Topic
Square root
About: Square root is a research topic. Over the lifetime, 2262 publications have been published within this topic receiving 33886 citations. The topic is also known as: 2nd root & second root.
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01 Jan 1993TL;DR: The principles of the algorithms available for performing arithmetic operations in digital computers, described independently of specific implementation technology and within the same framework, are explained.
Abstract: This text explains the fundamental principles of algorithms available for performing arithmetic operations on digital computers. These include basic arithmetic operations like addition, subtraction, multiplication, and division in fixed-point and floating-point number systems as well as more complex operations such as square root extraction and evaluation of exponential, logarithmic, and trigonometric functions. The algorithms described are independent of the particular technology employed for their implementation.
1,174 citations
TL;DR: The theory underlying these methods, along with an interpretation of factorization in terms of the graphical model associated with the SLAM problem, are presented, and both simulation results and actual SLAM experiments are presented that underscore the potential of these methods as an alternative to EKF-based approaches.
Abstract: Solving the SLAM (simultaneous localization and mapping) problem is one way to enable a robot to explore, map, and navigate in a previously unknown environment. Smoothing approaches have been investigated as a viable alternative to extended Kalman filter (EKF)-based solutions to the problem. In particular, approaches have been looked at that factorize either the associated information matrix or the measurement Jacobian into square root form. Such techniques have several significant advantages over the EKF: they are faster yet exact; they can be used in either batch or incremental mode; are better equipped to deal with non-linear process and measurement models; and yield the entire robot trajectory, at lower cost for a large class of SLAM problems. In addition, in an indirect but dramatic way, column ordering heuristics automatically exploit the locality inherent in the geographic nature of the SLAM problem. This paper presents the theory underlying these methods, along with an interpretation of factorization in terms of the graphical model associated with the SLAM problem. Both simulation results and actual SLAM experiments in large-scale environments are presented that underscore the potential of these methods as an alternative to EKF-based approaches.
979 citations
TL;DR: In this article, a deterministic analysis ensemble updates are implemented in Kalman square root filters, and the nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare ensemble data assimilation methods.
Abstract: Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.
848 citations
TL;DR: In this article, the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions was considered and a priori estimates of Gidas-Ni-Nirenberg type were obtained.
605 citations
TL;DR: This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required.
Abstract: A method for the inversion of a nonsymmetric matrix has been in use at ORNL and has proved to be highly stable numerically but to require a rather large number of arithmetic operations, including a total of N(N-1)/2 square roots. This note points out that the same result can be obtained with fewer arithmetic operations, and, in particular, for inverting a square matrix of order N, at most 2(N-1) square roots are required. For N > 4, this is a savings of (N-4)(N-1)/4 square roots. (T.B.A.)
577 citations