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PageRank is an important technique for determining the most important nodes in the Web. In this paper, we point out that the main result derived in Bao and Zhu [Acta Math Appl Sin (Engl Ser) 22:517–528, 2006] is incorrect, and establish new lower and upper bounds on the convergence of the minimally irreducible Markov chain method for PageRank. We show that the asymptotic convergence rates of the maximally...
We investigate two greedy strategies for finding an approximation to the minimum of a convex function E defined on a Hilbert space H. We prove convergence rates for these algorithms under suitable conditions on the objective function E. These conditions involve the behavior of the modulus of smoothness and the modulus of uniform convexity of E.
In this paper we propose and analyze a new fully-mixed finite element method for the stationary Boussinesq problem. More precisely, we reformulate a previous primal-mixed scheme for the respective model by holding the same modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid; and in contrast, we now introduce a...
We implement the sinc method to compute the eigenvalues of a second order boundary value problem with mixed type boundary conditions where the eigenparameter appears linearly in the boundary conditions. We investigate the behavior of the solutions as well as the characteristic determinant via successive iterations. The method is implemented by splitting the characteristic determinant into two parts,...
In this paper, we present a methodology for stabilizing the virtual element method applied to the convection-diffusion-reaction equation. The stabilization is carried out modifying the mesh inside the boundary layer so that the link-cutting condition is satisfied. The method provides a stable solution to all regimes. Numerical examples are presented for several regimes in which satisfactory results...
In this paper a novel operational matrix of derivatives of certain basis of Legendre polynomials is established. We show that this matrix is expressed in terms of the harmonic numbers. Moreover, it is utilized along with the collocation method for handling initial value problems of any order. The convergence and the error analysis of the proposed expansion are carefully investigated. Numerical examples...
In order to find the least squares solution of minimal norm to linear system $$Ax=b$$ A x = b with $$A \in \mathcal{C}^{m \times n}$$ A ∈ C m × n being a matrix of rank $$r< n \le m$$ r < n ≤ m , $$b \in \mathcal{C}^{m}$$ b ∈ C m , Zheng and Wang (Appl Math Comput 169:1305–1323, 2005) proposed a class of symmetric successive overrelaxation...
The Peaceman-Rachford splitting method (PRSM) is an efficient approach for two-block separable convex programming. In this paper we extend this method to the general case where the objective function consists of the sum of multiple convex functions without coupled variables, and present a generalized PRSM. Theoretically, we prove global convergence of the new method and establish the worst-case convergence...
The present study introduces results about unique solvability of Gaussian RBF interpolation with the different data sites and basis centers. For $$ N=2 $$ N = 2 , we show that the interpolation matrix is singular only when the vector of difference between basis centers and the vector of difference between data sites are perpendicular to each other. For $$N>2$$ N > 2 , we...
This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments,...
In this paper, based on the implicit fixed-point equation of the linear complementarity problem (LCP), a generalized Newton method is presented to solve the non-Hermitian positive definite linear complementarity problem. Some convergence properties of the proposed generalized Newton method are discussed. Numerical experiments are presented to illustrate the efficiency of the proposed method.
In the present work we study how the Aitken’s method can be applied to the sequence of moments $$\{c_k=(x, A^{k}x),~k \in {\mathbb {Z}}\},$$ { c k = ( x , A k x ) , k ∈ Z } , of a given symmetric positive definite matrix $$A \in {\mathbb {R}}^{p \times p},$$ A ∈ R p × p , for the prediction of possible unknown terms of this sequence. Direct estimation of $$c_k$$...
A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the...
The aim of the present paper is to analyze the behavior of Fiedler companion matrices in the polynomial root-finding problem from the point of view of conditioning of eigenvalues. More precisely, we compare: (a) the condition number of a given root $${\lambda }$$ λ of a monic polynomial p(z) with the condition number of $${\lambda }$$ λ as an eigenvalue of any Fiedler matrix of p...
The B-spline collocation methods and a new ODEs solver based on B-spline quasi-interpolation are developed to study the problem of forced convection over a horizontal flat plate, numerically. The problem is a system of nonlinear ordinary differential equations which arises in boundary layer flow. A more accurate value of $$\sigma =f^{\prime \prime }(0)$$ σ = f ″ ( 0 ) obtained by applying...
Although numerical methods of nonlinear stochastic differential delay equations (SDDEs) have been discussed by several authors, there is so far little work on the numerical approximation of SDDE with coefficients of polynomial growth. The main aim of the paper is to investigate convergence in probability of the Euler-Maruyama (EM) approximate solution for SDDE with one-sided polynomial growing drift...
A new class of objective functions and an associated fast descent algorithm that generalizes the K-means algorithm is presented. The algorithm represents clusters as unions of Voronoi cells and an explicit, but efficient, combinatorial search phase enables the algorithm to escape many local minima with guaranteed descent. The objective function has explicit penalties for gaps between clusters. Numerical...
Finding numerical solutions to the Troesch’s problem is known to be challenging, especially when the sensitivity parameter $$\lambda $$ λ is large. In this manuscript, we propose a numerical method for solving the Troesch’s problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin...
This paper deals with the choice of stabilization parameter for the grad-div stabilization applied to the generalized Oseen equations. In particular, inf-sup stable conforming pairs of finite element are used to derive the stabilization parameter on the basis of minimizing the $$H^1({\varOmega })$$ H 1 ( Ω ) error of the velocity. For the proposed choice of the parameter, the $$H^1({\varOmega...
We prove discrete Korn’s inequalities for Naghdi and Koiter shell models, which are applicable to discontinuous piecewise functions. They are useful in study of discontinuous finite element methods for shells.
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