Finding the "truncated" polynomial that is closest to a function

When implementing regular enough functions (e.g., elementary or special functions) on a computing system, we frequently use polynomial approximations. In most cases, the polynomial that best approximates (for a given distance and in a given interval) a function has coefficients that are not exactly representable with a finite number of bits. And yet, the polynomial approximations that are actually implemented do have coefficients that are represented with a finite - and sometimes small - number of bits: this is due to the finiteness of the floating-point representations (for software implementations), and to the need to have small, hence fast and/or inexpensive, multipliers (for hardware implementations). We then have to consider polynomial approximations for which the degree-$i$ coefficient has at most $m_i$ fractional bits (in other words, it is a rational number with denominator $2^{m_i}$). We provide a general method for finding the best polynomial approximation under this constraint. Then, we suggest refinements than can be used to accelerate our method.Comment: 14 pages, 1 figur

Deep R Programming

Deep R Programming is a comprehensive course on one of the most popular languages in data science (statistical computing, graphics, machine learning, data wrangling and analytics). It introduces the base language in-depth and is aimed at ambitious students, practitioners, and researchers who would like to become independent users of this powerful environment. This textbook is a non-profit project. Its online and PDF versions are freely available at . This early draft is distributed in the hope that it will be useful.Comment: Draft: v0.2.1 (2023-04-27

The OCaml system release 5.0: Documentation and user's manual

This manual documents the release 5.0 of the OCaml system. It is organized as follows. Part I, "An introduction to OCaml", gives an overview of the language. Part II, "The OCaml language", is the reference description of the language. Part III, "The OCaml tools", documents the compilers, toplevel system, and programming utilities. Part IV, "The OCaml library", describes the modules provided in the standard library. Part V, “Indexes”, contains an index of all identifiers defined in the standard library, and an index of keywords