3,552 research outputs found
An Analysis on Extracting Square and Cube Roots by Aryabhata’s Methods
Abstract— Finding accurate root of a number is still considered as challenging in computer science community. The only popularly known method to compute root is the long division method of finding square root. Aryabhata contributed two methods similar to long division method to compute square root and cube root. In recent years, his methods have also been studied, explained and tried to implement as computer algorithms.  The explained methods and proposed algorithms fail to give correct results. Some analyses have been made on these methods in order to ascertain why the algorithms fail. Improved algorithms have been provided to give correct result while computing square root or cube root using Aryabhata’s methods.   .Keywords—cube root, square root, Bino’s Model of multiplication, Large number manipulation, Long division method, Aryabhata’s methods.  (Article history: Received 12 November 2016 and accepted 30 December 2016
Computing cube root of a positive number
Proposed here is a new algorithm to compute the cube root of large positive integer. The algorithm is based on the implementation of long division method also known as manual method we usually use to find the square root of a number. To implement the long division method, the given number is first represented in a radix-10 representa and then Bino’s Model of Multiplication is used to systematically implement the long division method. A representa is a special array to represent a number in the form of an array so as to enable us to treat the representas in the same way as we treat numbers. This simplifies the difficulty of dealing large numbers in a computer. Also, at the same time it simplifies the implementation of long division method to find the cube root of positive number, ranging from single digit number to arbitrarily large positive number such as RSA challenge numbers. The algorithm can be used to compute cube root of a non-perfect cube number up to desired precision and each computed digit of cube root gives the best precision. Cube root of 2, 5, 10 up to 30 digits and integer parts of cube roots of first few and last few RSA challenge numbers are also provided in the experimental result to show that the algorithm works perfectly to compute the cube root of any positive integer, however small or large it may be.Keywords:Bino‘s Model of Multiplication, Convolution, Cube of a large number, Large number manipulation, Long division method, RSA challenge numbers, Representa, Cube root computatio
Some fast elliptic solvers on parallel architectures and their complexities
The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR
Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities
A complete method is proposed to compute a certified, or ambient isotopic,
meshing for an implicit algebraic surface with singularities. By certified, we
mean a meshing with correct topology and any given geometric precision. We
propose a symbolic-numeric method to compute a certified meshing for the
surface inside a box containing singularities and use a modified
Plantinga-Vegter marching cube method to compute a certified meshing for the
surface inside a box without singularities. Nontrivial examples are given to
show the effectiveness of the algorithm. To our knowledge, this is the first
method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure
The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems
We review the general problem of finding a global rotation that transforms a
given set of points and/or coordinate frames (the "test" data) into the best
possible alignment with a corresponding set (the "reference" data). For 3D
point data, this "orthogonal Procrustes problem" is often phrased in terms of
minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean
distance measure relating the two sets of matched coordinates. We focus on
quaternion eigensystem methods that have been exploited to solve this problem
for at least five decades in several different bodies of scientific literature
where they were discovered independently. While numerical methods for the
eigenvalue solutions dominate much of this literature, it has long been
realized that the quaternion-based RMSD optimization problem can also be solved
using exact algebraic expressions based on the form of the quartic equation
solution published by Cardano in 1545; we focus on these exact solutions to
expose the structure of the entire eigensystem for the traditional 3D spatial
alignment problem. We then explore the structure of the less-studied
orientation data context, investigating how quaternion methods can be extended
to solve the corresponding 3D quaternion orientation frame alignment (QFA)
problem, noting the interesting equivalence of this problem to the
rotation-averaging problem, which also has been the subject of independent
literature threads. We conclude with a brief discussion of the combined 3D
translation-orientation data alignment problem. Appendices are devoted to a
tutorial on quaternion frames, a related quaternion technique for extracting
quaternions from rotation matrices, and a review of quaternion
rotation-averaging methods relevant to the orientation-frame alignment problem.
Supplementary Material covers extensions of quaternion methods to the 4D
problem.Comment: This replaces an early draft that lacked a number of important
references to previous work. There are also additional graphics elements. The
extensions to 4D data and additional details are worked out in the
Supplementary Material appended to the main tex
Mapping and assessment of tree roots using ground penetrating radar with low-cost GPS
In this paper, we have presented a methodology combining ground penetrating radar (GPR) and a low-cost GPS receiver for three-dimensional detection of tree roots. This research aims to provide an effective and affordable testing tool to assess the root system of a number of trees. For this purpose, a low-cost GPS receiver was used, which recorded the approximate position of each GPR track, collected with a 500 MHz RAMAC shielded antenna. A dedicated post-processing methodology based on the precise position of the satellite data, satellite clock offsets data, and a local reference Global Navigation Satellite System (GNSS) Earth Observation Network System (GEONET) Station close to the survey site was developed. Firstly, the positioning information of local GEONET stations was used to filter out the errors caused by satellite position error, satellite clock offset, and ionosphere. In addition, the advanced Kalman filter was designed to minimise receiver offset and the multipath error, in order to obtain a high precision position of each GPR track. Kirchhoff migration considering near-field effect was used to identify the three-dimensional distribution of the root. In a later stage, a novel processing scheme was used to detect and clearly map the coarse roots of the investigated tree. A successful case study is proposed, which supports the following premise: the current scheme is an affordable and accurate mapping method of the root system architecture
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