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Recently, extensive researches on estimating the value of e have been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

This final number will be Graph HT ht . This final limit of this symbolic expression as symbolic variables approach the original values in matrix I A i will be Graph HT ht . Its determinant is denoted by Graph HT ht , also written Graph HT ht . Next, let us look at the following specific example: Graph HT ht where the I a i changes from 2 to -3, the I b i changes from 0 to 10. [Extracted from the article]

Traditional clustering algorithms such as K-means are highly sensitive to the initial centroid selection and perform poorly on non-convex dataset. To address these problems, a novel data adaptive Bregman clustering algorithm (DBGSA) is proposed in this paper, which combines the universal gravitational algorithm to bring similar points closer in the dataset. In addition, we introduce the Bregman divergence generalized power mean information loss minimization to identify cluster centers and build a hyperparameter identification optimization model, which effectively solves the problems of manual adjustment and uncertainty in the improved dataset. Extensive experiments are conducted on four simulated datasets and six real datasets. The results demonstrate that DBGSA significantly improves the accuracy of various clustering algorithms by an average of 63.8% compared to the contrasted algorithms, making it applicable to diverse datasets. [ABSTRACT FROM AUTHOR]

Most calculations in model reduction involve the solutions of a sequence of dual linear systems with multiple right-hand sides. To solve such systems efficiently, a new deflated BiCG method is explored in this paper. The proposed algorithm uses harmonic Ritz vectors to approximate left and right invariant subspaces inexpensively via small descenting direction vectors found by subsequent runs of deflated BiCG and then derives the deflated subspaces for the next pair of dual linear systems. This process leads to faster convergence for the next pair of systems. Numerical examples illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this work, the sign distribution for all inverse elements of general tridiagonal H-matrices is presented. In addition, some computable upper and lower bounds for the entries of the inverses of diagonally dominant tridiagonal matrices are obtained. Based on the sign distribution, these bounds greatly improve some well-known results due to Ostrowski (1952) , Shivakumar and Ji (1996) , Nabben (1999) and recently given by Peluso and Politi (2001) , Peluso and Popolizio (2008) and so forth. It is also stated that the inverse of a general tridiagonal matrix may be described by parameters ( and ) instead of ones as given by El-Mikkawy (2004) , El-Mikkawy and Karawia (2006) and Huang and McColl (1997) . According to these results, a new symbolic algorithm for finding the inverse of a tridiagonal matrix without imposing any restrictive conditions is presented, which improves some recent results. Finally, several applications to the preconditioning technology, the numerical solution of differential equations and the birth–death processes together with numerical tests are given. [Copyright &y& Elsevier]

Abstract: A new criterion for identifying generalized diagonally dominant matrices (GDDM, or -matrices) was given in [K. Ojiro, H. Niki, M. Usui, A note on a new criterion for the -matrix property, J. Comput. Appl. Math. 150 (2003) 293–302], where the criterion is effective when the given matrix is irreducible. However, when the matrix is reducible, it may lead to a false answer. In this work, to overcome this drawback, we present an improved version. The new method is always convergent and needs fewer iterations than the earlier one. An interesting sufficient and necessary condition for any weakly diagonally dominant matrix (WDDM) to be an -matrix is obtained. Finally, some numerical examples for the effectiveness of the proposed algorithm are presented. [Copyright &y& Elsevier]