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In this paper we analyze the stability of the problem of performing a rational QZ step with a shift that is an eigenvalue of a given regular pencil H - λ K in unreduced Hessenberg–Hessenberg form. In exact arithmetic, the backward rational QZ step moves the eigenvalue to the top of the pencil, while the rest of the pencil is maintained in Hessenberg–Hessenberg form, which then yields a deflation of the given shift. But in finite-precision the rational QZ step gets "blurred" and precludes the deflation of the given shift at the top of the pencil. In this paper we show that when we first compute the corresponding eigenvector to sufficient accuracy, then the rational QZ step can be constructed using this eigenvector, so that the exact deflation is also obtained in finite-precision. [ABSTRACT FROM AUTHOR]

Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal boundaries. In this paper, alternative approaches are developed for some interface and internal layer problems based on non-matching grids, in which two mesh sizes are used to obtain comparable high order accuracy near and away from an interface or an internal layer. For one-dimensional, or two-dimensional problems with straight interfaces or boundary layers that are parallel to one of the axes, the discussion is relatively easy. One of the challenges is how to construct a fourth order compact finite difference scheme at border grid points that connect two meshes. The idea is to employ a second order discretization near the interface in the fine mesh and a fourth order discretization away from the interface in the coarse and border grid points. For two-dimensional problems with a curved interface or an internal layer, a level set representation is utilized for which we can build a fine mesh within a tube | φ (x) | ≤ δ h of the interface. A new super-third seven-point discretization that can guarantee the discrete maximum principle has been developed at hanging nodes. The coefficient matrices of the finite difference equations developed in this paper are M-matrices, which leads to the convergence of the finite difference schemes. Non-trivial numerical examples have confirmed the desired accuracy and convergence of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs). A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions. A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a continuous approximate solution with a corresponding error estimate that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions. [ABSTRACT FROM AUTHOR]

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation y ′ ′ + (a - 2 q cos 2 x) y = 0 is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual O (m 2) . This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. [ABSTRACT FROM AUTHOR]

Modeling complex systems that consist of different types of objects leads to multilayer networks, in which vertices are connected by both inter-layer and intra-layer edges. In this paper, we investigate multiplex networks, in which vertices in different layers are identified with each other, and the only inter-layer edges are those that connect a vertex with its copy in other layers. Let the third-order adjacency tensor A ∈ R N × N × L and the parameter γ ≥ 0 , which is associated with the ease of communication between layers, represent a multiplex network with N vertices and L layers. To measure the ease of communication in a multiplex network, we focus on the average inverse geodesic length, which we refer to as the multiplex global efficiency e A (γ) by means of the multiplex path length matrix P ∈ R N × N . This paper generalizes the approach proposed in [15] for single-layer networks. We describe an algorithm based on min-plus matrix multiplication to construct P, as well as variants P K that only take into account multiplex paths made up of at most K intra-layer edges. These matrices are applied to detect redundant edges and to determine non-decreasing lower bounds e A K (γ) for e A (γ) , for K = 1 , 2 , ⋯ , N - 2 . Finally, the sensitivity of e A K (γ) to changes of the entries of the adjacency tensor A is investigated to determine edges that should be strengthened to enhance the multiplex global efficiency the most. [ABSTRACT FROM AUTHOR]

Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision. [ABSTRACT FROM AUTHOR]

In this paper, we intend to utilize the generalized scalar auxiliary variable (GSAV) approach proposed in recent paper (Ju et al., SIAM J. Numer. Anal., 60 (2022), 1905–1931) for the nonlocal coupled sine-Gordon equation to construct a class of fully decoupled, linear, and second-order accurate energy-preserving scheme. The unconditional unique solvability and discrete energy conservation law of the proposed scheme are rigorously discussed, and the unconditional convergence is then proved by the mathematical induction. Particularly, the convergence analysis covers the proposed scheme in multiple dimensions due to the corresponding nonlinear terms satisfy the global Lipschitz continuity straightforwardly. Finally, time evolution of dynamical behavior of the governing equation with different nonlocal parameters are observed, and ample numerical comparisons demonstrate that the proposed scheme manifests high efficiency in long-time computations. [ABSTRACT FROM AUTHOR]

In the present paper, we study the approximation of solutions (if any) of a system of equations, where the only condition required to the involved functions is to be continuous. To be more precise, by using the so called α -dense curves, if the given system has some solution in a densifiable set we propose a method, which only use evaluations of single variable functions, to approximate at least one solution of the system in such set. The feasibility and reliability of the proposed method is illustrated by several examples, and its drawbacks are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a branch and bound algorithm for globally solving linear multiplicative programs (LMP). The problem LMP is firstly converted to an equivalent problem (EP2), and then a novel linear relaxation technique is constructed for (EP2) to get the lower bound of the optimum of LMP. Subsequently, the presented region pruning technique aims to eliminate as many areas as possible where the optimal solution does not exist. The convergence analysis of the algorithm is provided, and the number of worst-case iterations required is estimated to obtain an ε -optimal solution. Finally, the numerical experiments show the effectiveness of our presented algorithm. [ABSTRACT FROM AUTHOR]

This paper presents a new conjugate gradient method on Riemannian manifolds and establishes its global convergence under the standard Wolfe line search. The proposed algorithm is a generalization of a Wei-Yao-Liu-type Hestenes-Stiefel method from Euclidean space to the Riemannian setting. We prove that the new algorithm is well-defined, generates a descent direction at each iteration, and globally converges when the step lengths satisfy the standard Wolfe conditions. Numerical experiments on the matrix completion problem demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper studies a stabilized virtual element method for the unsteady Navier-Stokes problems on polygonal meshes. Using "equal-order" virtual elements in space and the Crank-Nicolson scheme in time, we give a fully discrete formula. By introducing the local-projection type stabilizations, the method can not only circumvent the discrete inf-sup condition but also control spurious oscillations caused by high Reynolds numbers. Stability and error estimates for velocity and pressure are analyzed. Particularly, error estimates are derived in which the constants are independent of the negative powers of the viscosity. Several numerical experiments are simulated to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents three tensor randomized extended Kaczmarz methods for solving a tensor inconsistent linear system of equations under the t-product between tensors. The randomized extended Kaczmarz, the randomized extended block Kaczmarz, and the randomized extended greedy block Kaczmarz methods in tensor form are proposed to solve inconsistent tensor linear system of equations, respectively. The convergence of each method is proved. Several numerical examples are given to show the efficiency and effectiveness of our methods. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite-dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language. [ABSTRACT FROM AUTHOR]

Solving large-scale systems of nonlinear equations (SoNE) is a central task in mathematics that traverses different areas of applications. There are several derivative-free methods for finding SoNE solutions. However, most of the methods contributed to find SoNE solutions involve a monotone cost function. Methods dealing with pseudomonotone cost function remain rare. In this paper, we introduce two classes of derivative-free spectral three-term methods to solve large-scale continuous pseudomonotone SoNE. We combine the projection method of Solodov and Svaiter with the structure of the recently developed spectral three-term conjugate gradient method for unconstrained optimization by Amini and Faramarzi. We prove that the proposed methods possess sufficient descent property, trust region property, and global convergence without relying on Lipschitz continuity. Numerical experiments show that the proposed methods are efficient and competitive with existing methods. Finally, the proposed methods have been successfully applied to recover a sparse signal from incomplete and contaminated sampling measurements, yielding promising results. [ABSTRACT FROM AUTHOR]

In this paper, we consider solutions for discrete systems arising from multi-term time-fractional diffusion equations. Using discrete sine transform techniques, we find that all-at-once systems of such equations have a structure similar to that of diagonal-plus-Toeplitz matrices. We establish a generalized circulant approximate inverse preconditioner for the all-at-once systems. Through a detailed analysis of the preconditioned matrices, we show that the spectrum of the obtained preconditioned matrices is clustered around one. We give some numerical examples to demonstrate the effectiveness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

Applying parallel-in-time algorithms to multiscale Hamiltonian systems to obtain stable long-time simulations is very challenging. In this paper, we present novel data-driven methods aimed at improving the standard parareal algorithm developed by Lions et al. in 2001, for multiscale Hamiltonian systems. The first method involves constructing a correction operator to improve a given inaccurate coarse solver through solving a Procrustes problem using data collected online along parareal trajectories. The second method involves constructing an efficient, high-fidelity solver by a neural network trained with offline generated data. For the second method, we address the issues of effective data generation and proper loss function design based on the Hamiltonian function. We show proof-of-concept by applying the proposed methods to a Fermi-Pasta-Ulam (FPU) problem. The numerical results demonstrate that the Procrustes parareal method is able to produce solutions that are more stable in energy compared to the standard parareal. The neural network solver can achieve comparable or better runtime performance compared to numerical solvers of similar accuracy. When combined with the standard parareal algorithm, the improved neural network solutions are slightly more stable in energy than the improved numerical coarse solutions. [ABSTRACT FROM AUTHOR]

This article presents the discovery of an explicit 16-stage Runge–Kutta method that numerically satisfies the Runge–Kutta order conditions (in Butcher form) to 10th order, conjecturally improving the best-known number of stages in an explicit 10th-order Runge–Kutta method from 17 to 16. Unlike the vast majority of published Runge–Kutta methods, the method presented in this paper was not constructed via symbolic analysis or the use of simplifying assumptions, but instead by directly applying numerical optimization and root-finding algorithms to the order conditions. A naïve implementation of these algorithms would be made computationally infeasible by the considerable size and complex structure of the Butcher equations. However, we present a collection of software optimizations that greatly accelerate the evaluation of the Butcher equations and their derivatives while mitigating the effects of destructive cancellation, pushing these techniques within the realm of feasibility. While we do not have a formal proof of order, we present the results of numerical experiments to demonstrate that our method satisfies the Butcher equations to an accuracy of over 3000 decimal digits. [ABSTRACT FROM AUTHOR]

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]

In this paper, we introduce a new framework for deriving partitioned implicit-exponential integrators for stiff systems of ordinary differential equations and construct several time integrators of this type. The new approach is suited for solving systems of equations where the forcing term is comprised of several additive nonlinear terms. We analyze the stability, convergence, and efficiency of the new integrators and compare their performance with existing schemes for such systems using several numerical examples. We also propose a novel approach to visualizing the linear stability of the partitioned schemes, which provides a more intuitive way to understand and compare the stability properties of various schemes. Our new integrators are A-stable, second-order methods that require only one call to the linear system solver and one exponential-like matrix function evaluation per time step. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new method based on a library of chemical reactions for constructing a system of ordinary differential equations from stochastic simulations arising from an agent-based model. The advantage of this approach is that this library respects any coupling between systems components, whereas the SINDy algorithm (introduced by Brunton et al.) treats the individual components as decoupled from one another. Another advantage of our approach is that we can use a non-negative least squares algorithm to find the non-negative rate constants in a very robust, stable and simple manner. We illustrate our ideas on an agent-based model of tumour growth on a 2D lattice. [ABSTRACT FROM AUTHOR]

Considering validated Runge-Kutta schemes requires the computation of local truncation error. Computation of such error is expensive in term of complexity, thus few techniques have been proposed to compute or at least estimate it. For example, Runge-Kutta pairs provide an error estimation by computing the difference of two schemes of different orders while sharing the same stages. With the goal of proposing a validated scheme, an approximation is not sufficient, but could be helpful after some adjustments. In this paper, we compute a new pair and show its performance on a non-trivial problem. Two new techniques for the local truncation error computation are then presented: a symbolic based approach dedicated to a specific problem and a preliminary study of pair's approximation. [ABSTRACT FROM AUTHOR]

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L ∞ (L 2) -norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests. [ABSTRACT FROM AUTHOR]

In this paper, we develop a convex splitting algorithm for the time-dependent Ginzburg-Landau equation, which can preserve both the energy stability and maximum bound principle. The basic idea of the convex splitting method is to decompose the energy functional into the convex part and the concave part. The term corresponding to the convex part of the equation is implicitly treated, and the concave part is explicitly processed. The backward Euler time discretizing method is chosen for the time-dependent Ginzburg-Landau equation. The theoretical analysis proves that the convex splitting method can preserve the maximum bound principle and energy stability. The numerical results show that the numerical algorithm is stable. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the adaptive procedure is to transform the given equation into an equivalent one with a sufficiently smooth behavior in order to ensure the convergence of the Legendre spectral collocation method without dividing the domain of the integral equation, as usual, into the sub intervals. Existence and uniqueness of solutions are discussed. Convergence rates and error estimates are given for the numerical scheme in both L ∞ -norm and L 2 -norm. Finally, several numerical examples are provided to show that the proposed method is preferable to its classical predecessor and some other existing approaches. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of singularly perturbed semilinear parabolic convection-diffusion initial-boundary-value problems exhibiting a boundary layer. This type of model problem often appears in modeling various physical phenomena, particularly, in mathematical biology; and thus, it requires effective numerical techniques for analyzing them computationally. For this purpose, we approximate the considered nonlinear problem by developing two efficient fitted mesh methods followed by the extrapolation technique. The first one is the fully implicit fitted mesh method which utilizes the implicit-Euler method for the temporal discretization; and the other one is the implicit-explicit (IMEX) fitted mesh method which utilizes the IMEX-Euler method for the temporal discretization. The spatial discretization for both the numerical methods is based on a new hybrid finite difference scheme. To accomplish this, the spatial domain is discretized by an appropriate layer-adapted mesh and the time domain by an equidistant mesh. At first, we analyze stability and study the asymptotic behavior of the analytical solution of the governing nonlinear problem. Then, we perform stability analysis and establish the parameter-uniform convergence of both the newly proposed methods in the discrete supremum norm. Thereafter, we analyze the Richardson extrapolation technique solely for the time variable to improve the order of convergence in the temporal direction. Hereby, we provide a comparative error analysis to achieve parameter-robust higher-order numerical approximations (concerning both space and time) for the considered nonlinear problem utilizing two new algorithms on a nonuniform grid. The theoretical outcomes are finally supported by the extensive numerical experiments, which also include comparison of the proposed numerical methods along with the fully-implicit upwind method in terms of the order of accuracy and the computational cost. [ABSTRACT FROM AUTHOR]

This paper considers the iterative solution of finite element discretizations of second-order elliptic boundary value problems. Mesh independent estimations are given for the rate of superlinear convergence of preconditioned Krylov methods, involving the connection between the convergence rate and the Lebesgue exponent of the data. Numerical examples demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a reliable Jacobi Galerkin method is developed and analyzed to solve a particular class of cordial Volterra integral equations. Existence and uniqueness theorems approve that these kinds of equations possess smooth solutions versus smooth data, so representing their Galerkin solutions based on suitable polynomial basis produces spectrally accurate approximations. Moreover, in order to control condition number growth, we designed a reliable approach that calculates the approximate solutions recursively without solving any high-conditioning systems. Indeed, this approach deals with solving equations in a long integration domain with even high oscillatory solutions. The convergence analysis of the presented approach is established, and the familiar spectral accuracy is justified in L ∞ -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method. Finally, we provide an application of this method to approximate solution of a two-dimensional case. [ABSTRACT FROM AUTHOR]

Formerly the geometry was based on shapes, but since the last centuries this founding mathematical science deals with transformations, projections, and mappings. Projective geometry identifies a line with a single point, like the perspective on the horizon line and, due to this fact, it requires a restructuring of the real mathematical and numerical analysis. In particular, the problem of interpolating data must be refocused. In this paper, we define a linear structure along with a metric on a projective space, and prove that the space thus constructed is complete. Then, we consider an iterated function system giving rise to a fractal interpolation function of a set of data. [ABSTRACT FROM AUTHOR]

In this paper, we focus on solving the generalized absolute value equations (GAVE). We present a new method named as modulus-based block triangular splitting iteration (MBTS) method based on the block matrix structure resulting from the transformation of the GAVE into two equations. This method is developed by decomposing the matrix into diagonal and triangular matrices, as well as applying a series of suitable combination and modification techniques. The advantage of the MBTS method is that it is not necessary to solve the inverse of the coefficient matrix of the linear equation system during each iteration, which greatly improves the computational speed and reduces its storage requirements. In addition, we present some convergent theorems proving by different techniques and the estimate of the required number of iteration steps. Furthermore, in the accompanying corollaries, we provide some estimations for choosing appropriate parameter values. Finally, we validated the effectiveness and efficiency of our newly developed method through two numerical examples of the GAVE. [ABSTRACT FROM AUTHOR]

In this paper, we present several remarks on the paper by Yao et al. (citeyearcite.nine). The results presented in the present paper are interesting improvements on the main results of Yao et al. [ABSTRACT FROM AUTHOR]

Singular value decomposition plays a prominent role in the theoretical study and numerical calculation of a quaternion matrix in applied sciences. This paper, by means of a complex representation of a quaternion matrix, studies the algorithm for the singular value decomposition of a quaternion matrix, and derives a complex structure-preserving algorithm for the singular value decomposition of a quaternion matrix. This paper also gives two examples to demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Algorithmic differentiation is a tool used in several branches of computational science, both in conjunction with the interval calculus, and apart of it. This paper presents the ADHC library [14] developed by the author, and identified as package "na60" by Numerical Algorithms journal. This library makes intensive use of the C++ template meta-programming, and it has several unique features. ADHC seems particularly useful for interval-related applications. The library has been used by some solvers, also developed by the author, including HIBA_USNE. The paper describes the library, presenting its features, focusing on the new ones, added in version 2.0; in particular, we describe bounding subdifferentials of non-smooth functions and computing derivatives over various datatypes. Efficiency comparison with respect to other packages is also presented. Then some examples of ADHC applications are given, involving the use of HIBA_USNE and standard benchmark problems for solving nonlinear systems. A particular emphasis is put on examples related to modern machine learning, but not limited to them. Planned extensions and possible directions for future development of ADHC are also outlined and discussed. [ABSTRACT FROM AUTHOR]

The parameterized block splitting (PBS) is a convergent and efficient iterative method to solve the large complex symmetric linear systems. In this paper, by using PBS iterative technique, the Newton equation is approximately solved, then we establish the modified Newton-PBS iterative method to solve the complex nonlinear systems whose Jacobian matrices are large, sparse, and complex symmetric. Subsequently, the local convergence analysis are explored under appropriate conditions. Ultimately, we apply the new method and several known methods to experimental numerical examples, and experimental results verify the superiority and efficiency of our new method. Especially, in terms of CPU time and iteration steps, our method is obviously better. [ABSTRACT FROM AUTHOR]

In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form T (z) = z n + m z + r , where n ≥ 2 and m , r ∈ C , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear. [ABSTRACT FROM AUTHOR]

In this paper, we provide high order hybrid asymptotic augmented finite volume schemes on a uniform grid for nonlinear weakly degenerate and strongly degenerate wave equations. The whole domain is divided into singular and regular subdomains by introducing an intermediate point. Puiseux series asymptotic technique is used in singular subdomain, and augmented numerical method is used in regular subdomain. The key to the method are the recovery of Puiseux series for the nonlinear degenerate wave equation in singular subdomain and the organic combination between the singular and regular subdomains by means of augmented variables related to singularity. In particular, through imposing a condition at the intermediate point, we can not only improve the accuracy of the augmented variables, but also avoid the restriction conditions when the mesh is divided in regular subdomain. The advantage of this method is that the global convergence order of the degenerate wave equation is determined by the augmented numerical scheme in regular subdomain. A rigorous error estimate is conducted for the solution of the degenerate wave equation. Numerical examples on weakly degenerate and strongly degenerate problems are provided to illustrate the effectiveness of the proposed method. Especially, we use the method to solve an interesting example of a degenerate wave equation with coefficient blow-up. [ABSTRACT FROM AUTHOR]

This paper is mainly focused on the solution of Sylvester matrix differential equations with time-independent coefficients. We propose a new approach based on the construction of a particular constant solution which allows to construct an approximate solution of the differential equation from that of the corresponding algebraic equation. Moreover, when the matrix coefficients of the differential equation are large, we combine the constant solution approach with Krylov subspace methods for obtaining an approximate solution of the Sylvester algebraic equation, and thus form an approximate solution of the large-scale Sylvester matrix differential equation. We establish some theoretical results including error estimates and convergence as well as relations between the residuals of the differential and its corresponding algebraic Sylvester matrix equation. We also give explicit benchmark formulas for the solution of the differential equation. To illustrate the efficiency of the proposed approach, we perform numerous numerical tests and make various comparisons with other methods for solving Sylvester matrix differential equations. [ABSTRACT FROM AUTHOR]

This work develops the Milstein scheme for commutative stochastic differential equations with piecewise continuous arguments (SDEPCAs), which can be viewed as stochastic differential equations with time-dependent and piecewise continuous delay. As far as we know, although there have been several papers investigating the convergence and stability for different numerical methods on SDEPCAs, all of these methods are Euler-type methods and the convergence orders do not exceed 1/2. Accordingly, we first construct the Milstein scheme for SDEPCAs in this work and then show its convergence order can reach 1. Moreover, we prove that the Milstein method can preserve the stability of SDEPCAs. In the last section, we provide several numerical examples to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a comprehensive framework for studying Krylov subspace methods used to solve the linear system A x = f . These methods aim to achieve convergence within a specified number of iterations, denoted by m, given a particular initial estimate vector x 0 and its corresponding residual r 0 = f - A x 0 . Our analysis focuses on the minimal polynomial Φ m of degree m of A for the vector r 0 . We establish that these methods encompass Petrov-Galerkin methods and minimal seminorm methods as special cases. Additionally, we demonstrate that minimal seminorm methods satisfy implicit Petrov-Galerkin conditions. We provide a general formulation for the iterates based on generalized inverses. The choice of a specific left inverse and the method of constructing the Krylov basis are crucial distinguishing factors among different Krylov subspace methods. We describe and analyze the mathematical properties of these methods, emphasizing their dependency on two matrices. Notably, we prove that CMRH and QMR, as specific instances, also satisfy implicit Petrov-Galerkin orthogonality conditions. Furthermore, we explore techniques to improve the convergence behavior of these methods by carefully selecting vectors in their implementations. Through our investigation, we aim to deepen the understanding of Krylov subspace methods, provide insights into their convergence properties, and identify potential enhancements. We also consider some Krylov subspace methods, which are of product-type methods. In this case, the kth residual r k associated with the approximation x k of the exact solution is given by r k = Ψ k (A) Φ k (A) r 0 , and Ψ k is a polynomial of fixed or variable degree. We will examine particular choices of Ψ k involving local convergence, smoothing, fixed memory, and cost for each iteration. We will also give an enhancement of some product-type methods such as CGS. To illustrate the performance of the derived algorithms, we provide some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the forward rounding errors of the partial pivoting quaternion LU decomposition. To carry out error analysis of algorithms in quaternion arithmetic, we first propose the accuracy model for the basic quaternion arithmetic operations by the real structure-preserving method. And then, we study the forward rounding errors of the partial pivoting quaternion LU decomposition. Finally, by using the results of the forward rounding errors of the partial pivoting quaternion LU decomposition, we analyze the forward rounding errors of the nonsingular quaternion linear system. [ABSTRACT FROM AUTHOR]

A novel local and parallel multigrid method is proposed in this study for solving the semilinear Neumann problem with nonlinear boundary condition. Instead of solving the semilinear Neumann problem directly in the fine finite element space, we transform it into a linear boundary value problem defined in each level of a multigrid sequence and a small-scale semilinear Neumann problem defined in a low-dimensional correction subspace. Furthermore, the linear boundary value problem can be efficiently solved using local and parallel methods. The proposed process derives an optimal error estimate with linear computational complexity. Additionally, compared with existing multigrid methods for semilinear Neumann problems that require bounded second order derivatives of nonlinear terms, ours only needs bounded first order derivatives. A rigorous theoretical analysis is proposed in this paper, which differs from the maturely developed theories for equations with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we establish a modulus-based nonsmooth Newton's method for solving a class of horizontal nonlinear complementarity problems and prove the nearly quadratic convergence of the method under certain assumptions. Based on the analysis of convergence, we give the selection strategy for the approximate iteration vector and propose the improved modulus-based nonsmooth Newton's method. Finally, numerical experiments show that the new methods are superior to the modulus-based matrix iterative method regarding computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proximal variable smoothing gradient (ProxVSG) method for solving this kind of problem. The ProxVSG can be implemented efficiently, thanks to the fact that at each iteration, one just separately computes the proximal mapping of each nonsmooth function, rather than that of the sum of these nonsmooth functions. Furthermore, within our broad and flexible analysis framework, we propose a new proximal variable smoothing incremental aggregated gradient (ProxVSIAG) generalizations of the ProxVSG. In ProxVSIAG, an incremental aggregated estimate of the gradient is used, instead of the full gradient. Under suitable assumptions, we prove a complexity of O (ϵ - 3) to achieve an ϵ -approximate solution. Preliminary numerical experiments show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Recently, Norvidas has introduced the general multidimensional Hermite sampling series, which involves samples from a function and its mixed and non-mixed partial derivatives. The convergence of this sampling series is slow unless the sample values |f(x)| rapidly decay as | x j | → ∞ for all 1 ≤ j ≤ n . In this paper, we investigate a modified version of this sampling series that utilizes a multivariate Gaussian multiplier to approximate functions from two classes of multivariate analytic functions using a complex approach. The first class comprises entire functions of exponential type in n variables that fulfill a decay condition, while the second class includes analytic functions in n variables defined on a multidimensional horizontal strip. It has a significantly higher convergence rate compared to the general multidimensional Hermite sampling series. [ABSTRACT FROM AUTHOR]

In this paper, we employ Tikhonov regularization method with the projection methods using Jacobi polynomial bases to the first kind of Fredholm integral equations to find the approximate solution. We discuss the convergence analysis and obtain the convergence rates in L w α , β 2 norm under a priori parameter choice strategy. We also consider the Engl-type discrepancy principle as a posteriori parameter strategy for finding the regularization parameter and also evaluate the convergence rate which is of optimal order. Finally, we provide the numerical experiments to justify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach. [ABSTRACT FROM AUTHOR]

Runge–Kutta embedded pairs of high algebraic order are frequently utilized when strict tolerances are required. When creating such pairings of orders nine and eight for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients' magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders 9 and 8 with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems. [ABSTRACT FROM AUTHOR]

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

In this paper, we develop a polygonal mesh adaptation algorithm for a fully implicit scheme based on discontinuous Galerkin (DG) finite element methods in space and backward Euler method in time to solve the Allen-Cahn equation. The mathematical framework and a procedure for solving this nonlinear equation are given. We extend the DG discretization with a polygonal mesh adaptation method to save computation time and capture the thin interfaces more accurately. A criterion based on the local value of the phase field function gradient is used to select the target element for refinement and coarsening, and then a 4-node polygonal mesh refinement strategy is adopted by connecting the midpoint of each edge to the barycenter of the target element. Using numerical tests, including motion by the mean curvature, curvature-driven flow, the Allen-Cahn equation with a logarithmic free energy, the Allen-Cahn equation with advection, and the application for image segmentation, we verify the accuracy, efficiency, and capabilities of the adaptive DG on polygonal meshes and confirm the decreasing property of the discrete energy. [ABSTRACT FROM AUTHOR]

In this paper, we propose SGEM, stochastic gradient with energy and momentum, to solve a class of general non-convex stochastic optimization problems, based on the AEGD method introduced in AEGD (adaptive gradient descent with energy) Liu and Tian (Numerical Algebra, Control and Optimization, 2023). SGEM incorporates both energy and momentum so as to inherit their dual advantages. We show that SGEM features an unconditional energy stability property and provide a positive lower threshold for the energy variable. We further derive energy-dependent convergence rates in the general non-convex stochastic setting, as well as a regret bound in the online convex setting. Our experimental results show that SGEM converges faster than AEGD and generalizes better or at least as well as SGDM in training some deep neural networks. [ABSTRACT FROM AUTHOR]