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Both barycentric Lagrange interpolation and barycentric rational interpolation are thought to be stable and effective methods for approximating a given function on some special point sets. A direct evaluation of these interpolants due to N interpolation points at M sampling points requires $\mathcal {O}(NM)$ arithmetic operations. In this paper, we introduce two fast multipole methods to reduce the complexity to $\mathcal {O}(\max \left \{N,M\right \})$ . The convergence analysis is also presented in this paper. [ABSTRACT FROM AUTHOR]

This paper presents a method for solving nonlinear system with singular Jacobian at the solution. The convergence rate in the case of singularity deteriorates and one way to accelerate convergence is to form bordered system. A local algorithm, with finite-difference approximations, for forming and solving such system is proposed in this paper. To overcome the need that initial approximation has to be very close to the solution, we also propose a method which is a combination of descent method with finite-differences and local algorithm. Some numerical results obtained on relevant examples are presented. [ABSTRACT FROM AUTHOR]

This paper focuses on spectral distribution of kernel matrices related to radial basis functions. By relating a contemporary finite-dimensional linear algebra problem to a classical problem on infinite-dimensional linear integral operator, the paper shows how the spectral distribution of a kernel matrix relates to the smoothness of the underlying kernel function. The asymptotic behaviour of the eigenvalues of a infinite-dimensional kernel operator are studied from a perspective of low rank approximation-approximating an integral operator in terms of Fourier series or Chebyshev series truncations. Further, we study how the spectral distribution of interpolation matrices of an infinite smooth kernel with flat limit depends on the geometric property of the underlying interpolation points. In particularly, the paper discusses the analytic eigenvalue distribution of Gaussian kernels, which has important application on stably computing of Gaussian radial basis functions. [ABSTRACT FROM AUTHOR]

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)3-α), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(hk+1 + (Δt)3-α), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O( h + (Δ t)), where k, h, and Δ t are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this paper, trigonometrically fitted multi-step Runge-Kutta (TFMSRK) methods for the numerical integration of oscillatory initial value problems are proposed and studied. TFMSRK methods inherit the frame of multi-step Runge-Kutta (MSRK) methods and integrate exactly the problem whose solutions can be expressed as the linear combinations of functions from the set of $\{\exp (\mathrm {i}wt),\exp (-\mathrm {i}wt)\},$ or equivalently the set $\{\cos (wt),\sin (wt)\}$ , where w represents an approximation of the main frequency of the problem. The general order conditions are given and four new explicit TFMSRK methods with order three and four, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces. Our results are obtained under mild conditions on the iterative parameters. We apply our result to nonlinear Hammerstein integral equations and finally provide some numerical experiments to illustrate our proposed algorithm. [ABSTRACT FROM AUTHOR]

We propose in this paper a fully discrete Legendre-Galerkin method for solving general Volterra functional integral equations. The focus of this paper is the stability analysis of this method. Based on this stability result, we prove that the approximation equation has a unique solution, and then show that the Legendre-Galerkin method gives the optimal convergence order $\mathcal {O}(n^{-m})$ , where m denotes the degree of the regularity of the exact solution and n+1 denotes the dimensional number of the approximation space. Moreover, we establish that the spectral condition constant of the coefficient matrix relative to the corresponding linear system is uniformly bounded for sufficiently large n. Finally, we use numerical examples to confirm the theoretical prediction. [ABSTRACT FROM AUTHOR]

The topic of the present paper are certain approximation operators acting on the space of continous functions on [0,+ ∞) having polynomial growth. The operators which were defined by Jain in 1972 are based on a probability distribution which is called generalized Poisson distribution. As a main result we derive a complete asymptotic expansion for the sequence of these operators. [ABSTRACT FROM AUTHOR]

Projection based methods are a family of efficient and applicable derivative free methods for solving systems of nonlinear monotone equations. These methods, at each iteration, use a backtracking line search to generate a hyperplane which strictly separates the current approximation from the solution set of the problem. Numerical experiments indicate that choosing an appropriate line search highly affects the efficiency of projection based methods. In this paper we introduce a new line search procedure for generating the separating hyperplane. The convergence properties of the new procedure is established in a simple and short way. Numerical results show that the new line search is very effective and increases the efficiency of projection based methods. [ABSTRACT FROM AUTHOR]

The main object of this paper is to propose a product approximation for the gamma function. The estimates derived here are studied via the theory of completely monotonic functions. Sharp inequalities are stated. [ABSTRACT FROM AUTHOR]

Krylov subspace methods for approximating a matrix function f( A) times a vector v are analyzed in this paper. For the Arnoldi approximation to e v, two reliable a posteriori error estimates are derived from the new bounds and generalized error expansion we establish. One of them is similar to the residual norm of an approximate solution of the linear system, and the other one is determined critically by the first term of the error expansion of the Arnoldi approximation to e v due to Saad. We prove that each of the two estimates is reliable to measure the true error norm, and the second one theoretically justifies an empirical claim by Saad. In the paper, by introducing certain functions ϕ( z) defined recursively by the given function f( z) for certain nodes, we obtain the error expansion of the Krylov-like approximation for f( z) sufficiently smooth, which generalizes Saad's result on the Arnoldi approximation to e v. Similarly, it is shown that the first term of the generalized error expansion can be used as a reliable a posteriori estimate for the Krylov-like approximation to some other matrix functions times v. Numerical examples are reported to demonstrate the effectiveness of the a posteriori error estimates for the Krylov-like approximations to e v, cos( A) v and sin( A) v. [ABSTRACT FROM AUTHOR]

In this paper, a special point is found for the interpolation approximation of the distributed order fractional derivatives to achieve at least second-order accuracy. Then, two H1-Galerkin mixed finite element schemes combined with the higher accurate interpolation approximation are introduced and analyzed to solve the distributed order fractional sub-diffusion equations. The stable results, which just depend on initial value and source item, are derived. Some a priori estimates with optimal order of convergence both for the unknown function and its flux are established rigorously. It is shown that the H1-Galerkin mixed finite element approximations have the same rates of the convergence as in the classical mixed finite element method, but without LBB consistency condition and quasiuniformity requirement on the finite element mesh. Finally, some numerical experiments are presented to show the efficiency and accuracy of H1-Galerkin mixed finite element schemes. [ABSTRACT FROM AUTHOR]

In the present paper, we consider large-scale differential Lyapunov matrix equations having a low rank constant term. We present two new approaches for the numerical resolution of such differential matrix equations. The first approach is based on the integral expression of the exact solution and an approximation method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix problem is then solved by the Backward Differentiation Formula method (BDF) and the obtained solution is used to build a low rank approximate solution of the original problem. The process is being repeated, increasing the dimension of the projection space until some prescribed accuracy is achieved. We give some new theoretical results and present numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a model-hybrid approach for nonlinear optimization that employs both trust region method and quasi-Newton method, which can avoid possibly resolve the trust region subproblem if the trial step is not acceptable. In particular, unlike the traditional trust region methods, the new approach does not use a single approximate model from beginning to the end, but instead employs quadratic model or conic model at every iteration adaptively. We show that the new algorithm preserves the strong convergence properties of trust region methods. Numerical results are also presented. [ABSTRACT FROM AUTHOR]

We consider difference schemes for nonlinear time fractional Klein-Gordon type equations in this paper. A linearized scheme is proposed to solve the problem. As a result, iterative method need not be employed. One of the main difficulties for the analysis is that certain weight averages of the approximated solutions are considered in the discretization and standard energy estimates cannot be applied directly. By introducing a new grid function, which further approximates the solution, and using ideas in some recent studies, we show that the method converges with second-order accuracy in time. [ABSTRACT FROM AUTHOR]

In the literature, most known sequence transformations can be written as a ratio of two determinants. But, it is not always this case. One exception is that the sequence transformation proposed by Brezinski, Durbin, and Redivo-Zaglia cannot be expressed as a ratio of two determinants. Motivated by this, we will introduce a new algebraic tool—pfaffians, instead of determinants in the paper. It turns out that Brezinski-Durbin-Redivo-Zaglia’s transformation can be expressed as a ratio of two pfaffians. To the best of our knowledge, this is the first time to introduce pfaffians in the expressions of sequence transformations. Furthermore, an extended transformation of high order is presented in terms of pfaffians and a new convergence acceleration algorithm for implementing the transformation is constructed. Then, the Lax pair of the recursive algorithm is obtained which implies that the algorithm is integrable. Numerical examples with applications of the algorithm are also presented. [ABSTRACT FROM AUTHOR]

In this paper, based on the second-order compact approximation of firstorder derivative, the numerical algorithm with second-order temporal accuracy and fourth-order spatial accuracy is developed to solve the Stokes' first problem for a heated generalized second grade fluid with fractional derivative; the solvability, convergence, and stability of the numerical algorithm are analyzed in detail by algebraic theory and Fourier analysis, respectively; the numerical experiment support our theoretical analysis results. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to improve the order of approximation of the Bernstein operators. Three new operators of the Bernstein-type with the degree of approximations one, two, and three are obtained. Also, some theoretical results concerning the rate of convergence of the new operators are proved. Finally, some applications of the obtained operators such as approximation of functions and some new quadrature rules are introduced and the theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

The Hermite-Gauss sampling method is established to approximate the eigenvalues of the continuous Sturm-Liouville problems in 2016. In the present paper, we apply this method to approximate the eigenvalues of the Dirac system with transmission conditions at several points of discontinuity. This method gives us a higher accuracy results in comparison with the results of other sampling methods (classical sinc, regularized sinc, Hermite, sinc-Gaussian). The error of this method decays exponentially in terms of number of involved samples. Illustrative examples have been discussed to show the effectiveness of the presented method. We compare our results with the results of sinc-Gaussian sampling method which was the best sampling method before the presented method. [ABSTRACT FROM AUTHOR]

In a recent paper (Appl. Math. Comput. 215:1622-1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the convergence acceleration theorem in the case of summation of generalized hypergeometric series. Some new theoretical results and illustrative numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we propose piecewise homotopy analysis method (PHAM) to extend the convergence region of the homotopy analysis solution for well-posed initial value problems. A convergence theorem of the homotopy analysis solution in the expression of polynomials in the form of Cauchy-Kowalevskaya theorem is given for the nonlinear equation which is analytical near the initial point. We also discuss the influence of the convergence-control parameter h. Our convergence result supports Liao's conjecture (1): the convergence region increases with the increasing of h(<0). PHAM solution with the symbol of arbitrary initial values has a general form on every convergent subinterval, since the formal series solution is determined by governing equation and initial values. So, we only should substitute the initial values to general form of PHAM solution to obtain the analytical expression on a target subintervals. We also present three typical examples to illustrate the effectiveness and validity of PHAM and convergence results. [ABSTRACT FROM AUTHOR]

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance. [ABSTRACT FROM AUTHOR]

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method (HAM). In the present paper, we demonstrate that such a view is only valid in very special cases, and in general, the HAM is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the HAM. In particular, the HAM can be used to obtain solutions which are global (defined on the whole domain) rather than local (defined on some restriction of the domain). The HAM can also be used to obtain non-analytic solutions, which by their nature can not be expressed through the generalized Taylor series approach. We demonstrate these properties of the HAM by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence, it will not agree with the generalized Taylor series over the domain. Doing so, we show that the generalized Taylor series approach is not as robust as the HAM, and hence, the HAM is more general. Such results have important implications for how iterative solutions are calculated when approximating solutions to nonlinear differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive Filon-type method to approximate oscillatory integrals. By using some S-shaped functions connecting Gauss-Legendre nodes and Filon-type method, the adaptive schemes behave well for small as well as for large frequencies. Adding frequency-dependent nodes can improve the asymptotic order at the same time. Moreover, there exist some complex nodes leading to a higher asymptotic order method approximating to the integral, and the special nodes work in other methods as well. The error can be further decreased by taking extra Chebyshev nodes. The efficiency and accuracy are tested by some experiments. [ABSTRACT FROM AUTHOR]

Complex moment-based eigensolvers for solving interior eigenvalue problems have been studied because of their high parallel efficiency. Recently, we proposed the block Arnoldi-type complex moment-based eigensolver without a low-rank approximation. A low-rank approximation plays a very important role in reducing computational cost and stabilizing accuracy in complex moment-based eigensolvers. In this paper, we develop the method and propose block Krylov-type complex moment-based eigensolvers with a low-rank approximation. Numerical experiments indicate that the proposed methods have higher performance than the block SS-RR method, which is one of the most typical complex moment-based eigensolvers. [ABSTRACT FROM AUTHOR]

Krylov subspace methods often use short-recurrences for updating approximations and the corresponding residuals. In the bi-conjugate gradient (Bi-CG) type methods, rounding errors arising from the matrix-vector multiplications used in the recursion formulas influence the convergence speed and the maximum attainable accuracy of the approximate solutions. The strategy of a groupwise update has been proposed for improving the convergence of the Bi-CG type methods in finite-precision arithmetic. In the present paper, we analyze the influence of rounding errors on the convergence properties when using alternative recursion formulas, such as those used in the bi-conjugate residual (Bi-CR) method, which are different from those used in the Bi-CG type methods. We also propose variants of a groupwise update strategy for improving the convergence speed and the accuracy of the approximate solutions. Numerical experiments demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a meshless collocation method is considered to solve the multi-term time fractional diffusion-wave equation in two dimensions. The moving least squares reproducing kernel particle approximation is employed to construct the shape functions for spatial approximation. Also, the Caputo's time fractional derivatives are approximated by a scheme of order O( τ ), 1< α < 2. Stability and convergence of the proposed scheme are discussed. Some numerical examples are given to confirm the efficiency and reliability of the proposed method. [ABSTRACT FROM AUTHOR]

The generalized sampling expansion which uses samples from a bandlimited function f and its first r derivatives was first introduced by Linden and Abramson (Inform. Contr. 3, 26-31, 1960) and it was extended in different situations by some authors through the last fifty years. The use of the generalized sampling series in approximation theory is limited because of the slow convergence. In this paper, we derive a modification of a generalized sampling involving derivatives, which is studied by Shin (Commun. Korean Math. Soc. 17, 731-740, 2002), using a Gaussian multiplier. This modification is introduced for wider classes, the class of entire functions including unbounded functions on ℝ and the class of analytic functions in a strip. It highly improves the convergence rate of the generalized sampling which will be of exponential order. We will show that many known results included in Sampl. Theory Signal Image Process. 9, 199-221 (2007) and Numer. Funct. Anal. Optim. 36, 419-437 (2015) are special cases of our results. Numerical examples show a rightly good agreement with our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new two-level implicit compact operator method of order two in time ( t) and four in space ( x) for the solution of time dependent Burgers-Huxley equation with appropriate initial and boundary conditions. The presence of Reynolds number and nonlinear terms in the problem leads to severe difficulties in the numerical approximation. To overcome such difficulties, the method based on operators is constructed. We use only 3-spatial grid points and the obtained tridiagonal nonlinear system has been solved by Newton's iteration method. The test problems considered in the literature have been discussed to demonstrate the strength and utility of the proposed method. The computed numerical solutions are in good agreement with the exact solutions. We show that the proposed method enables us to obtain high accuracy solution for high Reynolds number. [ABSTRACT FROM AUTHOR]

In this paper, a new nonlinear conjugate gradient method is proposed, whose search direction can be viewed as a simple approximation to that of the memoryless BFGS method. The search direction of the proposed method satisfies the sufficient descent property regardless of line search. Global convergence properties of the new method are explored on uniformly convex functions and general functions with the standard Wolfe line search. Numerical experiments are done to test the efficiency of the new method, which implies the new method is promising. [ABSTRACT FROM AUTHOR]

Fast convolution algorithms on unstructured grids have become a well established subject. Algorithms such as Fast Multipole Method (FMM), Adaptive Cross Approximation (ACA) or $\mathcal {H}$-matrices for instance are, by now, classical and reduce the complexity of the matrix-vector product from O( N) to O( N log N) with a broad range of applications in e.g. electrostatics, magnetostatics, acoustics or electromagnetics. In this paper we describe a new algorithm of which we would like to explore the potential. Based on the Non Uniform FFT algorithm, it is at the same time simple, efficient and versatile. [ABSTRACT FROM AUTHOR]

The paper presents a trigonometrically-fitted implicit third derivative Runge-Kutta-Nystöm method (TTRKNM) whose coefficients depend on the frequency and stepsize for periodic initial value problems. The TTRKNM is a pair of methods which is obtained from its continuous version and applied to produce simultaneous approximations to the solution and its first derivative at each point in the interval of interest. A discussion of the stability property of the method is given. Numerical experiments are performed to demonstrate the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a Barzilai-Borwein (BB) type method for minimizing the sum of a smooth function and a convex but possibly nonsmooth function. At each iteration, our method minimizes an approximation function of the objective and takes the difference between the minimizer and the current iteration as the search direction. A nonmonotone strategy is employed for determining the step length to accelerate the convergence process. We show convergence of our method to a stationary point for nonconvex functions. Consequently, when the objective function is convex, the proposed method converges to a global solution. We establish sublinear convergence of our method when the objective function is convex. Moreover, when the objective function is strongly convex the convergence rate is R-linear. Preliminary numerical experiments show that the proposed method is promising. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive weak scheme for stochastic delay differential equations (SDDEs) based on the weak continuous Euler-Maruyama method which is a special member of the family of continuous weak Runge-Kutta schemes is introduced. The framework of the analysis of the global error is to embed the SDDE into a series of interrelated SDEs each defined on a separate interval in order to consider the error of SDE method and that of the interpolation. We perform the error estimation in a priori form based on the rooted tree theory of Rößler and then analyze the global error of the scheme by obtaining a computable expression of the principal terms of that which is useful for controlling it which contains both the numerical and statistical errors. Adopting the idea presented in Szepessy et al. (Commun. Pure Appl. Math. 54:1169-1214, ), we determine the optimal discretization points using the deterministic time-step mechanism and also the necessary number of realizations based on the standard deviation of the approximate solution. We show that this technique leads to increased accuracy of the expected value of the required functionals. By presenting some numerical experiments, the effectiveness of utilizing the adaptive idea with holding the tolerance proportionality property is illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we are dealing with q analogue of Durrmeyer type modified the Baskakov operators with two parameter α and β, which introduces a new sequence of positive linear q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous function defined on the interval [0, ∞). We study moments, weighted approximation properties, the rate of convergence using a weighted modulus of smoothness, asymptotic formula and better error estimation for these operators. [ABSTRACT FROM AUTHOR]

This paper presents methods for shape preserving spline interpolation. These methods are based on discrete weighted cubic splines. The analysis results in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Discrete weighted cubic B-splines and control point approximation are also considered. [ABSTRACT FROM AUTHOR]

We consider strong global approximation of SDEs driven by a homogeneous Poisson process with intensity λ > 0. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson process. We consider two classes of methods using equidistant or nonequidistant sampling of the Poisson process, respectively. We provide a construction of optimal schemes, based on the classical Euler scheme, which asymptotically attain the established minimal errors. It turns out that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh. [ABSTRACT FROM AUTHOR]

Padé approximation is the rational generalization of Hermite interpolating polynomial. On its own merits, it has earned a relevant place in the theory of constructive approximation. In this article, we will develop an exhaustive analysis of two-point Padé approximations to Herglotz-Riesz transforms. We study the convergence problem when the poles are partially preassigned. In this analysis, the Stieltjes polynomials on the unit circle naturally arise. Finally, some illustrative numerical examples are discussed. [ABSTRACT FROM AUTHOR]

We show that the Wynn recurrence (the missing identity of Frobenius of the Padé approximation theory) can be incorporated into the theory of integrable systems as a reduction of the discrete Schwarzian Kadomtsev–Petviashvili equation. This allows, in particular, to present the geometric meaning of the recurrence as a construction of the appropriately constrained quadrangular set of points. The interpretation is valid for a projective line over arbitrary skew field what motivates to consider non-commutative Padé theory. We transfer the corresponding elements, including the Frobenius identities, to the non-commutative level using the quasideterminants. Using an example of the characteristic series of the Fibonacci language we present an application of the theory to the regular languages. We introduce the non-commutative version of the discrete-time Toda lattice equations together with their integrability structure. Finally, we discuss application of the Wynn recurrence in a different context of the geometric theory of discrete analytic functions. [ABSTRACT FROM AUTHOR]

For Linear Complementarity Problems (LCP) with a positive semidefinite matrix M, iterative solvers can be derived by a process of regularization. In [] the initial LCP is replaced by a sequence of positive definite ones, with the matrices M + αI. Here we analyse a generalization of this method where the identity I is replaced by a positive definite diagonal matrix D. We prove that the sequence of approximations so defined converges to the minimal D-norm solution of the initial LCP. This extension opens the possibility for interesting applications in the field of rigid multibody dynamics. [ABSTRACT FROM AUTHOR]

Dual Bernstein polynomials find many applications in approximation theory, computational mathematics, numerical analysis, and computer-aided geometric design. In this context, one of the main problems is fast and accurate evaluation both of these polynomials and their linear combinations. New simple recurrence relations of low order satisfied by dual Bernstein polynomials are given. In particular, a first-order non-homogeneous recurrence relation linking dual Bernstein and shifted Jacobi orthogonal polynomials has been obtained. When used properly, it allows to propose fast and numerically efficient algorithms for evaluating all n + 1 dual Bernstein polynomials of degree n with O(n) computational complexity. [ABSTRACT FROM AUTHOR]

A new subspace minimization conjugate gradient algorithm with a nonmonotone Wolfe line search is proposed and analyzed. In the scheme, we propose two choices of the search direction by minimizing a quadratic approximation of the objective function in special subspaces, and state criterions on how to choose the direction. Under given conditions, we obtain the significant conclusion that each choice of the direction satisfies the sufficient descent property. Based on the idea on how the function is close to a quadratic function, a new strategy for choosing the initial stepsize is presented for the line search. With the used nonmonotone Wolfe line search, we prove the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising. [ABSTRACT FROM AUTHOR]

The best L1 approximation of the Heaviside function and the best ℓ1 approximation of multiscale univariate datasets by a cubic spline have a Gibbs phenomenon near the discontinuity. We show by numerical experiments that the Gibbs phenomenon can be reduced by using L1 spline fits which are the best L1 approximations in an appropriate spline space obtained by the union of L1 interpolation splines. We prove here the existence of L1 spline fits for function approximation which has never previously been done to the best of our knowledge. A major disadvantage of this technique is an increased computation time. Thus, we propose a sliding window algorithm on seven nodes which is as efficient as the global method both for functions and datasets with abrupt changes of magnitude, but within a linear complexity on the number of spline nodes. [ABSTRACT FROM AUTHOR]

We consider two numerical solution approaches for the Dym initial value problem using the reproducing kernel Hilbert space method. For each solution approach, the solution is represented in the form of a series contained in the reproducing kernel space, and a truncated approximate solution is obtained. This approximation converges to the exact solution of the Dym problem when a sufficient number of terms are included. In the first approach, we avoid to perform the Gram-Schmidt orthogonalization process on the basis functions, and this will decrease the computational time. Meanwhile, in the second approach, working with orthonormal basis elements gives some numerical advantages, despite the increased computational time. The latter approach also permits a more straightforward convergence analysis. Therefore, there are benefits to both approaches. After developing the reproducing kernel Hilbert space method for the numerical solution of the Dym equation, we present several numerical experiments in order to show that the method is efficient and can provide accurate approximations to the Dym initial value problem for sufficiently regular initial data after relatively few iterations. We present the absolute error of the results when exact solutions are known and residual errors for other cases. The results suggest that numerically solving the Dym initial value problem in reproducing kernel space is a useful approach for obtaining accurate solutions in an efficient manner. [ABSTRACT FROM AUTHOR]

Linear multistep methods (LMMs) applied to approximate the solution of initial value problems—typically arising from method-of-lines semidiscretizations of partial differential equations—are often required to have certain monotonicity or boundedness properties (e.g., strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties). These properties can be guaranteed by imposing step-size restrictions on the methods. To qualitatively describe the step-size restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) or its generalization, the step-size coefficient for boundedness (SCB). An LMM with larger SCM or SCB is more efficient, and the computation of the maximum SCM for a particular LMM is now straightforward. However, it is more challenging to decide whether a positive SCB exists, or determine if a given positive number is a SCB. Theorems involving sign conditions on certain linear recursions associated to the LMM have been proposed in the literature that allow us to answer the above questions: the difficulty with these theorems is that there are in general infinitely many sign conditions to be verified. In this work, we present methods to rigorously check the sign conditions. As an illustration, we confirm some recent numerical investigations concerning the existence of positive SCBs in the BDF and in the extrapolated BDF (EBDF) families. As a stronger result, we determine the optimal values of the SCBs as exact algebraic numbers in the BDF family (with 1 ≤ k ≤ 6 steps) and in the Adams-Bashforth family (with 1 ≤ k ≤ 3 steps). [ABSTRACT FROM AUTHOR]

For several applications, it is important to know the location of the singularities of a complex function: just for example, the rightmost singularity of a Laplace Transform is related to the exponential order of its inverse function. We discuss a numerical method to approximate, within an input accuracy tolerance, a finite sequence of Laurent coefficients of a function by means of the Discrete Fourier Transform (DFT) of its samples along an input circle. The circle may also enclose some singularities, since the method works with the Laurent expansion. The DFT is computed by the FFT algorithm so that, from a computational point of view, the efficiency is guaranteed. The function samples may be obtained by solving a numerical problem such as, for example, a differential problem. We derive, as consequences of the method, some new outcomes able to detect those singularities which are close to the circle and to discover if the singularities are all external or internal to the circle so that the Laurent expansion reduces to its regular or singular part, respectively. Other singularities may be located by means of a repeated application of the method, as well as an analytic continuation. Some examples and results, obtained by a first implementation, are reported. [ABSTRACT FROM AUTHOR]

We describe an algorithm for minimizing convex, not necessarily smooth, functions of several variables, based on a descent direction finding procedure that inherits some characteristics both of standard bundle method and of Wolfe's conjugate subgradient method. This is obtained by allowing appropriate upward shifting of the affine approximations of the objective function which contribute to the classic definition of the cutting plane function. The algorithm embeds a proximity control strategy. Finite termination is proved at a point satisfying an approximate optimality condition and some numerical results are provided. [ABSTRACT FROM AUTHOR]

We discuss the generation of polynomials with two bounds -an upper bound and a lower bound- on compact sets $\mathcal C=[0,1]^{d}\subset \mathbb {R}^{d}$ in view on numerical approximation and scientific computing. The presentation is restricted to d = 1,2. We show that a composition formula based on a weighted 4-squares Euler identity generates all such polynomials in dimension d=1. It yields a new algebraic or compositional approach to the classical problem related to polynomials with minimal uniform norm. The theoretical generalization to the multivariate case is discussed in dimension d=2 by means of the 8-squares Degen identity and the connection with quaternions algebras is made explicit. Numerical results in dimension d=1 illustrate the potentialities of this approach and some implementation details are provided. [ABSTRACT FROM AUTHOR]

The first 60 coefficients in the three-term recurrence relation for monic polynomials orthogonal with respect to cardinal B-splines φ as the weight functions on [0, m] ( m ∈ ℕ) are obtained in a symbolic form. They enable calculation of parameters, nodes, and weights, in the corresponding Gaussian quadrature up to 60 nodes. The efficiency of these Gaussian quadratures is shown in some numerical examples. Finally, two interesting conjectures are stated. [ABSTRACT FROM AUTHOR]