

Positioning Effect on Free Convection with an Elliptical Plate Placed in a Circular Enclosure

2020 ◽  
Vol 12 (8) ◽  
pp. 1054-1062
Author(s):  
Parth Patpatiya ◽  
Soumya ◽  
Bhavya Shaan ◽  
Bhavana Yadav
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
The Body ◽  
Elliptical Plate ◽  
Numerical Studies ◽  
First Case ◽  
Varied Temperature ◽  
Laminar Natural Convection ◽  
The Given ◽  
Plate System

In this analysis we have examined the process of the steady state laminar natural convection around heated elliptical plate with Rayleigh number 10^6 positioned inside a circular enclosure. The purpose of the numerical analysis is to analyze the behavior of isotherms, streamlines and heat transfer rate in enclosure plate system due to the variation in the position of elliptical plate (r/D =0.00, 0.05, and 0.2) and aspect ratio, where the given diameter of the enclosure is D and r is the distance between the centre of elliptical plate and centre of circle. Elliptical plate is inclined at different angles and results are summed up in relative manner. There are two cases, in first case aspect ratio a/D and b/D is varied and D is kept constant, whereas in second case aspect ratio a/D and b/D is kept constant and D is varied. Temperature difference between the enclosure and the inner body (i.e., temperature of inner body is kept high as compared to the enclosure) is maintained. Two dimensional study is followed by considering air as a fluid in enclosure. The effects of the Heat Transfer and Flow of Fluid are analyzed by the streamlines and isotherms plotted for the body placed inside enclosure. Value of local Nusselt number (Nu) is also plotted along the wall of elliptical plate and along the surface of the circular enclosure. For every aspect ratio isotherms and streamlines had been plotted. This work has been validated with various other numerical studies and was in good conciliation.

scholarly journals Numerical Solutions for Multi-Term Fractional Order Differential Equations with Fractional Taylor Operational Matrix of Fractional Integration

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 96 ◽  
Author(s):  
İbrahim Avcı ◽  
Nazim I. Mahmudov
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Main Idea ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Given

In this article, we propose a numerical method based on the fractional Taylor vector for solving multi-term fractional differential equations. The main idea of this method is to reduce the given problems to a set of algebraic equations by utilizing the fractional Taylor operational matrix of fractional integration. This system of equations can be solved efficiently. Some numerical examples are given to demonstrate the accuracy and applicability. The results show that the presented method is efficient and applicable.

scholarly journals Numerical solutions of multi-order fractional differential equations by Boubaker polynomials

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 226-230 ◽  
Author(s):  
A. Bolandtalat ◽  
E. Babolian ◽  
H. Jafari
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Boubaker Polynomials ◽  
Fractional Differential ◽  
The Given

AbstractIn this paper, we have applied a numerical method based on Boubaker polynomials to obtain approximate numerical solutions of multi-order fractional differential equations. We obtain an operational matrix of fractional integration based on Boubaker polynomials. Using this operational matrix, the given problem is converted into a set of algebraic equations. Illustrative examples are are given to demonstrate the efficiency and simplicity of this technique.

scholarly journals W. von Tschirnhaus and algebraic equations with roots in arithmetical progression

1940 ◽  
Vol 6 (3) ◽  
pp. 181-184
Author(s):  
Gino Loria
Keyword(s):  
Algebraic Equation ◽  
Algebraic Equations ◽  
Arithmetical Progression ◽  
German Mathematician ◽  
Ad Libitum

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

2012 ◽  
Vol 12 (04) ◽  
pp. 1250002 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
NGUYEN THI THE
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Exponents ◽  
Algebraic Equation ◽  
Differential Algebraic Equations ◽  
Lyapunov Spectrum ◽  
Stochastic Flow ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Two Parameter

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

scholarly journals Analytical-Numerical Calculation Algorithm of Algebraic Equations Roots with Specified Limits of Errors

2019 ◽  
Vol 18 (6) ◽  
pp. 1491-1514
Author(s):  
Yuri Bychkov ◽  
Elena Solovyeva ◽  
Sergei Scherbakov
Keyword(s):  
Numerical Method ◽  
Algebraic Equation ◽  
Initial Conditions ◽  
Absolute Error ◽  
Variable Order ◽  
Algebraic Equations ◽  
Adaptive Procedure ◽  
Calculation Algorithm ◽  
Mathematical Basis ◽  
Calculation Step

This paper proposes an algorithm for calculating approximate values of  roots of algebraic equations with a specified limit of absolute errors. A mathematical basis of the algorithm is an analytical-numerical method of solving nonlinear integral-differential equations with non-stationary coefficients. The analytical-numerical method belongs to the class of one-step continuous methods of variable order with an adaptive procedure for choosing a calculation step, a formalized estimate of the error of the performed calculations at each step and the error accumulated during the calculation. The proposed algorithm for calculating the approximate values of the roots of an algebraic equation with specified limit absolute errors consists of two stages. The results of the first stage are numerical intervals containing the unknown exact values of the roots of the algebraic equation. At the second stage, the approximate values of these roots with the specified limit absolute errors are calculated. As an example of the use of the proposed algorithm, defining the roots of the fifth-order algebraic equation with three different values of the limiting absolute error is presented. The obtained results allow drawing the following conclusions. The proposed algorithm enables to select numeric intervals that contain unknown exact values of the roots. Knowledge of these intervals facilitates the calculation of the approximate root values under any specified limiting absolute error. The algorithm efficiency, i.e., the guarantee of achieving the goal, does not depend on the choice of initial conditions. The algorithm is not iterative, so the number of calculation steps required for extracting a numerical interval containing an unknown exact value of any root of an algebraic equation is always restricted. The algorithm of determining a certain root of the algebraic equation is computationally completely autonomous.

scholarly journals Small Majorana fermion codes

2017 ◽  
Vol 17 (13&14) ◽  
pp. 1191-1205
Author(s):  
Mathew B. Hastings
Keyword(s):  
Upper Bound ◽  
Majorana Fermion ◽  
Hamming Code ◽  
Stabilizer Codes ◽  
Numerical Studies ◽  
Logical Qubits ◽  
Stabilizer Code ◽  
Majorana Modes ◽  
The Given

We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 4 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 4 and 6 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.

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