System Of Recurrence Relations Research Articles

Location of Solutions of Fredholm\u2013Nemytskii Integral Equations from a Whittaker-Type Operator

We analyse the global convergence of a Whittaker-type iterative method and obtain restricted global convergence domains, so that we can locate and separate solutions of Fredholm–Nemytskii nonlinear integral equations by means of balls. For this, we use two techniques, one based on the well-known fixed point theorem and the other on a system of recurrence relations. In both techniques, we use the Whittaker-type operator involved and auxiliary functions.

A Compartmental Epidemiological Model Applied to the Covid-19 Epidemic

The objective of this work is to provide a sophisticated but accessible compartmental epidemic model. Our algorithm is highly inspired from the compartmental model developed by Sofonea and al. in 2020. This model has been used as a reference for several working groups in France during the Covid-19 crisis. Each individual is allocated to a compartment according to her age, her current state with respect to the disease, as well as the length of time she has been in that state. The model then reproduces the mechanisms of transition from one state to another: mathematically, this translates into a system of recurrence relations. It captures how much individuals interact with one another through a parameter that estimates compliance with hygiene measures and lifestyle habits. The present work aims to make the model implementation fully transparent as well as the corresponding code available and give control to users so that they are able to test the model in total transparency. Focus has been put on reproducibility and explanation of the various parameters. The hard-coded parameters correspond to the data for the Covid-19 epidemic in France.

External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms

<p style='text-indent:20px;'>We deal with the reachability problem for linear and bilinear discrete-time uncertain systems under integral non-quadratic constraints on additive input terms and set-valued constraints on initial states. The bilinearity is caused by an interval type uncertainty in coefficients of the system. Algorithms for constructing external parallelepiped-valued (shorter, polyhedral) estimates of reachable sets are presented. For linear time-invariant systems, two techniques for constructing touching external estimates with constant orientation matrices are described and compared. For time-dependant bilinear systems, parallelepiped-valued estimates are constructed. For bilinear systems with constant coefficients, nonconvex estimates are proposed in the form of unions of parallelepipeds. Evolution of all estimates is determined by systems of recurrence relations.</p>

A fast and efficient composite Newton–Chebyshev method for systems of nonlinear equations

An iterative method with fifth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of double Newton’s method with frozen derivative and third step is second derivative-free modification of Chebyshev’s method. The semilocal convergence of the method in Banach spaces is established by using a system of recurrence relations. Then an existence and uniqueness theorem is given to show the R-order of the method to be five and a priori error bounds. Computational complexity is discussed and compared with existing methods. Numerical results are included to confirm theoretical results. A comparison with the existing methods shows that the new method is more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.

Non-linear reasoning for invariant synthesis

Automatic generation of non-linear loop invariants is a long-standing challenge in program analysis, with many applications. For instance, reasoning about exponentials provides a way to find invariants of digital-filter programs, and reasoning about polynomials and/or logarithms is needed for establishing invariants that describe the time or memory usage of many well-known algorithms. An appealing approach to this challenge is to exploit the powerful recurrence-solving techniques that have been developed in the field of computer algebra, which can compute exact characterizations of non-linear repetitive behavior. However, there is a gap between the capabilities of recurrence solvers and the needs of program analysis: (1) loop bodies are not merely systems of recurrence relations---they may contain conditional branches, nested loops, non-deterministic assignments, etc., and (2) a client program analyzer must be able to reason about the closed-form solutions produced by a recurrence solver (e.g., to prove assertions). This paper presents a method for generating non-linear invariants of general loops based on analyzing recurrence relations. The key components are an abstract domain for reasoning about non-linear arithmetic, a semantics-based method for extracting recurrence relations from loop bodies, and a recurrence solver that avoids closed forms that involve complex or irrational numbers. Our technique has been implemented in a program analyzer that can analyze general loops and mutually recursive procedures. Our experiments show that our technique shows promise for non-linear assertion-checking and resource-bound generation.

Convergence of a parameter based iterative method for solving nonlinear equations in Banach spaces

In this paper, we introduce a new class of a family of third order iterative methods which depend on a real parameter for solving nonlinear equations in Banach spaces. The semilocal convergence of iterative method under the assumption that the second order Fŕechet derivative of the operator satisfies the Holder continuity condition and the R-order \(2+p\) of the iteration family is established. An existence-uniqueness theorem and error estimate in terms of a parameter \(\alpha \in [0,1]\) are provided for these iterations using a technique based on a new system of recurrence relations. Numerical application of nonlinear Hammerstein integral equations of the mixed type is provided.

Generalized Pascal’s triangles and singular elements of modules of Lie algebras

We consider the problem of determining the multiplicity function \(m_\xi ^{{ \otimes ^p}\omega }\) in the tensor power decomposition of a module of a semisimple algebra g into irreducible submodules. For this, we propose to pass to the corresponding decomposition of a singular element Ψ((Lgω )⊗p) of the module tensor power into singular elements of irreducible submodules and formulate the problem of determining the function \(M_\xi ^{{ \otimes ^p}\omega }\). This function satisfies a system of recurrence relations that corresponds to the procedure for multiplying modules. To solve this problem, we introduce a special combinatorial object, a generalized (g,ω) pyramid, i.e., a set of numbers (p, {mi})g,ω satisfying the same system of recurrence relations. We prove that \(M_\xi ^{{ \otimes ^p}\omega }\) can be represented as a linear combination of the corresponding (p, {mi})g,ω. We illustrate the obtained solution with several examples of modules of the algebras sl(3) and so(5).

Convergence for a class of multi-point modified Chebyshev-Halley methods under the relaxed conditions

In this paper, the semilocal convergence for a class of multi-point modified Chebyshev-Halley methods in Banach spaces is studied. Different from the results in reference [11], these methods are more general and the convergence conditions are also relaxed. We derive a system of recurrence relations for these methods and based on this, we prove a convergence theorem to show the existence-uniqueness of the solution. A priori error bounds is also given. The R-order of these methods is proved to be 5+q with ??conditioned third-order Frechet derivative, where ?(μ) is a non-decreasing continuous real function for μ > 0 and satisfies ?(0) ? 0, ?(tμ) ≤ t q ?(μ) for μ > 0,t ? [0,1] and q ? [0,1]. Finally, we give some numerical results to show our approach.

Revision of Engesser’s Approach to the Problem of Euler Stability for Built-Up Columns with Batten Plates

Solving the buckling problem for a battened column as a statically indeterminate structure yields a Euler critical load that is significantly higher than the buckling load being obtained on the basis of Engesser’s assumption. The stability problem is reduced to numerically solving a two-point boundary value problem for a system of recurrence relations between deformation parameters of adjacent joint cross sections of the column. The expression of deformation parameters for each further joint cross section of the column as a linear function of deformation parameters for the preceding joint cross section is possible using the initial-value method. For columns with any degree of static indeterminacy, the critical force is determined as the smallest eigenvalue of the fourth-order system of homogeneous linear algebraic equations. A computer realization of the present method requires implementing reorthogonalization of the vectors of particular solutions for the system of recurrence relations. Plots of the buckling load for columns with any number of panels can be constructed as a function of the batten rigidity parameter. Applying these plots can be useful in selecting a cross section of the columns being designed.

A system of recurrence relations for rational approximations of the Euler constant

We obtain the system of recurrence relations for rational approximations of the Euler constant generalizing the recurrence relations obtained earlier by Aptekarev with coauthors. The leading coefficient of the recurrence relations of this system is 1, which can be used to verify that the generated numbers are integers.

Numerical-analytical method of solution of a nonlinear unsteady heat-conduction equation

A method of solution of a one-dimensional nonlinear unsteady heat-conduction equation has been proposed. The use of the method of Green’s functions made it possible to transform the resulting equation to a nonlinear Volterra integral equation of the second kind for temperature, which is solved by the quadratic-form method. A system of recurrence relations, which is solved numerically, has been obtained. The influence of the nonlinearity on the temperature profiles has been analyzed. A comparison to the numerical finite-element method has shown that the numerical-analytical technique allows a reduction of more than 103 times in the calculation time.

On the application of a fourth-order two-point method to Chandrasekhar's integral equation

In this study, a new iteration of order four is introduced on Banach spaces. We establish, using a new system of recurrence relations, a semilocal convergence theorem under the same conditions as for Newton's method. Next, we use this theorem to obtain the existence of a unique solution of a Chandrasekhar equation. Finally, this solution is approximated by the fourth-order method.

The structured coalescent process with weak migration

The aim of this paper is to study genealogical processes in a geographically structured population with weak migration. The coalescence time for sampled genes from different colonies diverges to infinity as the migration rates among colonies are close to zero. We investigate the moment generating functions of the coalescence time, the number of segregating sites and the number of allele types in sampled genes when there is low migration. Employing a perturbation method, we obtain a system of recurrence relations for the approximate solutions of these moment generating functions and solve them in some cases.

Chebyshev's approximation algorithms and applications

We introduce a new family of multipoint methods to approximate a solution of a nonlinear operator equation in Banach spaces. An existence-uniqueness theorem and error estimates are provided for these iterations using a technique based on a new system of recurrence relations. To finish, we apply the results obtained to some nonlinear integral equations of the Fredholm type.

A Super-Halley Type Approximation in Banach Spaces

The Super-Halley method is one of the best known third-order iteration for solving nonlnear equations. A Newton-like method which is an approximation of this method is studied. Our approach yields a fourth R-order iterative process which is more efficient than its classical predecessor. We establish a Newton-Kantorovich-type convergence theorem using a new system of recurrence relations, and give an explicit expression for the a priori error bound of the iteration.

Some Recurrence Relations Between Hansen Coefficients

A new system of recurrence relations for Hansen coefficients is obtained. This system gives a connection between only those coefficients which are included in the disturbing function of planetary or satellite motion and allows to compute efficiently the Hansen coefficients for perturbations both from internal and external bodies. The recurrence process can be realized both from high to low and from low to high harmonical terms of the disturbing function. The corresponding algorithms of evaluation of Hansen coefficients are presented. The efficiency of the obtained system of recurrence relations is discussed.

Recurrence relations for the midpoint method

In this paper, we present a new convergence analysis and error estimates for the Midpoint method in Banach spaces by using Newton-Kantorovich-type assumptions and a technique based on a new system of recurrence relations. Finally, we give three examples where we improve the error bounds are better given by other authors.

Secant-like methods for solving nonlinear integral equations of the Hammerstein type

We consider a one-parametric family of secant-type iterations for solving nonlinear equations in Banach spaces. We establish a semilocal convergence result for these iterations by means of a technique based on a new system of recurrence relations. This result is then applied to obtain existence and uniqueness results for nonlinear integral equations of the Hammerstein type. We also present a numerical example where the solution of a particular Hammerstein integral equation is approximated by different secant-type methods.

A modification of the super-Halley method under mild differentiability conditions

A new two-point iteration of order three is introduced to approximate a solution of a nonlinear operator equation in Banach spaces. Under the same assumptions as for Newton's method, we provide a result on the existence of a unique solution for the nonlinear equation, which is based on a technique consisting of a new system of recurrence relations.

Multipoint Super-Halley Type Approximation Algorithms in Banach Spaces

We use a new multipoint iteration of at least R-order three to solve a nonlinear operator equation in a Banach space. Under standard Newton-Kantorovich assumptions, we establish two Kantorovich convergence theorems using a new system of recurrence relations. We also give some explicit error bounds for this iteration.