共查询到20条相似文献,搜索用时 31 毫秒
1.
Nizar Demni 《Comptes Rendus Mathematique》2009,347(19-20):1125-1128
We give shorter proofs of the following known results: the radial Dunkl process associated with a reduced system and a strictly positive multiplicity function is the unique strong solution for all times t of a stochastic differential equation with a singular drift, the first hitting time of the Weyl chamber by a radial Dunkl process is finite almost surely for small values of the multiplicity function. The proof of the first result allows one to give a positive answer to a conjecture announced by Gallardo–Yor while that of the second shows that the process hits almost surely the wall corresponding to the simple root with a small multiplicity value. To cite this article: N. Demni, C. R. Acad. Sci. Paris, Ser. I 347 (2009). 相似文献
2.
Pierre Fraigniaud 《BIT Numerical Mathematics》1991,31(1):112-123
The convergence of the Durand-Kerner algorithm is quadratic in case of simple roots but only linear in case of multiple roots. This paper shows that, at each step, the mean of the components converging to the same root approaches it with an error proportional to the square of the error at the previous step. Since it is also shown that it is possible to estimate the multiplicity order of the roots during the algorithm, a modification of the Durand-Kerner iteration is proposed to preserve a quadratic-like convergence even in case of multiple zeros.This work is supported in part by the Research Program C3 of the French CNRS and MEN, and by the Direction des Recherches et Etudes Techniques (DGA). 相似文献
3.
We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed.
4.
A problem concerning the perturbation of roots of a system of homogeneous algebraic equations is investigated. The question of conservation and decomposition of a multiple root into simple roots are discussed. The main theorem on the conservation of the number of roots of a deformed (not necessarily homogeneous) algebraic system is proved by making use of a homotopy connecting initial roots of the given system and roots of a perturbed system. Hereby we give an estimate on the size of perturbation that does not affect the number of roots. Further on we state the existence of a slightly deformed system that has the same number of real zeros as the original system in taking the multiplicities into account. We give also a result about the decomposition of multiple real roots into simple real roots.
5.
N. N. Kalitkin I. P. Poshivailo 《Computational Mathematics and Mathematical Physics》2008,48(7):1113-1118
Newton’s method is most frequently used to find the roots of a nonlinear algebraic equation. The convergence domain of Newton’s method can be expanded by applying a generalization known as the continuous analogue of Newton’s method. For the classical and generalized Newton methods, an effective root-finding technique is proposed that simultaneously determines root multiplicity. Roots of high multiplicity (up to 10) can be calculated with a small error. The technique is illustrated using numerical examples. 相似文献
6.
M. T. Teryokhin 《Russian Mathematics (Iz VUZ)》2009,53(8):60-68
In this paper we investigate the existence of limit cycles of a system of the second-order differential equations with a vector parameter.We propose a method for representing a solution as a sum of forms with respect to the initial value and the parameter; we call this technique the method of small forms. We establish the conditions under which a sufficiently small neighborhood of the equilibrium point contains no limit cycles. We construct a polynomial, whose positive roots of odd multiplicity define the lower bound for the number of cycles, and simple positive roots (other positive roots do not exist) define the number of limit cycles in a sufficiently small neighborhood of the equilibrium point.We prove theorems, whose conditions guarantee that a positive root of odd multiplicity defines a unique limit cycle, but a positive root of even multiplicity defines exactly two limit cycles.We propose a method for defining the type of the stability of limit cycles. 相似文献
7.
Georges Comte Pierre Milman David Trotman 《Proceedings of the American Mathematical Society》2002,130(7):2045-2048
We show that to answer affirmatively Zariski's question concerning the topological invariance of the multiplicity of complex analytic hypersurfaces at isolated singular points, it suffices to prove two combined statements, each of which may be obtained separately.
8.
Qiang Ye. 《Mathematics of Computation》2008,77(264):2195-2230
For a (row) diagonally dominant matrix, if all of its off-diagonal entries and its diagonally dominant parts (which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off-diagonal entries in that row) are accurately known, we develop an algorithm that computes all the singular values, including zero ones if any, with relative errors in the order of the machine precision. When the matrix is also symmetric with positive diagonals (i.e. a symmetric positive semi-definite diagonally dominant matrix), our algorithm computes all eigenvalues to high relative accuracy. Rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm.
9.
Ramandeep Behl Alicia Cordero Sandile S. Motsa Juan R. Torregrosa Vinay Kanwar 《Numerical Algorithms》2016,71(4):775-796
There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the principle focus of this paper is on developing a new fourth-order optimal family of iterative methods. From the computational point of view, the conjugacy maps and the strange fixed points of some iterative methods are discussed, their basins of attractions are also given to show their dynamical behavior around the multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development. 相似文献
10.
Prof. Dr. Wolfgang Börsch-supan 《Numerische Mathematik》1970,14(3):287-296
Summary If, for each zero of a polynomial, an approximation is known, estimates for the errors of these approximations are given, based on the evaluation of the polynomial at these points. The procedure can be carried over to the case of multiple roots and root clusters using derivatives up to the orderk - 1, wherek is the multiplicity of the cluster. 相似文献
11.
Beong In Yun 《Applied mathematics and computation》2010,217(2):599-606
In this paper, to estimate a multiple root p of an equation f(x) = 0, we transform the function f(x) to a hyper tangent function combined with a simple difference formula whose value changes from −1 to 1 as x passes through the root p. Then we apply the so-called numerical integration method to the transformed equation, which may result in a specious approximate root. Furthermore, in order to enhance the accuracy of the approximation we propose a Steffensen-type iterative method, which does not require any derivatives of f(x) nor is quite affected by an initial approximation. It is shown that the convergence order of the proposed method becomes cubic by simultaneous approximation to the root and its multiplicity. Results for some numerical examples show the efficiency of the new method. 相似文献
12.
Weishi Liu 《Journal of Differential Equations》2009,246(1):428-451
The one-dimensional Poisson-Nernst-Planck (PNP) system is a basic model for ion flow through membrane channels. If the Debye length is much smaller than the characteristic radius of the channel, the PNP system can be treated as a singularly perturbed system. We provide a geometric framework for the study of the steady-state PNP system involving multiple types of ion species with multiple regions of piecewise constant permanent charge. Special structures of this particular problem are revealed, which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations. Near each singular orbit, an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of a solution of the singular boundary value problem. A new phenomenon on multiplicity and spatial behavior of steady-states involving three or more types of ion species is discovered in an example. (The phenomenon cannot occur when only two types of ion species are involved.) 相似文献
13.
《Journal of Mathematical Analysis and Applications》1987,123(1):199-221
The Behavior of the Newton-Raphson method at the singular roots has been studied by a number of authors and the convergence of the Newton-Raphson sequence has been shown to be linear. In this paper a new method with symbolic and numerical manipulations, termed the modified deflation algorithm, is proposed for the singular root of a system of nonlinear algebraic equations. The basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root. According to the method, both convergency and accuracy can greatly be improved. In addition it is often possible to obtain analytically the singular root from the new equations. In order to show the effectiveness of the present method two illustrative examples are solved. 相似文献
14.
Shigefumi Mori Yuri Prokhorov 《Proceedings of the Steklov Institute of Mathematics》2009,264(1):131-145
We prove that a terminal three-dimensional del Pezzo fibration has no fibers of multiplicity >6. We also obtain a rough classification
of possible configurations of singular points on multiple fibers and give some examples.
To the memory of Professor Vasily Alekseevich Iskovskikh 相似文献
15.
Laurent Smoch 《Advances in Computational Mathematics》2007,27(2):151-166
The purpose of this paper is to develop a spectral analysis of the Hessenberg matrix obtained by the GMRES algorithm used
for solving a linear system with a singular matrix. We prove that the singularity of the Hessenberg matrix depends on the
nature of A and some other criteria such as the zero eigenvalue multiplicity and the projection of the initial residual on particular
subspaces. We also show some new results about the distinct kinds of breakdown which may occur in the algorithm when the system
is singular.
相似文献
16.
An iteration method for roots of algebraic functions with roots of multiplicity greater than one is established using tools and techniques from interval arithmetic. The method is based on an interval iteration functions for multiple roots and it retains the convergence order of the underlying iteration method while preserving global convergence over an initial interval. A number of simple examples are provided to show that the method is feasible and that it produces reasonable results. 相似文献
17.
P. Díez 《Applied Mathematics Letters》2003,16(8):1211-1215
The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is superlinear: the rate of convergence is set by the gold number. Nevertheless, this property holds only for simple roots. If the multiplicity of the root is larger than one, the convergence of the secant method becomes linear. This communication includes a detailed analysis of the secant method when it is used to approximate multiple roots. Thus, a proof of the linear convergence is shown. Moreover, the values of the corresponding asymptotic convergence factors are determined and are found to be also related with the golden ratio. 相似文献
18.
Emre Mengi 《Numerische Mathematik》2011,118(1):109-135
The Wilkinson distance of a matrix A is the two-norm of the smallest perturbation E so that A + E has a multiple eigenvalue. Malyshev derived a singular value optimization characterization for the Wilkinson distance. In
this work we generalize the definition of the Wilkinson distance as the two-norm of the smallest perturbation so that the
perturbed matrix has an eigenvalue of prespecified algebraic multiplicity. We provide a singular value characterization for
this generalized Wilkinson distance. Then we outline a numerical technique to solve the derived singular value optimization
problems. In particular the numerical technique is applicable to Malyshev’s formula to compute the Wilkinson distance as well
as to retrieve a nearest matrix with a multiple eigenvalue. 相似文献
19.
Determining dimension of the solution component that contains a computed zero of a polynomial system 总被引:1,自引:0,他引:1
Y.C. Kuo 《Journal of Mathematical Analysis and Applications》2008,338(2):840-851
When the Jacobian of a computed numerical solution of a polynomial system in Cn allows very small singular values, the solution could be isolated with a multiple multiplicity or may belong to a solution component with positive dimension. The algorithm constructed in this article intends to differentiate those cases by determining the dimension of the solution component M in which the solution lies. Of particular interest is the case when dim(M)=0, then the solution is of course isolated. While the proposed algorithm is experimental, it has been tested successfully on the class of problems with the solution in question belonging to a reduced component. Numerical results are provided to illustrate the accuracy of the algorithm. 相似文献
20.
For complex linear homogeneous recursive sequences with constant coefficients we find a necessary and sufficient condition for the existence of the limit of the ratio of consecutive terms. The result can be applied even if the characteristic polynomial has not necessarily roots with modulus pairwise distinct, as in the celebrated Poincaré’s theorem. In case of existence, we characterize the limit as a particular root of the characteristic polynomial, which depends on the initial conditions and that is not necessarily the unique root with maximum modulus and multiplicity. The result extends to a quite general context the way used to find the Golden mean as limit of ratio of consecutive terms of the classical Fibonacci sequence. 相似文献