Many of the currently popular \block algorithms" are scalar algorithms in which the operations have been grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown here that block LU factorization is stable if A is block diagonally dominant by c...

We propose an incomplete Cholesky factorization for the solution of large-scale trust region subproblems and positive definite systems of linear equations. This factorization depends on a parameter p that specifies the amount of additional memory (in multiples of n, the dimension of the problem) that is available; there is no need to specify a drop tolerance. Our numerical results show that the...

Integer factorization is one of the vital algorithms discussed as a part of analysis of any black-box cipher suites where the cipher algorithm is based on number theory. The origin of the problem is from Discrete Logarithmic Problem which appears under the analysis of the cryptographic algorithms as seen by a cryptanalyst. The integer factorization algorithm poses a potential in computational s...

Using Zeilberger’s factorization of two-stack-sortable permutations, we write a functional equation — of a strange sort — that defines their generating function according to five statistics: length, number of descents, number of right-to-left and left-to-right maxima, and a fifth statistic that is closely linked to the factorization. Then, we show how one can translate this functional equation ...

The factorization of the Legendre polynomial of degree (p− e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism √ −2p, by proving an analogue of Deuring’s theorem on supersingular curves with multiplier √ −p. This is used to count th...

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ of the symmetric group $s_n$ such that each element in $s_n$‎ ‎lies in some conjugate of one of the $h_i.$ in this paper we‎ ‎conjecture that $$gamma(s_n)=frac{n}{2}left(1-frac{1}{p_1}right)‎ ‎left(1-frac{1}{p_2}right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $ninmathbb{n}$ an...

We address a number of theoretical and phenomenological issues in two-body charmless non-leptonic B decays in the context of a factorization model. A classification of the exclusive decays involving tree and penguin amplitudes is reviewed. The role of QCD anomaly in exclusive decays involving an η or η ′ is elucidated and comparison is made with the existing data. We argue that the factorizatio...

We pursue the scalable parallel implementation of the factorization of band matrices with medium to large bandwidth targeting SMP and multi-core architectures. Our approach decomposes the computation into a large number of fine-grained operations exposing a higher degree of parallelism. The SuperMatrix run-time system allows an out-of-order scheduling of operations that is transparent to the pr...

Let L = Q[α] be an abelian number field of prime degree q, and let a be a nonzero rational number. We describe an algorithm which takes as input a and the minimal polynomial of α over Q, and determines if a is a norm of an element of L. We show that, if we ignore the time needed to obtain a complete factorization of a and a complete factorization of the discriminant of α, then the algorithm run...

We pose and solve the existence of 2-factorizations of complete graphs and complete bipartite graphs that have the number of cycles per 2-factor varying, called pancomponented. Such 2-factorizations exist for all such graphs. The pancomponented problem requires a slight generalization of the methods used to solve pancyclic 2-factorization problem, by building 2-factors from cyclically generated...