Search

Abstract: We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials. [Copyright &y& Elsevier]

The two-point Padé approximation problem is to find a ratio of two coprime polynomials with some constraints on their degrees to approximate a function whose power series expansions at the origin and at infinity are given. In this paper, we introduce the Hankel vector for the two-point Padé approximation problem and establish the intrinsic connections between the two-point Padé approximation problem and a certain Padé approximation problem at infinity determined by the Hankel vector of the former. These connections provide us with a new way to study the structural characteristics of the two-point Padé table and to deduce the three-term recursive relations for the numerators and denominators of three adjacent entries in the two-point Padé table. [ABSTRACT FROM AUTHOR]

In this paper, we propose a lattice factorization based symmetric paraunitary matrix extension method to design a causal symmetric paraunitary multifilter banks(PUMFBs) and construct compactly supported symmetric orthogonal multiwavelets by using the method. First, we transform a Laurent polynomial matrix consisting of polyphase components of a given symmetric orthogonal multifilter into a Laurent polynomial matrix with a simple structure, and then obtain a lattice factorization based symmetric paraunitary extension matrix by factorizing the transformed Laurent polynomial matrix and extending the constant matrix of the factorization. We also construct symmetric orthogonal multiwavelets by the proposed symmetric paraunitary matrix extension method. Finally, some examples are given to illustrate the proposed symmetric paraunitary matrix extension method. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to present efficient and stable methods to compute Caputo fractional derivative (CFD) of the Müntz–Legendre polynomials based on three-term recurrence relations and Gauss–Jacobi quadrature rules. This approach with collocation method at Chebyshev–Gauss–Lobatto points has been applied for solving linear and nonlinear fractional multi-order differential equations (FDEs) described in Caputo sense. The main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations. In this work, for the first time, we present the new rates of convergence for projection error which are more accurate than the rate presented by Shen and Wang in Shen and Wang (2016). Moreover, we present convergence rate for spectral collocation method for linear FDEs with initial value on a finite interval and endpoint singularities. Also, we propose an error analysis for Jacobi–Gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper. Finally, the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a weak Galerkin finite element method for numerically solving the coupling of fluid flow with porous media flow. Flows are governed by the Stokes equations in primal velocity–pressure formulation and Darcy equation in the second order primary formulation, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. By using the weak Galerkin approach, we consider the two-dimensional problem with the usual polynomials of degree k ≥ 1 for the velocity and hydraulic head, while polynomials of degree k − 1 for the pressure, the velocity and hydraulic head is enhanced by polynomials of degree k − 1 on the edge of a finite element partition. This new method has a lot of attractive computational features: more general finite element partitions of arbitrary polygons with certain shape regularity, fewer degrees of freedom and parameter free. Stability and error estimates of optimal order are obtained. Moreover, numerical experiences are presented to illustrate the good performance, confirm the optimal order of convergence and verify the efficiency of the proposed weak Galerkin method in this paper. We also deal with finite element partitions consisting of general meshes, such as triangular mesh, quadrilateral mesh, hexagonal-dominant mesh and Voronoi mesh for the numerical weak Galerkin approximation. [ABSTRACT FROM AUTHOR]

This paper presents a numerical procedure based on a distribution approach for doing parameter estimation in nonlinear dynamical models. The originality of the paper is first, to present a complete study of the errors due to the method and due to the noise on the signal then, to apply it to a recent model describing the transmission of the chikungunya virus to the human population. The advantage of this numerical procedure is not to require any knowledge about the value of the parameters or about the statistics of measurement uncertainties. Furthermore, it attenuates a part of the noise improving consequently the results of the parameter estimation. The numerical results attest the relevance of this approach. [ABSTRACT FROM AUTHOR]

In a recent paper the authors studied a generic problem determined by the one-dimensional porous-thermoelasticity with microtemperatures in the context of the dual-phase-lag theory. Energy decay rates for that system were obtained. However, there were several cases when the coupling between the different components defining the system of equations is weaker that were not treated. In this paper we are interested in these singular cases and we will prove the polynomial decay of the solutions depending on the relations of the relaxation parameters. [ABSTRACT FROM AUTHOR]

This paper presents a hybridized formulation for the weak Galerkin mixed finite element method (WG-MFEM) which was introduced and analyzed in Wang and Ye (2014) for second order elliptic equations. The WG-MFEM method was designed by using discontinuous piecewise polynomials on finite element partitions consisting of polygonal or polyhedral elements of arbitrary shape. The key to WG-MFEM is the use of a discrete weak divergence operator which is defined and computed by solving inexpensive problems locally on each element. The hybridized formulation of this paper leads to a significantly reduced system of linear equations involving only the unknowns arising from the Lagrange multiplier in hybridization. Optimal-order error estimates are derived for the hybridized WG-MFEM approximations. Some numerical results are reported to confirm the theory and a superconvergence for the Lagrange multiplier. [ABSTRACT FROM AUTHOR]

Optimal averaged Gauss quadrature rules provide estimates for the quadrature error in Gauss rules, as well as estimates for the error incurred when approximating matrix functionals of the form u T f (A) v with a large matrix A ∈ R N × N by low-rank approximations that are obtained by applying a few steps of the symmetric or nonsymmetric Lanczos processes to A ; here u , v ∈ R N are vectors. The latter process is used when the measure associated with the Gauss quadrature rule has support in the complex plane. The symmetric Lanczos process yields a real tridiagonal matrix, whose entries determine the recursion coefficients of the monic orthogonal polynomials associated with the measure, while the nonsymmetric Lanczos process determines a nonsymmetric tridiagonal matrix, whose entries are recursion coefficients for a pair of sets of bi-orthogonal polynomials. Recently, it has been shown, by applying the results of Peherstorfer, that optimal averaged Gauss quadrature rules, which are associated with a nonnegative measure with support on the real axis, can be expressed as a weighted sum of two quadrature rules. This decomposition allows faster evaluation of optimal averaged Gauss quadrature rules than the previously available representation. The present paper provides a new self-contained proof of this decomposition that is based on linear algebra techniques. Moreover, these techniques are generalized to determine a decomposition of the optimal averaged quadrature rules that are associated with the tridiagonal matrices determined by the nonsymmetric Lanczos process. Also, the splitting of complex symmetric tridiagonal matrices is discussed. The new splittings allow faster evaluation of optimal averaged Gauss quadrature rules than the previously available representations. Computational aspects are discussed. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a recent investigation by Zhan and Dyachenko (2021) on the Hurwitz stability of monic matrix polynomials with algebraic techniques. By improving an inertia formula for matrix polynomials with respect to the imaginary axis, we show that, under some conditions, the quasi-stability of a monic matrix polynomial can be tested via the Hermitian nonnegative definiteness of two block Hankel matrices built from its matricial Markov parameters. Moreover, for the so-called doubly monic matrix polynomials, the quasi-stability criteria can be formulated in a much simpler form. In particular, the relationship between Hurwitz stable matrix polynomials and Stieltjes positive definite matrix sequences established in Zhan and Dyachenko (2021) is included as a special case. [ABSTRACT FROM AUTHOR]

While a lot is known about the classical orthogonal polynomials, their counterparts with respect to non-classical weights are not as well explored. Nevertheless, sometimes such weights come handy as well. For example, the famous Abel–Plana summation formula offers a convenient method of summing an infinite series, reducing the sum to an integral with the Abel weight function on the real line, w (x) = x 2 sin h π x . Orthogonal polynomials with respect to this weight naturally arise when we have to numerically evaluate this integral using the Gauss quadrature rule. These orthogonal polynomials are the object of study in this paper. We obtain a number of explicit formulas and algebraic relations between these and related polynomials, including the associated polynomials. In particular, for many of these polynomials we obtain Fourier expansions with the orthogonal polynomials as the basis. We also determine the weight functions whose orthogonal polynomials are the polynomials we discussed. At the end, we briefly discuss the asymptotic and perform numerical experiments. [ABSTRACT FROM AUTHOR]

It is well known that for a general class of measures the sequence of paraorthogonal polynomials { B n } n ∈ N satisfies lim n → ∞ | B n (z) | 1 n = max (| z | , 1) , uniformly on compact subsets of ℂ ˆ ∖ T. This is an essential property to obtain the exact rate of convergence for two-point Padé approximants to the Herglotz–Riesz transform of the measure when these paraorthogonal polynomials are used as denominators. In this paper we obtain a similar result when we appropriately preselect some of the zeros of these paraorthogonal polynomials. As an application, we obtain the corresponding exact rate of convergence of the approximants to the Herglotz–Riesz transform with rational perturbation when these polynomials are used as denominators. [ABSTRACT FROM AUTHOR]

In this paper we consider weighted integrals with respect to a modification of the generalized Laguerre weight functions w α (x) = x α e − x , α > − 1 , on (0 , + ∞) by the oscillatory factor exp (i ζ x) , where ζ > 0. For calculation of such integrals for analytic real-valued functions in D = { z ∈ ℂ ∣ Re z ≥ 0 , Im z ≥ 0 } we construct polynomials orthogonal with respect to the linear functional L : P → ℂ , given by L [ p ] = ∫ 0 + ∞ p (x) w α (x) exp (i ζ x) d x , where P is a linear space of all algebraic polynomials, as well as the corresponding quadrature rules of Gaussian type. The existence of such orthogonal polynomials and the corresponding three-term recurrence relations are proved. A few numerical examples are presented, including computation of the Cauchy principal value integrals. [ABSTRACT FROM AUTHOR]

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam’s method with that of this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) of constructing the matrices A i ∈ R n × n for i = 0 , 1 , 2 , ... , (k − 1) of specified linear structure such that the matrix polynomial P (λ) = λ k I n + ∑ i = 0 k − 1 λ i A i has the m (1 ⩽ m ⩽ k n) prescribed eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to linearly structured matrix polynomials. Typical linearly structured matrices are symmetric, skew-symmetric, tridiagonal, diagonal, pentagonal, Hankel, Toeplitz, etc. Therefore, construction of the matrix polynomial with the aforementioned structures is an important but challenging aspect of the polynomial inverse eigenvalue problem (PIEP). In this paper, a necessary and sufficient condition for the existence of solution to this problem is derived. Additionally, we characterize the class of all solutions to this problem by giving the explicit expressions of the solutions. It should be emphasized that the results presented in this paper resolve some important open problems in the area of PIEP namely, the inverse eigenvalue problems for structured matrix polynomials such as symmetric, skew-symmetric, alternating matrix polynomials as pointed out by De Terán et al. (2015). Further, we study sensitivity of solution to the perturbation of the eigendata. An attractive feature of our solution approach is that it does not impose any restriction on the number of eigendata for computing the solution of LPPIEP. Towards the end, the proposed method is validated with various numerical examples on a spring mass problem. [ABSTRACT FROM AUTHOR]

The weak Galerkin (WG) is a novel numerical method based on variational principles for weak functions and their weak partial derivatives defined as distributions. In the implementation, the weak partial derivatives and the weak functions are approximated by polynomials with various degrees of freedom. The accuracy and the computational complexity of the corresponding WG scheme is significantly impacted by the selection of such polynomials. This paper presents an optimal combination for the polynomial spaces that minimize the number of unknowns in the numerical scheme without compromising the accuracy of the numerical approximation. For illustrative purpose, the authors use the second order elliptic equation to demonstrate the basic ideas of polynomial reduction. Consequently, a new weak Galerkin finite element method is proposed and analyzed. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete H 1 norm and the standard L 2 norm. In addition, the paper presents some numerical results to demonstrate the power of the WG method in dealing with finite element partitions with arbitrary polygons in 2D or polyhedra in 3D. The numerical examples include various finite element partitions such as triangular mesh, quadrilateral mesh, honeycomb mesh in 2D and mesh with deformed cubes in 3D. [ABSTRACT FROM AUTHOR]

In this paper new algorithms are being proposed for evaluation of highly oscillatory integrals (HOIs) with stationary point(s). The algorithms are based on modified Levin quadrature (MLQ) with multiquadric radial basis functions (RBFs) coupled with quadrature rules based on hybrid functions of order 8 (HFQ8) and Haar wavelets quadrature (HWQ) (Aziz et al. 2011). Part of the new procedure presented in this paper is comprised of transplanting monomials (which are used in the conventional Levin method) by the RBFs. The linear and Hermite polynomials based quadratures (Xiang, 2007) are being replaced by the new methods based on HWQ and HFQ8 respectively. Both the methods are merged with MLQ to obtain the numerical solution of highly oscillatory integrals having stationary points. The accuracy of the new methods is neither dampened by presence of the stationary point(s) nor by the large value of frequency parameter ω . Theoretical facts about the error analysis of the new methods are analyzed and proved. Numerical examples are included to show efficiency and accuracy of the new methods. [ABSTRACT FROM AUTHOR]

This paper presents the polynomial-preserving recover (PPR) postprocessing technique for the weak Galerkin (WG) finite element methods on triangular meshes. The proposed technique involves a fine-tuning parameter in a stabilizer that improves the convergence order of the finite element methods. The supercloseness between the Lagrangian interpolation and the WG solution is analyzed, which leads to the main result about the superconvergence in the gradient of the WG solution. Numerical results are presented to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of iterative methods for finding simultaneously all zeros (multiple or simple) of a polynomial. The proposed family is constructed by combining the known Sakurai–Torii–Sugiura iteration function with an arbitrary iteration function. We provide a detailed convergence analysis in the following two directions: local convergence if the polynomial has multiple zeros with known multiplicity and semilocal convergence if the polynomial has only simple zeros. As an application, we study the convergence of several particular iterative methods with high order of convergence. [ABSTRACT FROM AUTHOR]

Abstract: A stationary subdivision scheme generates the full space of polynomials of degree up to if and only if its mask satisfies sum rules of order , or its symbol satisfies zero conditions of order . This property is often called the polynomial reproduction property of the subdivision scheme. It is a well-known fact that this property is, in general, only necessary for the associated refinable function to have approximation order . In this paper we study a different polynomial reproduction property of a multivariate scalar subdivision scheme with dilation matrix . Namely, we are interested in capability of a subdivision scheme to reproduce in the limit exactly the same polynomials from which the data is sampled. The motivation for this paper are the results in Levin (2003) [9] that state that such a reproduction property of degree of the subdivision scheme is sufficient for having approximation order . Our main result yields simple algebraic conditions on the subdivision symbol for computing the exact degree of such polynomial reproduction and also for determining the associated parametrization. The parametrization determines the grid points to which the newly computed values are attached at each subdivision iteration to ensure the higher degree of polynomial reproduction. We illustrate our results with several examples. [Copyright &y& Elsevier]

Abstract: This paper considers structured matrix methods for the calculation of the theoretically exact roots of a polynomial whose coefficients are corrupted by noise, and whose exact form contains multiple roots. The addition of noise to the exact coefficients causes the multiple roots of the exact form of the polynomial to break up into simple roots, but the algorithms presented in this paper preserve the multiplicities of the roots. In particular, even though the given polynomial is corrupted by noise, and all computations are performed on these inexact coefficients, the algorithms ‘sew’ together the simple roots that originate from the same multiple root, thereby preserving the multiplicities of the roots of the theoretically exact form of the polynomial. The algorithms described in this paper do not require that the noise level imposed on the coefficients be known, and all parameters are calculated from the given inexact coefficients. Examples that demonstrate the theory are presented. [Copyright &y& Elsevier]

Abstract In 1991, T. Sakurai, T. Torii and H. Sugiura presented a fourth-order iterative algorithm for finding all zeros of a polynomial simultaneously. In this paper, we provide a detailed convergence analysis (local and semilocal) of this method. The new results improve and complement existing results due to Petković et al. (2003) and Petković (2008). Numerical examples are given to show the applicability of our semilocal convergence results. [ABSTRACT FROM AUTHOR]

Abstract The subject of this paper is C 1 sign, monotonicity and convexity preserving spline interpolation to a set of ordered points from a real function of one real variable. Two solutions are proposed constructing, respectively, a Hermite parametric polynomial Cubic Spline (CS), and a Hermite Cubic Rational polynomial Spline (CRS). Both curves are based on the shape preserving Hermite Variable Degree Spline (VDS) Gabrielides and Sapidis (2018) (first introduced in Kaklis and Pandelis (1990)) and they use the Bézier representation of polynomials. Since the CS curve is parametric, the present problem also requires calculation of the y -component of CS for any specific x -value; a robust solution to this problem is discussed in detail. The CRS is non-parametric and it does solve the given interpolation-problem with its weights (which play the role of tension parameters) being directly computed using the properties of the VDS segments. [ABSTRACT FROM AUTHOR]

Abstract In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the determination of pseudospectra of matrix polynomials is very demanding computationally. This paper describes a new approach to computing approximations of pseudospectra of matrix polynomials by using rank-one or projected rank-one perturbations. These perturbations are inspired by Wilkinson's analysis of eigenvalue sensitivity. This approach allows the approximation of both structured and unstructured pseudospectra. Computed examples show the method to perform much better than a method based on random rank-one perturbations both for the approximation of structured and unstructured (i.e., standard) polynomial pseudospectra. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new combined approximating and interpolating ternary 4-point subdivision scheme with multiple parameters is proposed. A set of nice properties, such as support, continuity and polynomial generation, are briefly discussed. The new combined scheme not only contains a lot of classical ternary schemes as special cases, but also generates brand-new ternary schemes. Compared to other approximating subdivision schemes, limit curves generated by the new scheme are more consistent with the corresponding control polygons and keep detail features better. Examples are given to show the effectiveness of the scheme. Furthermore, fractal property is analyzed and fractal curves are also given. [ABSTRACT FROM AUTHOR]

Abstract In this paper we construct two new families of C 1 quartic quasi-interpolating splines on type-1 triangulations approximating regularly distributed data. The splines are directly determined by setting their Bernstein–Bézier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain. The quasi-interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation, reproduce cubic polynomials and yield approximation order four for smooth functions. We also propose some numerical tests that confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with general symmetric 2 n -point interpolatory subdivision scheme (ISS) with polynomial reproduction of arbitrary order m ≤ n. An explicit formulation is derived for the corresponding refinement symbol, and convergence intervals are obtained for the one-parameter case, the left hand endpoints of which improve on previous such lower parameter convergence bounds. [ABSTRACT FROM AUTHOR]

Abstract Although the numerical methods to stochastic differential equations with the coefficients of locally Lipschitz and polynomial growth have been discussed commonly by some authors, there are few works on the high strong order numerical methods. In this paper, the mean-square convergence of the general projected explicit Itô–Taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition, suitable local Lipschitz conditions and polynomial growth conditions. Our analysis follows the idea of stochastic C-stability and stochastic B-consistency. After giving the selection strategy of optimal parameters, we prove that the projected explicit Itô–Taylor methods, owning optimal parameters, share the same mean-square convergence orders with non-projected ones. Finally, two numerical experiments are presented to show the effectiveness of theoretical results. Highlights • The general projected explicit Itô–Taylor methods for SDEs are constructed. • A simple selection strategy for proper projection parameters is introduced. • The methods run well for SDEs under locally Lipschitz, polynomial growth condition. • The methods hold the same mean-square convergence order as non-projection ones. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the integral of x n log p (sin (x)) for natural numbers n and p. In doing so, we recover some well-known results and remark on some relations to the log-sine integral Ls n + p + 1 (n) (θ). Later, we use properties of Bell polynomials to find an expression for the derivative of the central binomial and shifted central binomial coefficients as finite sums of polygamma functions and harmonic numbers. [ABSTRACT FROM AUTHOR]

Abstract We consider a stochastic differential equations with time-dependent delay in this paper. We first obtain the existence, uniqueness and polynomial stability of exact solution to this equation under suitable conditions. Then for the numerical method of the corresponding SDDE, we present a so called modified truncated Euler–Maruyama(MTEM) method and consider the almost sure and mean square polynomial stability of this numerical method. By using the well known discrete semimartingale convergence theorem, sufficient conditions are obtained for both bounded and unbounded delay δ to ensure the polynomial stability of the corresponding numerical approximation. Results suggest that the MTEM method replicates the polynomial stability of given SDDE under suitable conditions. Examples are presented to illustrate the conclusion. [ABSTRACT FROM AUTHOR]

Abstract A numerical method for solving polynomial partial differential equations in two independent variables, defined in the paper, is presented. The technique is based on polynomial approximation. Properties and the operational matrices for partial derivatives for a polynomial in two variables are presented first. These properties are then used to reduce the solution of partial differential equations in two independent variables to a system of algebraic equations. Five illustrative examples are presented to prove the effectiveness of the present method. Results show that the numerical scheme is very convenient for solving polynomial partial differential equations. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the constructing methods of a recombined multinomial tree based on saddle-point approximation and its application to Levy models options pricing. Firstly, the Levy process and the European option pricing are introduced. Then, we used the characteristic function of Levy process to generate density function value at discrete point in a certain range, based on the saddle-point approximation method. Then, we provide the method to construct recombined multinomial tree and give the pricing formula of European option and path-dependent option pricing based on the backward iteration. Finally, we used the CGMY process to demonstrate its application to European option, American option and American barrier option pricing. It proves that saddle-point approximation turns the inverse Fourier integral transform problem to several function value calculation. Comparing to IFFT, the speed of calculation using this method is faster, and it avoid the negative probability density function value based on IFFT. Because of the linear growth of the number of nodes, we can extend the saddle-point approximation method to the calculation of path-dependent option. [ABSTRACT FROM AUTHOR]

Abstract This paper derives a superconvergence result for the modified weak Galerkin (MWG) finite element method of the second order elliptic problem. The convergence rate of the MWG approximation is improved by 30% after applying a low cost L 2 projection post-processing technique. These superconvergence phenomena are proved theoretically and confirmed numerically. [ABSTRACT FROM AUTHOR]

Abstract: Some regularity properties of the solution of linear multi-term fractional differential equations are derived. Based on these properties, the numerical solution of such equations by piecewise polynomial collocation methods is discussed. The results obtained in this paper extend the results of Pedas and Tamme (2011) where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer. In the present paper, we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using Caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points. Theoretical results are verified by some numerical examples. [Copyright &y& Elsevier]

Abstract: This paper aims at being a guide to understand polynomial transformations and polynomial reductions between NP-complete problems by presenting the methodologies for polynomial reductions/transformations and the differences between reductions and transformations. To this end the article shows examples of polynomial reductions/transformations and the restrictions to reduce/transform between NP-complete problems. Finally, this paper includes a digraph with the historical reductions/transformations between instances of NP-complete problems and introduces the term family of polynomial transformations. [Copyright &y& Elsevier]

Using Padé approximation, Sakurai, Torii and Sugiura derived in the paper (Sakurai et al., 1991) the generalized iterative method of order n + 2 for finding all zeros of a polynomial, where n is the highest order of a polynomial derivative involved in the presented iterative formula. In this note we give the determinantal representation of this method and analyze procedures for its implementation and some computational aspects. [ABSTRACT FROM AUTHOR]

A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of logarithmically singular integrals. The interpolation and integration errors are also analyzed. Numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature. [ABSTRACT FROM AUTHOR]

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring C [ x ] ∕ I associated to a zero-dimensional ideal I generated by n C -polynomials in n variables. We assume that the polynomials are generic in the sense that the number of solutions in C n equals the Bézout number. The main contribution of this paper is an automated choice of basis for C [ x ] ∕ I , which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and it depends on the given generators of I . [ABSTRACT FROM AUTHOR]

We present a new family of iterative methods for multiple and single roots of nonlinear equations. This family contains as a special case the authors’ family for finding simple roots from Herceg and Herceg (2015). Some well-known classical methods for simple roots, for example Newton, Potra–Pták, Newton–Steffensen, King and Ostrowski’s methods, belong to this family, which implies that our new family contains modifications of these methods suitable for finding multiple roots. Convergence analysis shows that our family contains methods of convergence order from 2 to 4. The new methods require two function evaluations and one evaluation of the first derivative per iteration, so all our fourth order methods are optimal in terms of the Kung and Traub conjecture. Several examples are presented and compared. Through various test equations, relevant numerical experiments strongly support the claimed theory in this paper. Extraneous fixed points of the iterative maps associated with the proposed methods are also investigated. Their dynamics is explored along with illustrated basins of attraction for various polynomials. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the qualitative behavior and exact travelling wave solutions of the Zhiber–Shabat equation are studied by using qualitative theory of polynomial differential system. The phase portraits of system are given under different parametric conditions. Some exact travelling wave solutions of the Zhiber–Shabat equation are obtained. The results presented in this paper improve the previous results. [Copyright &y& Elsevier]

Abstract: Multivariate Birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far. In this paper, we introduce a special new type of multivariate Birkhoff interpolation and present a Newton paradigm for it. Using the algorithms proposed in this paper, we can construct a Hermite system for any interpolation problem of this type and then obtain a Newton basis for the problem w.r.t. the Hermite system. [Copyright &y& Elsevier]

Abstract: In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the class of finite kernel functions by Y.Q. Bai, M.El Ghami and C. Roos [Y.Q. Bai, M. El Ghami, and C. Roos. A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (3) (2003) 766–782]. The proposed functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like the usual barrier functions. The goal of this paper is to investigate such a class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new arguments had to be used for the analysis. The iteration bound of large-update interior-point methods based on these functions and analyzed in this paper, is shown to be . For small-update interior-point methods the iteration bound is , which is currently the best-known bound for primal-dual IPMs. We also present some numerical results which show that by using a new kernel function, the best iteration numbers were achieved in most of the test problems. [Copyright &y& Elsevier]

Abstract: By using the extended Sturm–Liouville theorem for symmetric functions, we introduced a basic class of symmetric orthogonal polynomials (BCSOP) in a previous paper. The mentioned class satisfies a differential equation of the formand contains four main sequences of symmetric orthogonal polynomials. In this paper, again by using the mentioned theorem, we introduce a basic class of symmetric orthogonal functions (BCSOF) as a generalization of BCSOP and obtain its standard properties. We show that the latter class satisfies the equationin which is a free parameter and denotes eigenvalues corresponding to BCSOF. We then consider four sub-classes of defined orthogonal functions class and study their properties in detail. Since BCSOF is a generalization of BCSOP for , the four mentioned sub-classes respectively generalize the generalized ultraspherical polynomials, generalized Hermite polynomials and two other finite sequences of symmetric polynomials, which were introduced in the previous work. [Copyright &y& Elsevier]

Abstract: The determination of an approximate greatest common divisor (GCD) of two inexact polynomials and arises in several applications, including signal processing and control. This approximate GCD can be obtained by computing a structured low rank approximation of the Sylvester resultant matrix . In this paper, the method of structured total least norm (STLN) is used to compute a low rank approximation of , and it is shown that important issues that have a considerable effect on the approximate GCD have not been considered. For example, the established works only yield one matrix , and therefore one approximate GCD, but it is shown in this paper that a family of structured low rank approximations can be computed, each member of which yields a different approximate GCD. Examples that illustrate the importance of these and other issues are presented. [Copyright &y& Elsevier]

Abstract: The monomial basis for polynomials in N variables is labeled by compositions. To each composition there is associated a hook-length product, which is a product of linear functions of a parameter. The zeroes of this product are related to “critical pairs” of compositions; a concept defined in this paper. This property can be described in an elementary geometric way; for example: consider the two compositions and , then the respective ranks, permutations of the index set {1,2,…,6} sorting the compositions, are and , and the two vectors of differences (between the compositions and the ranks, respectively) are and , which are parallel, with ratio . For a given composition and zero of its hook-length product there is an algorithm for constructing another composition with the parallelism property and which is comparable to it in a certain partial order on compositions, derived from the dominance order. This paper presents the motivation from the theory of nonsymmetric Jack polynomials and the description of the algorithm, as well as the proof of its validity. [Copyright &y& Elsevier]

Abstract: Laplace''s method is one of the best-known techniques in the asymptotic approximation of integrals. The salient step in the technique''s historical development was Erdélyi''s use of Watson''s Lemma to obtain an infinite asymptotic expansion valid for any Laplace-type integral, published in 1956. Erdélyi''s expansion contains coefficients that must be calculated in each application of Laplace''s method, a tedious process that has traditionally involved the reversion of a series. This paper shows that the coefficients in fact have a very simple general form. In effect, we extend Erdélyi''s theorem. Our results greatly simplify calculation of the in any particular application and clarify the theoretical basis of Erdélyi''s expansion: it turns out that Faà di Bruno''s formula has always played a central role in it. We prove or derive the following: [ ] The correct dimensionless groups. Erdélyi''s expansion is properly expressed in terms of scaled coefficients . [ ] Two explicit expressions for in terms of combinatorial objects called partial ordinary Bell polynomials. This form is probably computationally optimal and makes checking for correctness a relatively straightforward process. [ ] A recursive expression for . [ ] Each coefficient can be expressed as a polynomial in , where and are quantities in Erdélyi''s formulation. The main insight that emerges is that the traditional approach to Laplace''s method, involving reversion of a series, is less efficient and need only be invoked if one is interested in the role of the reversion coefficients in Erdélyi''s expansion—a point which Erdélyi himself alluded to. We consider as an example an integral that occurs in a variational approach to finding the binding energy of helium dimers. We also present a three-line computer code to generate the coefficients exactly in the general case. In a sequel paper (to be published in SIAMReview), a new representation for the gamma function is obtained, and the link with Faà di Bruno''s formula is explained. [Copyright &y& Elsevier]

Abstract: In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric typeIn this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition. [Copyright &y& Elsevier]

Abstract: In this paper, we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete Sobolev inner product, that is, an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives. In a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence. Here, we study the numerical stability of these algorithms for the generation and evaluation of a finite series of Sobolev orthogonal polynomials. Besides, we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence. [Copyright &y& Elsevier]

Abstract: The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials. Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949, see (AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75–143, Dokl. Akad. Nauk SSSR 69(2) (1949) 125). In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n. The aim of this paper is to give an overview of the techniques that have led to these examples, a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples. [Copyright &y& Elsevier]