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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]

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, we consider the numerical treatment of singular eigenvalue problems supplied with eigenparameter dependent boundary conditions using spectral methods. On the one hand, such boundary conditions hinder the construction of test and trial space functions which could incorporate them and thus providing well-conditioned Galerkin discretization matrices. On the other hand, they can generate surprising behavior of the eigenvectors hardly detected by analytic methods. These singular problems are often indirectly approximated by regular ones. We argue that spectral collocation as well as tau method offer remedies for the first two issues and provide direct and efficient treatment to such problems. On a finite domain, we consider the so-called Petterson-König's rod eigenvalue problem and on the half line, we take into account the Charney's baroclinic stability problem and the Fourier eigenvalue problem. One boundary condition in these problems depends on the eigenparameter and additionally, this also could depend on some physical parameters. The Chebyshev collocation based on both, square and rectangular differentiation and a Chebyshev tau method are used to discretize the first problem. All these schemes cast the problems into singular algebraic generalized eigenvalue ones which are solved by the QZ and/or Arnoldi algorithms as well as by some target oriented Jacobi-Davidson methods. Thus, the spurious eigenvalues are completely eliminated. The accuracy of square Chebyshev collocation is roughly estimated and its order of approximation with respect to the eigenvalue of interest is determined. For the problems defined on the half line, we make use of the Laguerre-Gauss-Radau collocation. The method proved to be reliable, accurate, and stable with respect to the order of approximation and the scaling parameter. [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]

An expanded mixed element method is presented to solve a strongly nonlinear elliptic problem. Existence and uniqueness of approximation solution are analyzed. Error estimates in L and H norms are also obtained in this paper. To solve the resulting nonlinear system of equations efficiently, we use a two-grid algorithm to decompose the nonlinear system into a small nonlinear system on a coarse grid with mesh size H and a linear system on a fine grid with mesh size h. It's shown that the approximation still achieves asymptotically optimal as long as the mesh sizes satisfy H= O( h). Some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed numerical algorithm. [ABSTRACT FROM AUTHOR]

The multiple-sets split equality problem, a generalization and extension of the split feasibility problem, has a variety of specific applications in real world, such as medical care, image reconstruction, and signal processing. It can be a model for many inverse problems where constraints are imposed on the solutions in the domains of two linear operators as well as in the operators’ ranges simultaneously. Although, for the split equality problem, there exist many algorithms, there are but few algorithms for the multiple-sets split equality problem. Hence, in this paper, we present a relaxed two points projection method to solve the problem; under some suitable conditions, we show the weak convergence and give a remark for the strong convergence method in the Hilbert space. The interest of our algorithm is that we transfer the problem to an optimization problem, then, based on the model, we present a modified gradient projection algorithm by selecting two different initial points in different sets for the problem (we call the algorithm as two points algorithm). During the process of iteration, we employ subgradient projections, not use the orthogonal projection, which makes the method implementable. Numerical experiments manifest the algorithm is efficient. [ABSTRACT FROM AUTHOR]

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated. [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, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area. [ABSTRACT FROM AUTHOR]

This paper is concerned with the reformulation of Neumann-Neumann waveform relaxation (NNWR) methods and Dirichlet-Neumann waveform relaxation (DNWR) methods, a family of parallel space-time approaches to solving time-dependent PDEs. By changing the order of the operations, pipeline-parallel computation of the waveform iterates are possible, without changing the solution of each waveform iterate. The parallel efficiency of the pipeline implementation is analyzed, as well as the change in the communication pattern. Numerical studies are presented to show the effectiveness of the pipeline NNWR and DNWR algorithms. [ABSTRACT FROM AUTHOR]

We investigate the optimal solution of systems of initial-value problems with smooth right-hand side functions f from a Hölder class $F^{r,\varrho }_{\text {reg}}$, where r ≥ 0 is the number of continuous derivatives of f, and ϱ ∈ (0, 1] is the Hölder exponent of rth partial derivatives. We consider algorithms that use n evaluations of f, the ith evaluation being corrupted by a noise δ of deterministic or random nature. For δ ≥ 0, in the deterministic case the noise δ is a bounded vector, ∥ δ∥≤ δ. In the random case, it is a vector-valued random variable bounded in average, ( E(∥ δ∥)) ≤ δ, q ∈ [1, + ∞). We point out an algorithm whose L error ( p ∈ [0, + ∞]) is O( n + δ), independently of the noise distribution. We observe that the level n + δ cannot be improved in a class of information evaluations and algorithms. For ε > 0, and a certain model of δ-dependent cost, we establish optimal values of n( ε) and δ( ε) that should be used in order to get the error at most ε with minimal cost. [ABSTRACT FROM AUTHOR]

A large scale nonsymmetric algebraic Riccati equation X C X − X E − A X + B = 0 arising in transport theory is considered, where the n × n coefficient matrices B, C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop count of original SDA_ls by half. Numerical experiments illustrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in Després (Numer. Algorithms 76(3), 829–859, 2017) and Després and Herda (Numer. Algorithms 77(1), 309–311, 2018) where an interpretation of monovariate polynomials with two bounds is provided in terms of a quaternion algebra and the Euler four-squares formulas. Thanks to this structure, we generate a new nonlinear projection algorithm onto the set of polynomials with two bounds. The numerical analysis of the method provides theoretical error estimates showing stability and continuity of the projection. Some numerical tests illustrate this novel algorithm for constrained polynomial approximation. [ABSTRACT FROM AUTHOR]

We consider the problem of symbolic-numeric integration of symbolic functions, focusing on rational functions. Using a hybrid method allows the reliable yet efficient computation of symbolic antiderivatives while avoiding issues of ill-conditioning to which numerical methods are susceptible. We propose two alternative methods for exact input that compute the rational part of the integral using Hermite reduction and then compute the transcendental part two different ways using a combination of exact integration and efficient numerical computation of roots. The symbolic computation is done within bpas, or Basic Polynomial Algebra Subprograms, which is a highly optimized environment for polynomial computation on parallel architectures, while the numerical computation is done using the highly optimized multiprecision rootfinding package MPSolve. We provide for both algorithms computable expressions for the first-order term of a structured forward and backward error and show how, away from singularities, tolerance proportionality is achieved by adjusting the precision of the rootfinding tasks. [ABSTRACT FROM AUTHOR]

Based on the alternating direction method of multipliers (ADMM), we develop three numerical algorithms incrementally for solving the optimal control problems constrained by random Helmholtz equations. First, we apply the standard Monte Carlo technique and finite element method for the random and spatial discretization, respectively, and then ADMM is used to solve the resulting system. Next, combining the multi-modes expansion, Monte Carlo technique, finite element method, and ADMM, we propose the second algorithm. In the third algorithm, we preprocess certain quantities before the ADMM iteration, so that nearly no random variable is in the inner iteration. This algorithm is the most efficient one and is easy to implement. The error estimates of these three algorithms are established. The numerical experiments verify the efficiency of our algorithms. [ABSTRACT FROM AUTHOR]

We solve the Dirichlet and mixed Dirichlet-Neumann boundary value problems for the variable coefficient Cauchy-Navier equations of elasticity in a square using a Legendre spectral Galerkin method. The resulting linear system is solved by the preconditioned conjugate gradient (PCG) method with a preconditioner which is shown to be spectrally equivalent to the matrix of the resulting linear system. Numerical tests demonstrating the convergence properties of the scheme and PCG are presented. [ABSTRACT FROM AUTHOR]

Based on the modified relaxed splitting (MRS) preconditioner proposed by Fan and Zhu (Appl. Math. Lett. 55, 18-26 2016), an inexact modified relaxed splitting (IMRS) preconditioner is proposed for the generalized saddle point problems arising from the incompressible Navier-Stokes equations. The eigenvalues and eigenvectors of the preconditioned matrix are analyzed, and the convergence property of the corresponding iteration method is also discussed. Numerical experiments are presented to show the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES. [ABSTRACT FROM AUTHOR]

An algorithm for fast and accurate evaluation of band-limited functions at many scattered points on the unit 2-d sphere is developed. The algorithm is based on trigonometric representation of spherical harmonics in spherical coordinates and highly localized tensor-product trigonometric kernels (needlets). It is simple, fast, local, memory efficient, numerically stable and with guaranteed accuracy. Comparison of this algorithm with other existing algorithms in the literature is also presented. [ABSTRACT FROM AUTHOR]

New bounds for the infinity norm of the inverse of Nekrasov matrices, which involve a parameter, are given. And then we determine the optimal value of the parameter such that the new bounds are better than those in Cvetkovic et al. (Appl. Math. Comput. 219, 5020-5024, ). Numerical examples are given to illustrate the corresponding results. [ABSTRACT FROM AUTHOR]

We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing. [ABSTRACT FROM AUTHOR]

We propose a generalization of the limited-memory Cholesky factorization of Lin and Moré (SIAM J. Sci. Comput. 21(1), 24-45, ) to the symmetric indefinite case with special interest in symmetric quasi-definite matrices. We use this incomplete factorization to precondition two formulations of linear systems arising from regularized interior-point methods for quadratic optimization. An advantage of the limited-memory approach is predictable memory requirements. We establish existence of incomplete factors when the input matrix is an H-matrix but our numerical results illustrate that the factorization succeeds more generally. An appropriate diagonal shift is applied whenever the input matrix is not quasi definite. As the memory parameter increases an efficiency measure of the preconditioner suggested by Scott and Tůma () improves. The combination of the 3×3 block formulation analyzed by Greif, Moulding, and Orban (SIAM J. Optim. 24(1), 49-83, ), the SYMAMD ordering, and a moderate memory parameter results in encouraging performance. [ABSTRACT FROM AUTHOR]