In this paper the accuracy of LU factorization of tridiagonal matrices without pivoting is considered. Two types of componentwise condition numbers for the L and U factors of tridiadonal matrices are presented and compared. One type is a condition number with respect to small relative perturbations of each entry of the matrix. The other type is a condition number with respect to small component...

It is known that any local maximizer of an explicitly quasiconvex realvalued function is actually a global minimizer, whenever it belongs to the intrinsic core of the function’s domain. We show that a similar property holds for componentwise explicitly quasiconvex vector-valued functions, with respect to the optimality concepts of ideal, strong and weak efficiency. These new results are applied...

Controllability of componentwise nonnegative discrete-time linear systems is considered. The key difference here from the well-established positive systems theory is that we permit the case where the input takes negative values, provided that the state remains nonnegative. Such a framework is very natural, moreover necessary, in situations such as population ecology to describe the control acti...

The degree of a projective subscheme has an upper bound in term the codimension and reduction number. If variety almost maximal degree, that is, equals to minus one, then its Betti table been described explicitly. We build on this work by showing for most such varieties, defining ideals are componentwise linear particular linearity is suitable classifying tables varieties. As application, we co...

Research on the functional organization of the reticular core of vertebrate control nervous systems has reached a significant landmark. Our problem is as described in Quarterly Progress Report No. 76 (page 313), but our progress has been achieved with a somewhat different model than that mentioned there. The new model is enclosed within heavy lines in Fig. XXII-1, with everything outside only f...

For a square matrix normed to 1, the normwise distance to singularity is well known to be equal to the reciprocal of the condition number. In this paper we give an elementary and selfcontained proof for the fact that an ill-conditioned matrix is also not far from a singular matrix in a componentwise sense. This is shown to be true for any weighting of the componentwise distance. In words: Ill-c...

Three different kinds of condition numbers: normwise, mixed and componentwise, are discussed for the canonical generalized polar decomposition (CGPD) of real matrices. The technique used herein is different from the ones in previous literatures of the polar decomposition. With some modifications of the definition of the componentwise condition number, its application scope is extended. Explicit...