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statistical-and-mathematical-methods-for-ai/sections/_finite_numbers.tex
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@ -1,8 +1,8 @@
\section{Finite numbers}
\chapter{Finite numbers}
\subsection{Sources of error}
\section{Sources of error}
\begin{description}
\item[Measure error] \marginnote{Measure error}
@ -25,10 +25,10 @@
\subsection{Error measurement}
\section{Error measurement}
Let $x$ be a value and $\hat{x}$ its approximation. Then:
\begin{description}
\begin{descriptionlist}
\item[Absolute error]
\begin{equation}
E_{a} = \hat{x} - x
@ -40,11 +40,11 @@ Let $x$ be a value and $\hat{x}$ its approximation. Then:
E_{a} = \frac{\hat{x} - x}{x}
\marginnote{Relative error}
\end{equation}
\end{description}
\end{descriptionlist}
\subsection{Representation in base \texorpdfstring{$\beta$}{B}}
\section{Representation in base \texorpdfstring{$\beta$}{B}}
Let $\beta \in \mathbb{N}_{> 1}$ be the base.
Each $x \in \mathbb{R} \smallsetminus \{0\}$ can be uniquely represented as:
@ -66,7 +66,7 @@ where $0.d_1d_2\dots$ is the \textbf{mantissa} and $\beta^p$ the \textbf{exponen
\subsection{Floating-point}
\section{Floating-point}
A floating-point system $\mathcal{F}(\beta, t, L, U)$ is defined by the parameters: \marginnote{Floating-point}
\begin{itemize}
\item $\beta$: base
@ -86,7 +86,7 @@ Each $x \in \mathcal{F}(\beta, t, L, U)$ can be represented in its normalized fo
\end{example}
\subsubsection{Numbers distribution}
\subsection{Numbers distribution}
Given a floating-point system $\mathcal{F}(\beta, t, L, U)$, the total amount of representable numbers is:
\begin{equation*}
2(\beta-1) \beta^{t-1} (U-L+1)+1
@ -101,9 +101,9 @@ It must be noted that there is an underflow area around 0.
\end{figure}
\subsubsection{Numbers representation}
\subsection{Numbers representation}
Given a floating-point system $\mathcal{F}(\beta, t, L, U)$, the representation of $x \in \mathbb{R}$ can result in:
\begin{description}
\begin{descriptionlist}
\item[Exact representation]
if $p \in [L, U]$ and $d_i=0$ for $i>t$.
@ -117,16 +117,16 @@ Given a floating-point system $\mathcal{F}(\beta, t, L, U)$, the representation
\item[Overflow]
if $p > U$. In this case, an exception is usually raised.
\end{description}
\end{descriptionlist}
\subsubsection{Machine precision}
\subsection{Machine precision}
Machine precision $\varepsilon_{\text{mach}}$ determines the accuracy of a floating-point system. \marginnote{Machine precision}
Depending on the approximation approach, machine precision can be computes as:
\begin{description}
\begin{descriptionlist}
\item[Truncation] $\varepsilon_{\text{mach}} = \beta^{1-t}$
\item[Rounding] $\varepsilon_{\text{mach}} = \frac{1}{2}\beta^{1-t}$
\end{description}
\end{descriptionlist}
Therefore, rounding results in more accurate representations.
$\varepsilon_{\text{mach}}$ is the smallest distance among the representable numbers (\Cref{fig:finnum_eps}).
@ -143,9 +143,9 @@ In alternative, $\varepsilon_{\text{mach}}$ can be defined as the smallest repre
\end{equation*}
\subsubsection{IEEE standard}
\subsection{IEEE standard}
IEEE 754 defines two floating-point formats:
\begin{description}
\begin{descriptionlist}
\item[Single precision] Stored in 32 bits. Represents the system $\mathcal{F}(2, 24, -128, 127)$. \marginnote{float32}
\begin{center}
\small
@ -165,12 +165,12 @@ IEEE 754 defines two floating-point formats:
\hline
\end{tabular}
\end{center}
\end{description}
\end{descriptionlist}
As the first digit of the mantissa is always 1, it does not need to be stored.
Moreover, special configurations are reserved to represent \texttt{Inf} and \texttt{NaN}.
\subsubsection{Floating-point arithmetic}
\subsection{Floating-point arithmetic}
Let:
\begin{itemize}
\item $+: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ be a real numbers operation.