From 57cc17025113acce07d08af894845f68b54f515c Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Sat, 23 Sep 2023 10:12:11 +0200 Subject: [PATCH] Add margin notes --- .../main.tex | 17 ++++++++-- .../sections/_finite_numbers.tex | 33 ++++++++++--------- 2 files changed, 33 insertions(+), 17 deletions(-) diff --git a/statistical-and-mathematical-methods-for-ai/main.tex b/statistical-and-mathematical-methods-for-ai/main.tex index a27a904..28ccab3 100644 --- a/statistical-and-mathematical-methods-for-ai/main.tex +++ b/statistical-and-mathematical-methods-for-ai/main.tex @@ -1,6 +1,6 @@ \documentclass[11pt]{article} -\usepackage[margin=3cm]{geometry} -\usepackage{graphicx} +\usepackage[margin=3cm, lmargin=2cm, rmargin=4cm]{geometry} +\usepackage{graphicx, xcolor} \usepackage{amsmath, amsfonts, amssymb, amsthm, mathtools} \usepackage{hyperref} \usepackage[nameinlink]{cleveref} @@ -21,8 +21,16 @@ } \setlist[description]{labelindent=\parindent} % Indents `description` +\renewcommand*{\marginfont}{\color{gray}\footnotesize} \newtheorem{example}{Example}[section] +\newtheoremstyle{definition}{}{}{}{}{\bfseries}{.}{ }{\thmname{#1}\thmnumber{ #2}\thmnote{ (#3)}} +\theoremstyle{definition} +\newtheorem*{definition}{Def} + +\newcommand{\ubar}[1]{\text{\b{$#1$}}} +\renewcommand{\vec}[1]{\text{\textbf{#1}}} +\newcommand{\nullvec}[0]{\bar{\vec{0}}} \begin{document} @@ -47,5 +55,10 @@ \pagenumbering{arabic} \input{sections/_finite_numbers.tex} + \newpage + \input{sections/_linear_algebra.tex} + \newpage + \input{sections/_linear_systems.tex} + \end{document} \ No newline at end of file diff --git a/statistical-and-mathematical-methods-for-ai/sections/_finite_numbers.tex b/statistical-and-mathematical-methods-for-ai/sections/_finite_numbers.tex index bbd0419..51c9bdc 100644 --- a/statistical-and-mathematical-methods-for-ai/sections/_finite_numbers.tex +++ b/statistical-and-mathematical-methods-for-ai/sections/_finite_numbers.tex @@ -5,16 +5,16 @@ \subsection{Sources of error} \begin{description} - \item[Measure error] + \item[Measure error] \marginnote{Measure error} Precision of the measurement instrument. - \item[Arithmetic error] + \item[Arithmetic error] \marginnote{Arithmetic error} Propagation of rounding errors in each step of an algorithm. - \item[Truncation error] + \item[Truncation error] \marginnote{Truncation error} Approximating an infinite procedure into a finite number of iterations. - \item[Inherent error] + \item[Inherent error] \marginnote{Inherent error} Caused by the finite representation of the data (floating-point). \begin{figure}[h] \centering @@ -31,13 +31,15 @@ Let $x$ be a value and $\hat{x}$ its approximation. Then: \begin{description} \item[Absolute error] \begin{equation} - E_{a} = \hat{x} - x - \end{equation} + E_{a} = \hat{x} - x + \marginnote{Absolute error} + \end{equation} Note that, out of context, the absolute error is meaningless. \item[Relative error] \begin{equation} - E_{a} = \frac{\hat{x} - x}{x} - \end{equation} + E_{a} = \frac{\hat{x} - x}{x} + \marginnote{Relative error} + \end{equation} \end{description} @@ -56,17 +58,16 @@ where: \item starting from an index $i$, not all $d_j$ ($j \geq i$) are equal to $\beta-1$ \end{itemize} % -\Cref{eq:finnum_b_representation} can be represented using the normalized scientific notation as: +\Cref{eq:finnum_b_representation} can be represented using the normalized scientific notation as: \marginnote{Normalized scientific notation} \begin{equation} x = \pm (0.d_1d_2\dots) \beta^p \end{equation} -where $0.d_1d_2\dots$ is the \textbf{mantissa} and $\beta^p$ the \textbf{exponent}. +where $0.d_1d_2\dots$ is the \textbf{mantissa} and $\beta^p$ the \textbf{exponent}. \marginnote{Mantissa\\Exponent} \subsection{Floating-point} - -A floating-point system $\mathcal{F}(\beta, t, L, U)$ is defined by the parameters: +A floating-point system $\mathcal{F}(\beta, t, L, U)$ is defined by the parameters: \marginnote{Floating-point} \begin{itemize} \item $\beta$: base \item $t$: precision (number of digits in the mantissa) @@ -109,6 +110,7 @@ Given a floating-point system $\mathcal{F}(\beta, t, L, U)$, the representation \item[Approximation] if $p \in [L, U]$ but $d_i$ may not be 0 for $i>t$. In this case, the representation is obtained by truncating or rounding the value. + \marginnote{Truncation\\Rounding} \item[Underflow] if $p < L$. In this case, the values is approximated as 0. @@ -119,7 +121,7 @@ Given a floating-point system $\mathcal{F}(\beta, t, L, U)$, the representation \subsubsection{Machine precision} -Machine precision $\varepsilon_{\text{mach}}$ determines the accuracy of a floating-point system. +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} \item[Truncation] $\varepsilon_{\text{mach}} = \beta^{1-t}$ @@ -144,7 +146,7 @@ In alternative, $\varepsilon_{\text{mach}}$ can be defined as the smallest repre \subsubsection{IEEE standard} IEEE 754 defines two floating-point formats: \begin{description} - \item[Single precision] Stored in 32 bits. Represents the system $\mathcal{F}(2, 24, -128, 127)$. + \item[Single precision] Stored in 32 bits. Represents the system $\mathcal{F}(2, 24, -128, 127)$. \marginnote{float32} \begin{center} \small \begin{tabular}{|c|c|c|} @@ -154,7 +156,7 @@ IEEE 754 defines two floating-point formats: \end{tabular} \end{center} - \item[Double precision] Stored in 64 bits. Represents the system $\mathcal{F}(2, 53, -1024, 1023)$. + \item[Double precision] Stored in 64 bits. Represents the system $\mathcal{F}(2, 53, -1024, 1023)$. \marginnote{float64} \begin{center} \small \begin{tabular}{|c|c|c|} @@ -187,6 +189,7 @@ A floating-point operation causes a small rounding error: \end{equation} % Although, some operations may be subject to the \textbf{cancellation} problem which causes information loss. +\marginnote{Cancellation} \begin{example} Given $x = 1$ and $y = 1 \cdot 10^{-16}$, we want to compute $x + y$ in $\mathcal{F}(10, 16, U, L)$.\\ \begin{equation*}