Existing definitions of componentwise backward error and componentwise condi tion number for linear systems are extended to systems with multiple right-hand sides and to a general class of componentwise measure of perturbations involving Holder p-norms. It is shown that for a system of order n with r right-hand sides, the componentwise backward error can be computed by finding the minimum p-nor...

A graph is componentwise biconnected if every connected component either is an isolated vertex or is biconnected. We present a linear-time algorithm for the problem of adding the smallest number of edges to make a bipartite graph componentwise biconnected while preserving its bipartiteness. This algorithm has immediate applications for protecting sensitive information in statistical tables.

Five numerical methods for pricing American put options under Heston’s stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M -matrices is proposed. The projected SOR, a projected ...

Let R = k[x1, . . . , xn] be a polynomial ring over a field k. Let J = {j1, . . . , jt} be a subset of {1, . . . , n}, and let mJ ⊂ R denote the ideal (xj1 , . . . , xjt). Given subsets J1, . . . , Js of {1, . . . , n} and positive integers a1, . . . , as, we study ideals of the form I = m1 J1 ∩ · · · ∩ m as Js . These ideals arise naturally, for example, in the study of fat points, tetrahedral...

This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a diffe...

Classical condition numbers are normwise: they measure the size of both input perturbations and output errors using some norms. To take into account the relative of each data component, and, in particular, a possible data sparseness, componentwise condition numbers have been increasingly considered. These are mostly of two kinds: mixed and componentwise. In this paper, we give explicit expressi...

we consider the class $mathfrak m$ of $bf r$--modules where $bf r$ is an associative ring. let $a$ be a module over a group ring $bf r$$g$, $g$ be a group and let $mathfrak l(g)$ be the set of all proper subgroups of $g$. we suppose that if $h in mathfrak l(g)$ then $a/c_{a}(h)$ belongs to $mathfrak m$. we investigate an $bf r$$g$--module $a$ such that $g not = g'$, $c_{g}(a) = 1$. we stud...

How small can a stationary iterative method for solving a linear system Ax = b make the error and the residual in the presence of rounding errors? We give a componentwise error analysis that provides an answer to this question and we examine the implications for numerical stability. The Jacobi, Gauss-Seidel and successive overrelaxation methods are all found to be forward stable in a componentw...

Expressions are presented for the errors in individual components of the solution to systems of linear equations and linear least squares problems. No assumptions about the structure or distribution of the perturbations are made. The resulting "componentwise condition numbers" measure the sensitivity of each solution component to perturbations. It is shown that any linear system has at least on...