Compensated summation algorithms are designed to improve the accuracy of ill-conditioned sums. They are based on algorithms, such as FastTwoSum, which are proved to provide, with rounding to nearest, the sum of two floating-point numbers and the associated rounding error. Discrete stochastic arithmetic enables one to estimate rounding error propagation in numerical codes. It requires a random r...

Nowadays, any serious oating point unit design has to comply with the IEEE oating point standard. Many of these designs perform the rounding in four steps which are normalization, signiicand rounding, post normalization, and exponent rounding. Although it is commonly agreed upon that this rounding algorithm implements the rounding function speciied in the IEEE standard, this is actually a (folk...

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular, statistics. We improve earlier solutions of different authors in two ways. For rounding matrices of size m × n, we reduce the runtime from O((mn)) to O(mn lo...

Processing floating point division generally consists of SRT recurrence, quotient conversion, rounding, and normalization steps. In the rounding step, a high speed adder is required for increment operation, increasing the overall execution time. In this paper, a floating point divider performing quotient conversion and rounding in parallel is presented by analyzing the operational characteristi...

Karloo and Zwick obtained recently an optimal 7=8-approximation algorithm for MAX 3-SAT. In an attempt to see whether similar methods can be used to obtain a 7=8-approximation algorithm for MAX SAT, we consider the most natural generalization of MAX 3-SAT, namely MAX 4-SAT. We present a semideenite programming relaxation of MAX 4-SAT and a new family of rounding procedures that try to cope well...

The present investigation is concerned with estimating the rounding error in numerical solution of stochastic differential equations. A statistical rounding error analysis of Euler’s method for stochastic differential equations is performed. In particular, numerical evaluation of the quantities EjXðtnÞ2 Ŷnj and E1⁄2FðŶnÞ2 FðXðtnÞÞ is investigated, where X(tn) is the exact solution at the nth ti...

Background: Bedside interprofessional rounding is gaining ground as a means to improve collaboration and patient outcomes, yet little is known regarding patients’ perceptions of the practice. Methods: This descriptive study used individual patient interviews to elicit views on interprofessional rounding from 35 patients at a large, urban hospital. Results: The findings identified three major ca...

In the next few lectures we will talk about three new variants of the classical randomized rounding technique, namely the rounding by sampling method, the pipage rounding method and the iterative rounding method. We will use these techniques to design approximation algorithms for Asymmetric TSP (ATSP) and the Steiner tree problems. The materials of the this lecture are based on the works of Asa...