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Abstract: With a transformation matrix of B-spline bases (abbreviated to BSBT matrix), a B-spline basis can be represented by another B-spline basis. In this paper, we first give the existence conditions and some useful properties of BSBT matrices. Then we propose a recursive formula for BSBT matrices and an efficient method for the computation of BSBT matrices. Based on these results, we further probe into the applications of BSBT matrices in knot insertion and degree elevation of B-spline curves. Using BSBT matrices, we develop a new uniform algorithm for knot insertion and degree elevation of B-spline curves. This algorithm is efficient, general-purpose, and simple to implement. It can be used to insert one knot or multiple knots, raise one degree or multiple degrees, or to insert knot and raise degree simultaneously. Several examples are given to illustrate the stability of the algorithm. The results in the paper show that BSBT matrices can be applied to establish a uniform mathematical model for knot insertion, degree elevation, knot removal and degree reduction of B-spline curves and surfaces, and to provide a general tool for the conversions between different representations of B-spline curves and surfaces. [Copyright &y& Elsevier]

Abstract: In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix find, for assigned k, nonnegative matrices and such that V = WH. Exact, nontrivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned k, the factorization WH closest to V in I-divergence. An iterative algorithm, EM like, for the construction of the best pair (W, H) has been proposed in the literature. In this paper we interpret the algorithm as an alternating minimization procedure à la Csiszár–Tusnády and investigate some of its stability properties. NMF is widespreading as a data analysis method in applications for which the positivity constraint is relevant. There are other data analysis methods which impose some form of nonnegativity: we discuss here the connections between NMF and Archetypal Analysis. [Copyright &y& Elsevier]

Abstract: According to the hierarchical identification principle, a hierarchical gradient based iterative estimation algorithm is derived for multivariable output error moving average systems (i.e., multivariable OEMA-like models) which is different from multivariable CARMA-like models. As there exist unmeasurable noise-free outputs and unknown noise terms in the information vector/matrix of the corresponding identification model, this paper is, by means of the auxiliary model identification idea, to replace the unmeasurable variables in the information vector/matrix with the estimated residuals and the outputs of the auxiliary model. A numerical example is provided. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an inversion algorithm for a banded matrix is presented. The twisted decompositions of a banded matrix are given first; then the inverse of the matrix is obtained, one column at time. The method is about two times faster than the standard method based on the decomposition, as is shown with the analysis of computing complexity and the numerical experiments. [Copyright &y& Elsevier]

Abstract: In the present paper we study the computation of the rank of a block bidiagonal Toeplitz (BBT) sequence of matrices. We propose matrix-based, numerical and symbolical, updating and direct methods, computing the rank of BBT matrices and comparing them with classical procedures. The methods deploy the special form of the BBT sequence, significantly reducing the required flops and leading to fast and efficient algorithms. The numerical implementation of the algorithms computes the numerical rank in contrast with the symbolical implementation, which guarantees the computation of the exact rank of the matrix. The combination of numerical and symbolical operations suggests a new approach in software mathematical computations denoted as hybrid computations. [Copyright &y& Elsevier]

Abstract: The covering generalized rough sets are an improvement of traditional rough set model to deal with more complex practical problems which the traditional one cannot handle. It is well known that any generalization of traditional rough set theory should first have practical applied background and two important theoretical issues must be addressed. The first one is to present reasonable definitions of set approximations, and the second one is to develop reasonable algorithms for attributes reduct. The existing covering generalized rough sets, however, mainly pay attention to constructing approximation operators. The ideas of constructing lower approximations are similar but the ideas of constructing upper approximations are different and they all seem to be unreasonable. Furthermore, less effort has been put on the discussion of the applied background and the attributes reduct of covering generalized rough sets. In this paper we concentrate our discussion on the above two issues. We first discuss the applied background of covering generalized rough sets by proposing three kinds of datasets which the traditional rough sets cannot handle and improve the definition of upper approximation for covering generalized rough sets to make it more reasonable than the existing ones. Then we study the attributes reduct with covering generalized rough sets and present an algorithm by using discernibility matrix to compute all the attributes reducts with covering generalized rough sets. With these discussions we can set up a basic foundation of the covering generalized rough set theory and broaden its applications. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a new method to efficiently compute a representation of an orthogonal basis of the nullspace of a sparse matrix operator with , . We assume that B has full rank, i.e., rank. It is well-known that the last columns of the orthogonal matrix Q in a QR factorization form such a desired null basis. The orthogonal matrix Q can be represented either explicitly as a matrix, or implicitly as a matrix H of Householder vectors. Typically, the matrix H represents the orthogonal factor much more compactly than Q. We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity. This new algorithm may, e.g., be used to construct a null space basis of the discrete divergence operator in the finite element context, and we will provide numerical results for this particular application. [Copyright &y& Elsevier]

Abstract: This paper mainly discusses the following two problems: The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. Moreover, the optimal approximation solution, an algorithm and a numerical example of Problem II are provided. [Copyright &y& Elsevier]

Abstract: We discuss questions of eigenvalue conditioning. We study in some depth relationships between the classical theory of conditioning and the theory of the zero-structured conditioning, and we derive from the existing theory formulae for the mathematical objects involved. Then an algorithm to compare the zero-structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented. Numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero-structure. Patterned matrices (Toeplitz and Hankel) will be investigated in a forthcoming paper (Eigenvalue patterned condition numbers: Toeplitz and Hankel cases, Tech. Rep. 3, Mathematics Department, University of Rome ‘ La Sapienza’ , 2005.). [Copyright &y& Elsevier]

Abstract: Let F be a set of n × n fuzzy matrices. F is called simultaneously controllable if there exists a permutation matrix P such that for each A ε F, C = [cij ] = P ⊗ A ⊗ P T satisfies cij ≥ cji for i > j, where ⊗ is the max-min composition. In this paper, the necessary and sufficient conditions for a set of n × n fuzzy matrices to be simultaneously controllable will be established. A constructive algorithm which can determine a simultaneously controllable set of n × n fuzzy matrices is presented as well. [Copyright &y& Elsevier]

Abstract: We propose a “fast” algorithm for the construction of a data-sparse inverse of a generalToeplitz matrix. The computational cost for inverting an N × N Toeplitz matrix equals the cost of four length-N FFTs plus an O(N)-term. This cost should be compared to the O(N log2 N) cost of previously published methods. Moreover, while those earlier methods are based on algebraic considerations, the procedure of this paper is analysis-based; as a result, its stability does not depend on the symmetry and positive-definiteness of the matrix being inverted. The performance of the scheme is illustrated with numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a new inversion free variant of the basic fixed point iteration method for obtaining a maximal positive definite solution of the nonlinear matrix equation . It is more accurate than Zhan''s algorithm (J. Sci. Comput. 17 (1996) 1167) and has less number of operations than the algorithm of Guo and Lancaster (Math. Comput. 68 (1999) 1589). We derive convergence conditions of the iteration and existence conditions of a solution to the problem. Finally, we give some numerical results to illustrate the behavior of the considered algorithm. [Copyright &y& Elsevier]

Abstract: Koukouvinos et al. [C. Koukouvinos, M. Mitrouli, J. Seberry, Growth in Gaussian elimination for weighing matrices, W(n, n −1), Linear Algebra Appl. 306 (2000) 189–202], conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n −1 is n −1 and that the first and last few pivots are for n >14. In the present paper we study the growth problem for skew and symmetric conference matrices. An algorithm for extending a k × k matrix with elements 0, ±1 to a skew and symmetric conference matrix of order n is described. By using this algorithm we show that the unique W(8,7) has two pivot structures. We also prove that the unique W(10,9) has three pivot patterns. [Copyright &y& Elsevier]

Abstract: This paper proposes a second-order scheme of precision integration for dynamic analysis with respect to long-term integration. Rather than transforming into first-order equations, a recursive scheme is presented in detail for direct solution of the homogeneous part of second-order algebraic and differential equations. The sine and cosine matrices involved in the scheme are calculated using the so-called 2 N algorithm. Numerical tests show that both the efficiency and the accuracy of homogeneous equations can be improved considerably with the second-order scheme. The corresponding particular solution is also presented, incorporated with the second-order scheme where the excitation vector is approximated by the truncated Taylor series. [Copyright &y& Elsevier]

Abstract: This paper proposes a rank-one modified fitting algorithm. The main feature of the approach is that the update formula for the deflection matrix in an iteration is deduced from a quadratic fitting function, not from the traditional quasi-Newton equation. Under some mild assumptions, the algorithm is globally and linearly convergent. [Copyright &y& Elsevier]

This paper presents a unified approach to the solution of the truncated matrix Stieltjes moment problem: Sk=∫0∞uk dσ(u) (k=0,…,m), in both the cases m=2n and m=2n−1, based on the use of the iterative Schur algorithm. [Copyright &y& Elsevier]

A novel upper bound for the permanent of (0,1)-matrices is obtained in this paper, by using an unbiased estimator of permanent [Random Structures Algorithms 5 (1994) 349]. It is a refinement of Minc’s very famous result, and apparently tighter than the current best general bound in some cases. [Copyright &y& Elsevier]

One of the most common data structures, at least in scientific computing, is the multidimensional array. Some numerical algorithms may conveniently be expressed as a generalized matrix multiplication, which computes a multidimensional array from two other multidimensional arrays. By adopting index notation with the Einstein summation convention, an elegant tool for expressing generalized matrix multiplications is obtained. Index notation is the succinct and compact notation primarily used in tensor calculus.In this paper, we develop computer support for index notation as a domain specific language. Grammar and semantics are proposed, yielding an unambiguous interpretation algorithm. An object-oriented implementation of a C++ library that supports index notation is described.A key advantage with computer support of index notation is that the notational gap between a mathematical index notation algorithm and its implementation in a computer language is avoided. This facilitates program construction as well as program understanding. Program examples that demonstrate the close resemblance between code and the original mathematical formulation are presented. [Copyright &y& Elsevier]