We present an inversion algorithm for nonsingular n × n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and, when n is a power of two, requires O (̃n3d) field operations for a generic input; the soft-O notation O ̃indicates some missing log(nd) factors. Up to such logarithmic factors, this asymptotic complexity is of the same order as the number of dis...

We present an inversion algorithm for nonsingular n × n matrices whose entries are degree d polynomials over a field. The algorithm is deterministic and requires O (̃nd) field operations for a generic input. The “soft Oh” notation O ̃indicates some missing log(nd) factors. The polynomial matrix inverse can thus be computed by a straight-line program whose size is — asymptotically and up to logari...

Let F : C → C be an invertible map for which both F and F−1 are polynomials. Then deg F−1 ≤ (deg F )n−1. This is a well-known result. The proof that we give here, at least for low n, does not depend on advanced algebraic geometry.

We investigated an interpolation algorithm for computing the MoorePenrose inverse of a given polynomial matrix, based on the LeverrierFaddeev method. Also, a method for estimating the degrees of polynomial matrices arising from the Leverrier-Faddeev algorithm is given as the improvement of the interpolation algorithm. Algorithms are implemented in the symbolic programming language MATHEMATICA, ...

We study the complexity of the following problems in the streaming model. Membership testing for DLIN. We show that every language in DLIN can be recognised by a randomized one-passO(log n) space algorithm with inverse polynomial one-sided error, and by a deterministic p-pass O(n/p) space algorithm. We show that these algorithms are optimal. Membership testing for LL(k). For languages generated...

so far several methods have been developed for mapping and interpolation of isohyets.one of the recently accepted methods is geographically weighting regression which is suitable for evaluation of spatial heterogeneity of dependent variable by using local regressions. in order to evaluate annually precipitation spatial variation, this study was conducted in gilan province which precipitation is...

The condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. In their study of the complexity of Bézout’s theorem, M. Shub and S. Smale prove that the condition number of a polynomial system is equal to the inverse of the distance from this polynomial system to the nearest ill-conditioned one. Here we explain how this result can be extended t...

We explore the connection between ideals of fat points (which correspond to subschemes of Pn obtained by intersecting (mixed) powers of ideals of points), and piecewise polynomial functions (splines) on a d-dimensional simplicial complex ∆ embedded in R. Using the inverse system approach introduced by Macaulay [11], we give a complete characterization of the free resolutions possible for ideals...