Related papers: Modern Computer Arithmetic (version 0.5.1)
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
The ever-increasing computational complexity of deep learning models makes their training and deployment difficult on various cloud and edge platforms. Replacing floating-point arithmetic with low-bit integer arithmetic is a promising…
This book aims to provide an introduction to the topic of deep learning algorithms. We review essential components of deep learning algorithms in full mathematical detail including different artificial neural network (ANN) architectures…
We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…
Most of the digital signal processing applications performs operations like multiplication, addition, square-root calculation, solving linear equations etc. The physical implementation of these operations consumes a lot of hardware and,…
This chapter provides an overview of the different Artificial Intelligence (AI) systems that are being used in contemporary digital tools for Mathematics Education (ME). It is aimed at researchers in AI and Machine Learning (ML), for whom…
This book provides a comprehensive and accessible introduction to the emerging field of AI for mathematics. It covers the core principles and diverse applications of using artificial intelligence to advance mathematical research. Through…
For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
In an era where data-driven decision-making and computational efficiency are paramount, optimization plays a foundational role in advancing fields such as mathematics, computer science, operations research, machine learning, and beyond.…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
The greatest demand for today's computing is machine learning. This paper analyzes three machine learning algorithms: transformers, spatial convolution, and FFT. The analysis is novel in three aspects. First, it measures the cost of memory…
Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards…
Nowadays, parallel computing is ubiquitous in several application fields, both in engineering and science. The computations rely on the floating-point arithmetic specified by the IEEE754 Standard. In this context, an elementary brick of…
These notes begin in Chapter 1 with a review of linear algebra and the postulates of quantum mechanics, leading to an explanation of single- and multi-qubit gates. Chapter 2 explores the challenge of constructing arbitrary quantum states…
This chapter is concerned with the design and analysis of algorithms for minimizing finite automata. Getting a minimal automaton is a fundamental issue in the use and implementation of finite automata tools in frameworks like text…
Recent times are witnessing rapid development in machine learning algorithm systems, especially in reinforcement learning, natural language processing, computer and robot vision, image processing, speech, and emotional processing and…
Over the past thirty years or so the authors have been teaching various programming for mathematics courses at our respective Universities, as well as incorporating computer algebra and numerical computation into traditional mathematics…