An analysis on the efficiency of Euler's method for computing the matrix pth root

It is shown that the matrix sequence generated by Euler's method starting from the identity matrix converges to the principal pth root of a square matrix, if all the eigenvalues of the matrix are in a region including the one for Newton's method given by Guo in 2010. The convergence is cubic if the matrix is invertible. A modification version of Euler's method using the Schur decomposition is developed. Numerical experiments show that the modified algorithm has the overall good numerical behavior.     

Central Schur rings over the projective special linear groups

In [1 Bannai, E. (1991). Subschemes of some association schemes. J. Algebra 14:167–188.[Crossref], [Web of Science ®] , [Google Scholar]], Bannai presents a fusion condition and uses this to consider central Schur rings (S-rings) over the simple groups PSL(2,q) where q is a prime power. In this paper, we concretely describe all such S-rings in terms of symmetric S-rings over cyclic groups. The final section discusses counting these.     

A spectral element method using the modal basis and its application in solving second‐order nonlinear partial differential equations

We present a high‐order spectral element method (SEM) using modal (or hierarchical) basis for modeling of some nonlinear second‐order partial differential equations in two‐dimensional spatial space. The discretization is based on the conforming spectral element technique in space and the semi‐implicit or the explicit finite difference formula in time. Unlike the nodal SEM, which is based on the Lagrange polynomials associated with the Gauss–Lobatto–Legendre or Chebyshev quadrature nodes, the Lobatto polynomials are used in this paper as modal basis. Using modal bases due to their orthogonal properties enables us to exactly obtain the elemental matrices provided that the element‐wise mapping has the constant Jacobian. The difficulty of implementation of modal approximations for nonlinear problems is treated in this paper by expanding the nonlinear terms in the weak form of differential equations in terms of the Lobatto polynomials on each element using the fast Fourier transform (FFT). Utilization of the Fourier interpolation on equidistant points in the FFT algorithm and the enough polynomial order of approximation of the nonlinear terms can lead to minimize the aliasing error. Also, this approach leads to finding numerical solution of a nonlinear differential equation through solving a system of linear algebraic equations. Numerical results for some famous nonlinear equations illustrate efficiency, stability and convergence properties of the approximation scheme, which is exponential in space and up to third‐order in time. Copyright © 2014 John Wiley & Sons, Ltd.     

Prediction error sampling procedure based on dominant Schur decomposition. Application to state estimation in high dimensional oceanic model

Iterative procedure is described to generate patterns of dominant Schur vectors of the system dynamics. Their roles in estimating the filter gain is study. These patterns are produced by several integrations of the model from a set of perturbations. This approach is motivated by a number of interesting results on stability of the filter whose gain is approximated in a subspace of dominant Schur vectors. A simple method for the filter design is presented which is aimed at overcoming the most serious drawback of advanced filtering algorithms for high dimensional systems related to very high computational cost in evaluation of the filter gain.The resulting filter will be compared with the existing ones, showing its relevance from a practical point of view. In order to demonstrate its efficiency, the new filter is tested on various experiments. These experiments include the much studied problem of estimating the solution of the Lorenz system as well as that of assimilating sea surface height observations in a high dimensional oceanic model. It is shown that significant increases in efficiency can be obtained by using this filter and that the proposed filter is very promising for solving realistic assimilation problems in meteorology and oceanography.     

On perfect hashing of numbers with sparse digit representation via multiplication by a constant

Consider the set of vectors over a field having non-zero coefficients only in a fixed sparse set and multiplication defined by convolution, or the set of integers having non-zero digits (in some base b) in a fixed sparse set. We show the existence of an optimal (or almost-optimal, in the latter case) ‘magic’ multiplier constant that provides a perfect hash function which transfers the information from the given sparse coefficients into consecutive digits. Studying the convolution case we also obtain a result of non-degeneracy for Schur functions as polynomials in the elementary symmetric functions in positive characteristic.     

Robust multigrid preconditioners for the high‐contrast biharmonic plate equation

We study the high‐contrast biharmonic plate equation with Hsieh–Clough–Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11 :319–331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high‐contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high‐contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright © 2010 John Wiley & Sons, Ltd.     

Schur positivity and log-concavity related to longest increasing subsequences

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen’s log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. This also enables us to propose a new approach to Chen’s original conjecture.     

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ?‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ?‐Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n³r) algorithm for computing the (unique) solution.