Related papers: Numerical analysis of Givens rotation
We propose a new instruction (FPADDRE) that computes the round-off error in floating-point addition. We explain how this instruction benefits high-precision arithmetic operations in applications where double precision is not sufficient.…
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
Finite-precision arithmetic computations face an inherent tradeoff between accuracy and efficiency. The points in this tradeoff space are determined, among other factors, by different data types but also evaluation orders. To put it simply,…
We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are based on…
Geometric predicates are at the core of many algorithms, such as the construction of Delaunay triangulations, mesh processing and spatial relation tests. These algorithms have applications in scientific computing, geographic information…
We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss--Legendre quadrature has been conventionally…
Modern computer architectures support low-precision arithmetic, which present opportunities for the adoption of mixed-precision algorithms to achieve high computational throughput and reduce energy consumption. As a growing number of…
The recent hardware trend towards reduced precision computing has reignited the interest in numerical techniques that can be used to enhance the accuracy of floating point operations beyond what is natively supported for basic arithmetic…
Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this article, we present a thorough description of this function and analyze the…
Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…
The defects of the traditional strapdown inertial navigation algorithms become well acknowledged and the corresponding enhanced algorithms have been quite recently proposed trying to mitigate both theoretical and algorithmic defects. In…
An algorithm for obtaining all n\times n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has…
The problem of exactly summing n floating-point numbers is a fundamental problem that has many applications in large-scale simulations and computational geometry. Unfortunately, due to the round-off error in standard floating-point…
For most deep learning algorithms training is notoriously time consuming. Since most of the computation in training neural networks is typically spent on floating point multiplications, we investigate an approach to training that eliminates…
Low precision arithmetic, in particular half precision floating point arithmetic, is now available in commercial hardware. Using lower precision can offer significant savings in computation and communication costs with proportional savings…
In recent years, half precision floating-point arithmetic has gained wide support in hardware and software stack thanks to the advance of artificial intelligence and machine learning applications. Operating at half precision can…
Division algorithms have been developed to reduce latency and to improve the efficiency of the processors. Floating point division is considered as a high latency operation. This papers looks into one such division algorithm, examines the…
We present a simplified computational rule for the back-propagation formulas for artificial neural networks. In this work, we provide a generic two-step rule for the back-propagation algorithm in matrix notation. Moreover, this rule…
We present efficient realization of Generalized Givens Rotation (GGR) based QR factorization that achieves 3-100x better performance in terms of Gflops/watt over state-of-the-art realizations on multicore, and General Purpose Graphics…
Given a combinatorial decomposition for a counting problem, we resort to the simple scheme of approximating large numbers by floating-point representations in order to obtain efficient Fully Polynomial Time Approximation Schemes (FPTASes)…