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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]

Several methods have been proposed to calculate a rigorous error bound of an approximate solution of a linear system by floating-point arithmetic. These methods are called 'verification methods'. Applicable range of these methods are different. It depends mainly on the condition number and the dimension of the coefficient matrix whether such methods succeed to work or not. In general, however, the condition number is not known in advance. If the dimension or the condition number is large to some extent, then Oishi-Rump's method, which is known as the fastest verification method for this purpose, may fail. There are more robust verification methods whose computational cost is larger than the Oishi-Rump's one. It is not so efficient to apply such robust methods to well-conditioned problems. The aim of this paper is to choose a suitable verification method whose computational cost is minimum to succeed. First in this paper, four fast verification methods for linear systems are briefly reviewed. Next, a compromise method between Oishi-Rump's and Ogita-Oishi's one is developed. Then, an algorithm which automatically and efficiently chooses an appropriate verification method from five verification methods is proposed. The proposed algorithm does as much work as necessary to calculate error bounds of approximate solutions of linear systems. Finally, numerical results are presented. [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]

In this paper, we present an existence of solution for a functional integro-differential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics, and other field of physics and mathematical chemistry. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo's theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on Sinc approximation to solve the equation. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme. [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]

In this paper, Homotopy Analysis Method (HAM) is applied to numerically approximate the eigenvalues of the fractional Sturm-Liouville problems. The eigenvalues are not unique. These multiple solutions, i.e., eigenvalues, can be calculated by starting the HAM algorithm with one and the same initial guess and linear operator $\mathcal{L}$. It can be seen in this paper that the auxiliary parameter $\hbar,$ which controls the convergence of the HAM approximate series solutions, has another important application. This important application is predicting and calculating multiple solutions. [ABSTRACT FROM AUTHOR]

Abstract The paper addresses the problem of how to ensure existence of blossoms in the context of piecewise spaces built from joining different extended Chebyshev spaces by means of connection matrices. The interest of this issue lies in the fact that existence of blossoms is equivalent to existence of B-spline bases in all associated spline spaces. As is now classical, blossoms are defined in a geometric way by means of intersections of osculating flats. In such a piecewise context, intersecting a number of osculating flats is a tough proposition. In the present paper, we show that blossoms exist if an only if Bzier points exist, which significantly simplifies the problem. Existence of blossoms also proves to be equivalent to existence of Bernstein bases. In order to establish the latter results, we start by extending to the piecewise context some results which are classical for extended Chebyshev spaces. [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]

In the framework of the numerical solution of linear systems arising from image restoration, in this paper we present an adaptive approach based on the reordering of the image approximations obtained with the Arnoldi-Tikhonov method. The reordering results in a modified regularization operator, so that the corresponding regularization can be interpreted as problem dependent. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

The limited memory BFGS method (L-BFGS) is an adaptation of the BFGS method for large-scale unconstrained optimization. However, The L-BFGS method need not converge for nonconvex objective functions and it is inefficient on highly ill-conditioned problems. In this paper, we proposed a regularization strategy on the L-BFGS method, where the used regularization parameter may play a compensation role in some sense when the condition number of Hessian approximation tends to become ill-conditioned. Then we proposed a regularized L-BFGS method and established its global convergence even when the objective function is nonconvex. Numerical results show that the proposed method is efficient. [ABSTRACT FROM AUTHOR]

In this paper, we establish two families of approximations for the gamma function: where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$, $b$, $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes' formulae, which, to our knowledge, are better than other approximations in the literature. [ABSTRACT FROM AUTHOR]

Recently we have introduced a new technique for combining classical bivariate Shepard operators with three point polynomial interpolation operators (Dell'Accio and Di Tommaso, On the extension of the Shepard-Bernoulli operators to higher dimensions, unpublished). This technique is based on the association, to each sample point, of a triangle with a vertex in it and other ones in its neighborhood to minimize the error of the three point interpolation polynomial. The combination inherits both degree of exactness and interpolation conditions of the interpolation polynomial at each sample point, so that in Caira et al. (J Comput Appl Math 236:1691-1707, ) we generalized the notion of Lidstone Interpolation (LI) to scattered data sets by combining Shepard operators with the three point Lidstone interpolation polynomial (Costabile and Dell'Accio, Appl Numer Math 52:339-361, ). Complementary Lidstone Interpolation (CLI), which naturally complements Lidstone interpolation, was recently introduced by Costabile et al. (J Comput Appl Math 176:77-90, ) and drawn on by Agarwal et al. () and Agarwal and Wong (J Comput Appl Math 234:2543-2561, ). In this paper we generalize the notion of CLI to bivariate scattered data sets. Numerical results are provided. [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]

Blossoming is a useful technique to study bases and curve representations in computer-aided geometric design. Recently Simeonov et al. (Comput Aided Geom Des 28:549-565, ) have used a blossom generalization, namely the h-blossom, to derive new results about the h-Bernstein basis and h-Bézier curves that have previously been studied in approximation theory and computer-aided geometric design. This paper introduces a basis related to the h-Bernstein basis. There is a close relationship between this new basis and the h-Bernstein basis, between the new basis and the h-blossom, and between the new basis and 'progressive' curves. This paper explores these relationships and uses them to derive properties both of the new basis itself, and of curves represented in terms of the new basis. [ABSTRACT FROM AUTHOR]

Fractional reaction-subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider a variable-order nonlinear reaction-subdiffusion equation. A numerical approximation method is proposed to solve the equation. Its convergence and stability are analyzed by Fourier analysis. By means of the technique for improving temporal accuracy, we also propose an improved numerical approximation. Finally, the effectiveness of the theoretical results is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

Positive basis is an important concept in direct search methods. Although any positive basis can ensure the convergence in theory, the maximum positive bases are often used to construct direct search algorithms. In this paper, two direct search methods for computational expensive functions are proposed based on the minimal positive bases. The Coope-Price's frame-based direct search framework is employed to insure convergence. PRP+ method and a recently developed descent conjugate gradient method are employed respectively to accelerate convergence. The data profiles and the performance profiles of the numerical experiments show that the proposed methods are effective for computational expensive functions. [ABSTRACT FROM AUTHOR]

In this paper, we propose two local error estimates based on drift and diffusion terms of the stochastic differential equations in order to determine the optimal step-size for the next stage in an adaptive variable step-size algorithm. These local error estimates are based on the weak approximation solution of stochastic differential equations with one-dimensional and multi-dimensional Wiener processes. Numerical experiments are presented to illustrate the effectiveness of this approach in the weak approximation of several standard test problems including SDEs with small noise and scalar and multi-dimensional Wiener processes. [ABSTRACT FROM AUTHOR]

Abstract  The fourth conference on “Multivariate Approximation: Theory and Applications” was held April 26 to May 01, 2007 at Cancun, Mexico. This introductory paper describes the main topics in Approximation Theory presented during this conference, ranging from theoretical aspects to numerous applications. [ABSTRACT FROM AUTHOR]

Abstract  In this paper we consider the problem of the approximation of the integral of a smooth enough function f(x,y) on the standard simplex $${\int\limits_{\Delta _{2} } {f{\left( {x,y} \right)}dxdy} } = {\sum\limits_{\alpha = 1}^3 {{\sum\limits_{i,j} {A_{{\alpha ij}} \frac{{\alpha ^{{i j}} }}{{\alpha x^{i} \alpha y^{j} }}f{\left( {x_{\alpha } ,y_{\alpha } } \right)} E{\left( f \right)}} }} }$$ where the nodes are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples. [ABSTRACT FROM AUTHOR]

Abstract  Many radial basis function (RBF) methods contain a free shape parameter that plays an important role for the accuracy of the method. In most papers the authors end up choosing this shape parameter by trial and error or some other ad hoc means. The method of cross validation has long been used in the statistics literature, and the special case of leave-one-out cross validation forms the basis of the algorithm for choosing an optimal value of the shape parameter proposed by Rippa in the setting of scattered data interpolation with RBFs. We discuss extensions of this approach that can be applied in the setting of iterated approximate moving least squares approximation of function value data and for RBF pseudo-spectral methods for the solution of partial differential equations. The former method can be viewed as an efficient alternative to ridge regression or smoothing spline approximation, while the latter forms an extension of the classical polynomial pseudo-spectral approach. Numerical experiments illustrating the use of our algorithms are included. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, an iterative algorithm is constructed for solving linear matrix equation AXB = C over generalized centro-symmetric matrix X. We show that, by this algorithm, a solution or the least-norm solution of the matrix equation AXB = C can be obtained within finite iteration steps in the absence of roundoff errors; we also obtain the optimal approximation solution to a given matrix X 0 in the solution set of which. In addition, given numerical examples show that the iterative method is efficient. [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]