Conventional Floating-point Arithmetic Research Articles

Analysis of Posit and Bfloat Arithmetic of Real Numbers for Machine Learning

Modern computational tasks are often required to not only guarantee predefined accuracy, but get the result fast. Optimizing calculations using floating point numbers, as opposed to integers, is a non-trivial task. For this reason, there is a need to explore new ways to improve such operations. This paper presents analysis and comparison of various floating point formats - float, posit and bfloat. One of the promising areas in which the problem of using such values can be considered to be the most acute is neural networks. That is why we pay special attention to algorithms of linear algebra and artificial intelligence to assess efficiency of new data types in this area. The research results showed that software implementations of posit16 and posit32 have high accuracy, but they are not particularly fast; on the other hand, bfloat16 is not much different from float32 in accuracy, but significantly surpasses it in performance for large amounts of data and complex machine learning algorithms. Thus, posit16 can be used in systems with less stringent performance requirements, as well as in conditions of limited computer memory; and also in cases when bfloat16 cannot provide required accuracy. As for bfloat16, it can speed up systems based on the IEEE 754 standard, but it cannot solve all the problems of conventional floating point arithmetic. Thus, although posits and bfloats are not a full fledged replacement for float, they provide (under certain conditions) advantages that can be useful for implementation of machine learning algorithms.

A Modified Fixed-Point Chirp Scaling Algorithm Based on Updating Phase Factors Regionally for Spaceborne SAR Real-Time Imaging

To realize real-time imaging for the spaceborne synthetic aperture radar (SAR) using the chirp scaling (CS) algorithm, high generation rates of phase factors and heavy computation loads are unavoidable. For example, updating phase factors continuously in the 2-D time or frequency space is difficult for generating phase factors for real-time imaging. To solve this problem, a modified CS algorithm based on updating phase factors regionally is proposed. Moreover, conventional floating-point arithmetic is unaffordable for real-time imaging onboard. Therefore, fixed-point processing is adopted. In the proposed imaging algorithm, the phase factors are updated only at some specified times and frequencies, and fixed-point operation is adopted. Furthermore, based on the paired echo theory and the finite word length error mathematical model, the effect of the proposed imaging algorithm on the SAR image quality is studied. Moreover, based on the above analysis and the experiments of Chinese HJ-1C and GF-3 spaceborne SAR practical measured data, a hardware system including digital signal processor and field-programmable gate array boards is constructed.

Interval Versions of Central-Difference Method for Solving the Poisson Equation in Proper and Directed Interval Arithmetic

Abstract To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

Chemical equation balancing: An integer programming approach

Presented here is an integer linear program (ILP) formulation for automatic balancing of a chemical equation. Also described is a integer nonlinear programming (INP) algorithm for balancing. This special algorithm is polynomial time O( n 3), unlike the ILP approach, and uses the widely available conventional floating-point arithmetic, obviating the need for both rational arithmetic and multiple modulus residue arithmetic. The rational arithmetic is unsuitable due to intermediate number growth, while the residue arithmetic suffers from the lack of a priori knowledge of the set of prime bases that avoids a possible failure due to division by zero. Further, unlike the floating point arithmetic, both arithmetics are not built-in/standard and hence additional programming effort is needed. The INP algorithm has been tested on several typical chemical equations and found to be very successful for most problems in our extensive balancing experiments. This algorithm also has the capability to determine the feasibility of a new chemical reaction and, if it is feasible, then it will balance the equation and also provide the information if two or more linearly independent balancings exist through the rank information. Any general method to solve the ILP is fail-proof, but it is not polynomial time. Since we have not encountered truly large chemical equations having, say, 1000 products and reactants in a real-world situation, a non-polynomial ILP solver is also useful. A justification for the objective functions for ILP and INP algorithms, each of which produces a unique solution, is provided.

Floating-Point Arithmetics

Three types of floating-point arithmetics with error control are discussed and compared with conventional floating-point arithmetic. General multiplication and division shift criteria are derived (for any base) for Metropolis-type arithmetics. The limitations and most suitable range of application for each arithmetic are discussed.