Hermite–Padé approximation and simultaneous quadrature formulas

We study Hermite–Padé approximation of the so-called Nikishin systems of functions. In particular, the set of multi-indices for which normality is known to take place is considerably enlarged as well as the sequences of multi-indices for which convergence of the corresponding simultaneous rational approximants takes place. These results are applied to the study of the convergence properties of simultaneous quadrature rules of a given function with respect to different weights.     

Duality in Padé-type approximation

Padé and Padé-type approximants are usually defined by replacing the function (1 − xt)⁻¹ by its Hermite (that is confluent) interpolation polynomial and then applying the functional c defined by c(xⁱ) = c_i where the c_i's are the coefficients of the series to be approximated. In this paper the functional d which, applied to (1 − xt)⁻¹, gives the same Padé or Padé-type approximant as before is studied. It can be considered as the dual of the interpolation operator applied to the functional c.     

On the Drazin inverse of block matrices and generalized Schur complement

In this paper, we give an additive result for the Drazin inverse with its applications, we obtain representations for the Drazin inverse of a 2 × 2 complex block matrix having generalized Schur complement S=D-CA^DB equal to zero or nonsingular. Several situations are analyzed and recent results are generalized [R.E. Hartwig, X. Li, Y. Wei, Representations for the Drazin inverse of a 2×2 block matrix, SIAM J. Matrix Anal. Appl. 27 (3) (2006) 757-771].     

A note on the Drazin inverses with Banachiewicz-Schur forms

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M^⊥. According to the orthogonal decomposition H=M⊕M^⊥, every operator M∈B(H) can be written in a block-form . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz-Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore-Penrose inverse and the group inverse when they can be expressed in the Banachiewicz-Schur forms.     

Hermite矩阵广义Schur补的特征值交错性质

研究了Hermite矩阵的Schur补的特征值交错性质,将Smith建立的结果推广到Hermite矩阵的广义Sclmr补.     

Theoretical and numerical considerations about Padé approximants for the matrix logarithm

We show that for a vast class of matrix Lie groups, which includes the orthogonal and the symplectic, diagonal Padé approximants of log((1+x)/(1−x)) are structure preserving. The conditioning of these approximants is analyzed. We also present a new algorithm for the Briggs–Padé method, based on a strategy for reducing the number of square roots in the inverse scaling and squaring procedure.     

Computing the determinants of matrix Padé approximation

This paper is devoted to a computational problem of two special determinants which appear in the construction of generalized inverse matrix Padé approximants of type [n/2k] for the given power series with matrix coefficients. The main tools to be used are well-known Schur complement theorem and Arnoldi process for skew-symmetric systems.     

A note on the Drazin inverse of an anti-triangular matrix

In this paper we give formulae for the generalized Drazin inverse M^d of an anti-triangular matrix M under some conditions. Moreover, some particular cases of these results are also considered.     

A numerical solution of the elasticity problem of settled of the wronkler ground with variable coefficients

In this paper, we have given numerical solution of the elasticity problem of settled on the wronkler ground with variable coefficient. The approximation solution of boundary value problem which is pertinent to this has been converted to integral equations, and then by using the successive approximation methods, has been reached. In addition to this, the approximation solution of the problem was put into Padé series form. We applied these methods to an example which is the elasticity problem of unit length homogeny beam, which is a special form of boundary value problem. First we calculate the successive approximation of the given boundary value problem then transform it into Padé series form, which give an arbitrary order for solving differential equation numerically.     

Discrete non-local boundary conditions for split-step Padé approximations of the one-way Helmholtz equation

This paper deals with the efficient numerical solution of the two-dimensional one-way Helmholtz equation posed on an unbounded domain. In this case, one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of the so-called discrete transparent boundary conditions for state-of-the-art parabolic equation methods, namely a split-step discretization of the high-order parabolic approximation and the split-step Padé algorithm of Collins. Finally, several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach.     

Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem

In this paper, the left and right inverse eigenpairs problem for generalized centrosymmetric matrices is considered. We obtain the necessary and sufficient conditions for the solvability of the problem, and we present the general expression of the solution. The related optimal approximation problem to a given matrix over the solution set is solved. In addition, a numerical algorithm and examples to solve the problem are given.     

Perturbation analysis for the generalized Schur complement of a positive semi‐definite matrix

Let and S=C?B^HA^?B be the generalized Schur complement of A?0 in P. In this paper, some perturbation bounds of S are presented which generalize the result of Stewart (Technical Report TR‐95‐38, University of Maryland, 1995) and enrich the perturbation theory for the Schur complement. Also, an error estimate for the smallest perturbation of C, which lowers the rank of P, is discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A note on the approximability of the toughness of graphs

We show that, if NP≠ZPP, for any >0, the toughness of a graph with n vertices is not approximable in polynomial time within a factor of . We give a 4-approximation for graphs with toughness bounded by and we show that this result cannot be generalized to graphs with a bounded toughness. More exactly we prove that there is no constant approximation for graphs with bounded toughness, unless P=NP.     

On the inverse of a general tridiagonal matrix

In the current paper a new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented. The algorithm is suited for implementation using computer algebra systems such as MAPLE, MATHEMATICA, MATLAB and MACSYMA. Symbolic and numeric examples are given.     

A Padé family of iterations for the matrix sign function and related problems

In this paper we consider the Pad'e family of iterations for computing the matrix sign function and the Padé family of iterations for computing the matrix p‐sector function. We prove that all the iterations of the Padé family for the matrix sign function have a common convergence region. It completes a similar result of Kenney and Laub for half of the Padé family. We show that the iterations of the Padé family for the matrix p‐sector function are well defined in an analogous common region, depending on p. For this purpose we proved that the Padé approximants to the function (1?z)^?σ, 0<σ<1, are a quotient of hypergeometric functions whose poles we have localized. Furthermore we proved that the coefficients of the power expansion of a certain analytic function form a positive sequence and in a special case this sequence has the log‐concavity property. Copyright © 2011 John Wiley & Sons, Ltd.     

A note on the stability of a pth order iteration for finding generalized inverses

A number of papers intended for the numerical computation of generalized inverses by means of higher-order iterative methods have been published recently. This note investigates the numerical stability of a general family of iterative methods in order to complete the previous studies in this trend of research.     

A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#−A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D−A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.     

A note on the growth order of the inverse and product bases of special monogenic polynomials

The purpose of this paper is to generalize some theorems on the representation of certain classes of the entire special monogenic functions by inverse and product bases of special monogenic polynomials. We derive bounds for the order of inverse and product bases. Some examples are given showing that the resulting bounds are attainable. Our results improve and generalize some known results in Clifford setting concerning the order of inverse and product bases. Copyright © 2011 John Wiley & Sons, Ltd.