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This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. [ABSTRACT FROM AUTHOR]

We consider Nitsche's method for solving elliptic Dirichlet boundary control problems on curved domains with control constraints. By using Nitsche's method for the treatment of inhomogeneous Dirichlet boundary conditions, the L2 boundary control enters in the variational formulation in a natural sense. The idea was first used in Chang, et al. (Math. Anal. Appl.453, 529–557 2017) where the curved boundary was approximated by a broken line and a locally defined mapping was needed to obtain the numerical control on the curved boundary. In this paper, we develop a method defined on curved domains directly. We derive a priori estimates of quasi-optimal order for the control in the L2 norm, and quasi-optimal order for the state and adjoint state in energy norms. Numerical examples are provided to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

For the Helmholtz equation Δu + k2u = 0 in 2D domain S, there exists a unique solution if k2 is not exactly equal to an eigenvalue λ of the Laplace eigenvalue problem Δu + λu = 0 in S. One important criterion for numerical methods is that there must exist no spurious (i.e., superfluous) eigenvalues so that the above unique solution can be obtained correctly. For exterior problems, the method of fundamental solutions (MFS) using Hankel functions suffers from spurious eigenvalues. New modified Hankel functions have been proposed in Zhang et al. (Appl. Numer. Math. 145, 236–260, 2019) to eliminate all spurious eigenvalues. In this paper, we study bounded multiply connected domains. The MFS algorithms without spurious eigenvalues and their strict analysis are our goals. First, we study bounded simply connected domains by the MFS. The algorithms using Hankel functions are free from spurious eigenvalues. A brief error analysis is provided. Next, we focus on bounded multiply connected domains and choose an annular domain for analysis. The Hankel functions and the modified Hankel functions in [47] are chosen as the exterior fundamental solutions (FS) and the interior FS, respectively. Such combined FS eliminate all spurious eigenvalues, and new error and stability analyses are explored. Bounds of errors involve degeneracy, and those of the condition number involve a gap to eigenvalues. Numerical experiments are carried out to support the analysis made, and better pseudo-boundaries of source nodes are also investigated numerically. The new error and stability analyses in this paper are new and essential, thus providing a solid theoretical basis of the MFS for the Helmholtz equation in 2D bounded simply connected and multiply connected domains. [ABSTRACT FROM AUTHOR]

Theory of quasi-orthogonal polynomials is significantly related to constructing vector Padé approximations. The present paper introduces an efficient procedure to compute adjacent families of quasi-orthogonal polynomials as defined in Sadaka (Appl. Numer. Math. 21:57–70, 1996) and Sadaka (Appl. Numer. Math. 24:483–499, 1997). The derived algorithm uses short recursive relations whose coefficients are written in terms of lower triangular block determinants. The strategy of computing such determinants is given and analyzed. [ABSTRACT FROM AUTHOR]

There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005). [ABSTRACT FROM AUTHOR]

In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more. [ABSTRACT FROM AUTHOR]

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used. [ABSTRACT FROM AUTHOR]

We will refer to the paper "Convergence rates for Kaczmarz-type algorithms", published in Numerical Algorithms, 79(1)(2018), 1-17. It was observed by Dr. Mokhtar Abbasi (Department of Mathematics, University of Qom, Iran) that in the proof of Theorem 7, related to the linear convergence rate of the MREK algorithm it exists an inequality which is not always true. [ABSTRACT FROM AUTHOR]

The strong ℋ -tensors have important applications in many areas of science and engineering, e.g., the determination of positive definiteness for an even-order homogeneous polynomial form in the real field. In this paper, we propose two iterative algorithms with non-parameter for identifying strong ℋ -tensors, which overcome the drawback of choosing the best value of parameter 𝜖 in some existing algorithms given by Li et al. and Liu et al. (J. Comput. Appl. Math., 255, 1–14, 2014 and Comput. Appl. Math. 36, 1623–1635, 2017). Some numerical experiments are performed to illustrate the feasibility and effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

The sinc-Gaussian sampling formula is used to approximate an analytic function, which satisfies a growth condition, using only finite samples of the function. The error of the sinc-Gaussian sampling formula decreases exponentially with respect to N, i.e., N− 1/2e−αN, where α is a positive number. In this paper, we extend this formula to allow the approximation of derivatives of any order of a function from two classes of analytic functions using only finite samples of the function itself. The theoretical error analysis is established based on a complex analytic approach; the convergence rate is also of exponential type. The estimate of Tanaka et al. (Jpan J. Ind. Appl. Math. 25, 209–231 2008) can be derived from ours as an immediate corollary. Various illustrative examples are presented, which show a good agreement with our theoretical analysis. [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]

Based on the variant of the deteriorated positive-definite and skew-Hermitian splitting (VDPSS) preconditioner developed by Zhang and Gu (BIT Numer. Math. 56:587-604, 2016), a generalized VDPSS (GVDPSS) preconditioner is established in this paper by replacing the parameter α in (2,2)-block of the VDPSS preconditioner by another parameter β. This preconditioner can also be viewed as a generalized form of the VDPSS preconditioner and the new relaxed HSS (NRHSS) preconditioner which has been exhibited by Salkuyeh and Masoudi (Numer. Algorithms, 2016). The convergence properties of the GVDPSS iteration method are derived. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. We also study the upper bounds on the degree of the minimum polynomial of the preconditioned matrix. Numerical experiments are implemented to illustrate the effectiveness of the GVDPSS preconditioner and verify that the GVDPSS preconditioned generalized minimal residual method is superior to the DPSS, relaxed DPSS, SIMPLE-like, NRHSS, and VDPSS preconditioned ones for solving saddle point problems in terms of the iterations and computational times. [ABSTRACT FROM AUTHOR]

In this paper, we first present a family of iterative algorithms for simultaneous determination of all zeros of a polynomial. This family contains two well-known algorithms: Dochev-Byrnev's method and Ehrlich's method. Second, using Proinov's approach to studying convergence of iterative methods for polynomial zeros, we provide a semilocal convergence theorem that unifies the results of Proinov (Appl. Math. Comput. 284: 102-114, 2016) for Dochev-Byrnev's and Ehrlich's methods. [ABSTRACT FROM AUTHOR]

Abstract  The connection between Gauss quadrature rules and the algebraic eigenvalue problem for a Jacobi matrix was first exploited in the now classical paper by Golub and Welsch (Math. Comput. 23(106), 221–230, 1969). From then on many computational problems arising in the construction of (polynomial) Gauss quadrature formulas have been reduced to solving direct and inverse eigenvalue problems for symmetric tridiagonals. Over the last few years (rational) generalizations of the classical Gauss quadrature formulas have been studied, i.e., formulas integrating exactly in spaces of rational functions. This paper wants to illustrate that stable and efficient procedures based on structured numerical linear algebra techniques can also be devised for the solution of the eigenvalue problems arising in the field of rational Gauss quadrature. [ABSTRACT FROM AUTHOR]

Traces of inverse powers of a positive definite symmetric tridiagonal matrix give lower bounds of the minimal singular value of an upper bidiagonal matrix. In a preceding work, a formula for the traces which gives the diagonal entries of the inverse powers is presented. In this paper, we present another formula which gives the traces based on a quite different idea from the one in the preceding work. An efficient implementation of the formula for practice is also presented. [ABSTRACT FROM AUTHOR]

In this paper a particular partition on blocks of generalized ( h, r)-circulant matrices is determined. We obtain a characterization of generalized ( h, r)-circulant matrices and get some results on the values of the permanent and also on the determination of the eigenvalues of r-circulant matrices. At last, a lower bound for the permanent of these matrices is achieved. [ABSTRACT FROM AUTHOR]

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

This paper deals with a general type of linear matrix equation problem. It presents new iterative algorithms to solve the matrix equations of the form A X B = F. These algorithms are based on the incremental subgradient and the parallel subgradient methods. The convergence region of these algorithms are larger than other existing iterative algorithms. Finally, some experimental results are presented to show the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Abstract  Given a nonsingular complex matrix and complex vectors v and w of length N, one may wish to estimate the quadratic form w * A  − 1 v, where w * denotes the conjugate transpose of w. This problem appears in many applications, and Gene Golub was the key figure in its investigations for decades. He focused mainly on the case A Hermitian positive definite (HPD) and emphasized the relationship of the algebraically formulated problems with classical topics in analysis - moments, orthogonal polynomials and quadrature. The essence of his view can be found in his contribution Matrix Computations and the Theory of Moments, given at the International Congress of Mathematicians in Zürich in 1994. As in many other areas, Gene Golub has inspired a long list of coauthors for work on the problem, and our contribution can also be seen as a consequence of his lasting inspiration. In this paper we will consider a general mathematical concept of matching moments model reduction, which as well as its use in many other applications, is the basis for the development of various approaches for estimation of the quadratic form above. The idea of model reduction via matching moments is well known and widely used in approximation of dynamical systems, but it goes back to Stieltjes, with some preceding work done by Chebyshev and Heine. The algebraic moment matching problem can for A HPD be formulated as a variant of the Stieltjes moment problem, and can be solved using Gauss-Christoffel quadrature. Using the operator moment problem suggested by Vorobyev, we will generalize model reduction based on matching moments to the non-Hermitian case in a straightforward way. Unlike in the model reduction literature, the presented proofs follow directly from the construction of the Vorobyev moment problem. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan–Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266–277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we derive one-parameter families of Newton, Halley, Chebyshev, Chebyshev-Halley type methods, super-Halley, C-methods, osculating circle and ellipse methods respectively for finding simple zeros of nonlinear equations, permitting f ′ (x) = 0 at some points in the vicinity of the required root. Halley, Chebyshev, super-Halley methods and, as an exceptional case, Newton method are seen as the special cases of the family. All the methods of the family and various others are cubically convergent to simple roots except Newton’s or a family of Newton’s method. [ABSTRACT FROM AUTHOR]

Abstract  It is known that the unique thin plate spline interpolant to a function f ∈ C 3 IR sampled at the scaled integers h Z converges at an optimal rate of h 3. In this paper we present results from a recent numerical investigation of the case where the function is sampled at equally spaced points on the unit interval. In this setting the known theoretical error bounds predict a drop in the convergence rate from h 3 to h. However, numerical experiments show that the usual rate of convergence is h 3/2 and that the deterioration occurs near the end points of the interval. We will examine the effect of the boundary on the accuracy of the interpolant and also the effect of the smoothness of the target function. We will show that there exists functions which enjoy an even faster order of convergence of h 5/2. [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  In this paper we derive an analytical solution of the one-dimensional Boussinesq equations, in the case of waves relatively long, with small amplitudes, in water of varying depth. To derive the analytical solution we first assume that the solution of the model has a prescribed wave form, and then we obtain the wave velocity, the wave number and the wave amplitude. Finally a specific application for some realistic values of wave parameters is given and a graphical presentation of the results is provided. [ABSTRACT FROM AUTHOR]

Abstract  This paper is devoted to multiresolution schemes that use a stencil selection procedure in order to obtain adaptation to the presence of edges in the images. Since non adapted schemes, based on a centered stencil, are less affected by the presence of texture, we propose the introduction of some weight that leads to a more frequent use of the centered stencil in regions without edges. In these regions the different stencils have similar weights and therefore the selection becomes an ill-posed problem with high risk of instabilities. In particular, numerical artifacts appear in the decompressed images. Our attention is centered in ENO schemes, but similar ideas can be developed for other multiresolution schemes. A nonlinear multiresolution scheme corresponding to a nonlinear interpolatory technique is analyzed. It is based on a modification of classical ENO schemes. As the original ENO stencil selection, our algorithm chooses the stencil within a region of smoothness of the interpolated function if the jump discontinuity is sufficiently big. The scheme is tested, allowing to compare its performances with other linear and nonlinear schemes. The algorithm gives results that are at least competitive in all the analyzed cases. The problems of the original ENO interpolation with the texture of real images seem solved in our numerical experiments. Our modified ENO multiresolution will lead to a reconstructed image free of numerical artifacts or blurred regions, obtaining similar results than WENO schemes. Similar ideas can be used in multiresolution schemes based in other stencil selection algorithms. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we consider the symmetric part of the so-called ith right shift operator. We determine its eigenvalues as also the associated eigenvectors in a complete and closed form. The proposed proof is elementary, using only basical skills such as Trigonometry, Arithmetic and Linear algebra. The first section is devoted to the introduction of the tackled problem. Second and third parts contain almost all the “technical” stuff of the proof. Afterwards, we continue with the end of the proof, provide a graphical illustration of the results, as well as an application on the polyhedral “sandwiching” of a special compact of arising in Signal theory. [ABSTRACT FROM AUTHOR]