Description: In this note, we show that the proof of Remark 3 of Lemma 3.2 in “A three-term derivative-free projection method for nonlinear monotone system of equations” (Calcolo 53:427–450, 2016 ) is not correct, which implies that the conclusion of Remark 3 is not appropriate to prove Theorem 3.1. A new proof of Remark 3 is established, which guarantees the corresponding global convergence Theorem 3.1. Throughout, we use the same notations and equation numbers as in the above reference.

Description: In this paper, we establish a general theorem for iteration functions in a cone normed space over \({{\mathbb {R}}}^n\) . Using this theorem together with a general convergence theorem of Proinov (J Complex 33:118–144, 2016 ), we obtain a local convergence theorem with a priori and a posteriori error estimates as well as a theorem under computationally verifiable initial conditions for the Schröder’s iterative method considered as a method for simultaneous computation of polynomial zeros of unknown multiplicity. Numerical examples which demonstrate the convergence properties of the proposed method are also provided.

Description: In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp -version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p , provided that the number of smoothing steps, which depends on p , is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed multilevel solvers are shown to be convergent in practice, even when some of the theoretical assumptions are not fully satisfied.

Description: In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the data space and the solution space are measured by the weighted Frobenius norm and the Euclidean norm, respectively. Then, we investigate the corresponding structured partial condition number when the problem is structured. To estimate these condition numbers with high reliability, the probabilistic spectral norm estimator and the small-sample statistical condition estimation method are applied and two algorithms are devised. The obtained results are illustrated by numerical examples.

Description: An extended QR algorithm specifically tailored for Hamiltonian matrices is presented. The algorithm generalizes the customary Hamiltonian QR algorithm with additional freedom in choosing between various possible extended Hamiltonian Hessenberg forms. We introduced in Ferranti et al. (Calcolo, 2015 . doi: 10.1007/s10092-016-0192-1 ) an algorithm to transform certain Hamiltonian matrices to such forms. Whereas the convergence of the classical QR algorithm is related to classical Krylov subspaces, convergence in the extended case links to extended Krylov subspaces, resulting in a greater flexibility, and possible enhanced convergence behavior. Details on the implementation, covering the bidirectional chasing and the bulge exchange based on rotations are presented. The numerical experiments reveal that the convergence depends on the selected extended forms and illustrate the validity of the approach.

Description: In this article, a parameter-uniform hybrid numerical method is presented to solve a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms. Due to these discontinuities, interior layers appear in the solution of the problem considered. The hybrid numerical method uses the standard finite difference scheme in the coarse mesh region and the cubic spline difference scheme in the fine mesh region which is constructed on piecewise-uniform Shishkin mesh. Second order one sided difference approximations are used at the point of discontinuity. Error analysis is carried out and the method ensures that the parameter-uniform convergence of almost the second order. Numerical results are provided to validate the theoretical results.

Description: This paper studies the problem of approximating a function f in a Banach space \(\mathcal{X}\) from measurements \(l_j(f)\) , \(j=1,\ldots ,m\) , where the \(l_j\) are linear functionals from \(\mathcal{X}^*\) . Quantitative results for such recovery problems require additional information about the sought after function f . These additional assumptions take the form of assuming that f is in a certain model class \(K\subset \mathcal{X}\) . Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B , called the Chebyshev ball , which contains the set \(\bar{K}\) of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces \(\mathcal{X}\) such as the \(L_p\) spaces, \(1\le p\le \infty \) , and for K the unit ball of a smoothness space in \(\mathcal{X}\) . Our interest in this paper is in the model classes \(K=\mathcal{K}(\varepsilon ,V)\) , with \(\varepsilon 〉0\) and V a finite dimensional subspace of \(\mathcal{X}\) , which consists of all \(f\in \mathcal{X}\) such that \(\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon \) . These model classes, called approximation sets , arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933–965, 2015 ) for the case when \(\mathcal{X}\) is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015 ). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation.

Description: In this paper an exponentially fitted spline method is presented for solving singularly perturbed convection delay problems with boundary layer at left (or right) end of the domain. The error analysis of the scheme is investigated. It is shown that the proposed scheme provides second order accuracy, independent of the perturbation parameter. Numerical results are presented to illustrate the efficiency of the method.

Description: We consider the problem of solving a rational matrix equation arising in the solution of G-networks. We propose and analyze two numerical methods: a fixed point iteration and the Newton–Raphson method. The fixed point iteration is shown to be globally convergent with linear convergence rate, while the Newton method is shown to have a local convergence, with quadratic convergence rate. Numerical experiments show the effectiveness of the proposed methods.

Description: We study the asymptotic behavior of harmonic interpolation of harmonic functions based on Radon projections when the chords coalesce to some points, a chord and a point. We show that the limit is the Lagrange or Taylor-type interpolation at coalescing points or chords.