New Cube Root Algorithm Based on Third Order Linear Recurrence Relation in Finite Field

In this paper, we present a new cube root algorithm in finite field $\mathbb{F}_{q}$ with $q$ a power of prime, which extends the Cipolla-Lehmer type algorithms \cite{Cip,Leh}. Our cube root method is inspired by the work of Müller \cite{Muller} on quadratic case. For given cubic residue $c \in \mathbb{F}_{q}$ with $q \equiv 1 \pmod{9}$, we show that there is an irreducible polynomial $f(x)=x^{3}-ax^{2}+bx-1$ with root $\alpha \in \mathbb{F}_{q^{3}}$ such that $Tr(\alpha^{\frac{q^{2}+q-2}{9}})$ is a cube root of $c$. Consequently we find an efficient cube root algorithm based on third order linear recurrence sequence arising from $f(x)$. Complexity estimation shows that our algorithm is better than previously proposed Cipolla-Lehmer type algorithms

Trace Expression of r-th Root over Finite Field

Efficient computation of $r$-th root in $\mathbb F_q$ has many applications in computational number theory and many other related areas. We present a new $r$-th root formula which generalizes Müller\u27s result on square root, and which provides a possible improvement of the Cipolla-Lehmer algorithm for general case. More precisely, for given $r$-th power $c\in \mathbb F_q$, we show that there exists $\alpha \in \mathbb F_{q^r}$ such that $Tr\left(\alpha^\frac{(\sum_{i=0}^{r-1}q^i)-r}{r^2}\right)^r=c$ where $Tr(\alpha)=\alpha+\alpha^q+\alpha^{q^2}+\cdots +\alpha^{q^{r-1}}$ and $\alpha$ is a root of certain irreducible polynomial of degree $r$ over $\mathbb F_q$

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Fast algorithms for Brownian dynamics simulation with hydrodynamic interactions

In the Brownian dynamics simulation with hydrodynamic interactions, one needs to generate the total displacement vectors of Brownian particles consisting of two parts: a deterministic part which is proportional to the product of the Rotne-Prager-Yamakawa (RPY) tensor D [1, 2] and the given external forces F; and a hydrodynamically correlated random part whose covariance is proportional to the RPY tensor. To be more precise, one needs to calculate Du for a given vector u and compute √Dv for a normally distributed random vector v. For an arbitrary N-particle configuration, D is a 3N x 3N matrix and u, v are vectors of length 3N. Thus, classical algorithms require O(N2) operations for computing Du and O(N3) operations for computing √Dv, which are prohibitively expensive and render large scale simulations impossible since one needs to carry out these calculations many times in a Brownian dynamics simulation. In this dissertation, we first present two fast multipole methods (FMM) for computing Du. The first FMM is a simple application of the kernel independent FMM (KIFMM) developed by Ying, Biros, and Zorin [3], which requires 9 scalar FMM calls. The second FMM, similar to the FMM for Stokeslet developed by Tornberg and Greengard [4], decomposes the RPY tensor into harmonic potentials and its derivatives, and thus requires only four harmonic FMM calls. Both FMMs reduce the computational cost of Du from O(N2) to O(N) for an arbitrary N-particle configuration. We then discuss several methods of computing √Dv, which are all based on the Krylov subspace approximations, that is, replacing √Dv by p(D)v with p(D) a low degree polynomial in D. We first show rigorously that the popular Chebyshev spectral approximation method (see, for example, [5, 6]) requires √κ log 1/ε terms for a desired precision E, where K is the condition number of the RPY tensor D. In the Chebyshev spectral approximation method, one also needs to estimate the extreme eigenvalues of D. We have considered several methods: the classical Lanczos method, the Chebyshev-Davidson method, and the safeguarded Lanczos method proposed by Zhou and Li [7]. Our numerical experiments indicate that K is usually very small when the particles are distributed uniformly with low density, and that the safeguarded Lanczos method is most effective for our cases with very little additional computational cost. Thus, when combined with the FMMs we described earlier, the Chebyshev approximation method with safeguarded Lanczos method as eigenvalue estimators essentially reduces the cost of computing √Dv from O(N3) to O(N) for most practical particle configurations. Finally, we propose to combine the so-called spectral Lanczos decomposition method (SLDM) (see, for example, [8]) and the FMMs to compute √Dv. Our numerical experiments show that the SLDM is generally more efficient than the popular Chebyshev spectral approximation method. The fast algorithms developed in this dissertation will be useful for the study of diffusion limited reactions, polymer dynamics, protein folding, and particle coagulation as it enables large scale Brownian dynamics simulations. Moreover, the algorithms can be extended to speed up the computation involving the matrix square root for many other matrices, which has potential applications in areas such as statistical analysis with certain spatial correlations and model reduction in dynamic control theory

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Fractonic order in infinite-component Chern-Simons gauge theories

2+1D multi-component U(1) gauge theories with a Chern-Simons (CS) term provide a simple and complete characterization of 2+1D Abelian topological orders. In this paper, we extend the theory by taking the number of component gauge fields to infinity and find that they can describe interesting types of 3+1D "fractonic" order. "Fractonic" describes the peculiar phenomena that point excitations in certain strongly interacting systems either cannot move at all or are only allowed to move in a lower dimensional sub-manifold. In the simplest cases of infinite-component CS gauge theory, different components do not couple to each other and the theory describes a decoupled stack of 2+1D fractional Quantum Hall systems with quasi-particles moving only in 2D planes -- hence a fractonic system. We find that when the component gauge fields do couple through the CS term, more varieties of fractonic orders are possible. For example, they may describe foliated fractonic systems for which increasing the system size requires insertion of nontrivial 2+1D topological states. Moreover, we find examples which lie beyond the foliation framework, characterized by 2D excitations of infinite order and braiding statistics that are not strictly local

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed