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26
statistical-and-mathematical-methods-for-ai/main.tex
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\documentclass[11pt]{article} \documentclass[11pt]{scrreprt}
\usepackage[margin=3cm, lmargin=2cm, rmargin=4cm, marginparwidth=3cm]{geometry} \usepackage{geometry}
\usepackage{graphicx, xcolor} \usepackage{graphicx, xcolor}
\usepackage{amsmath, amsfonts, amssymb, amsthm, mathtools, bm} \usepackage{amsmath, amsfonts, amssymb, amsthm, mathtools, bm}
\usepackage{hyperref} \usepackage{hyperref}
@ -7,20 +7,16 @@
\usepackage[all]{hypcap} % Links hyperref to object top and not caption \usepackage[all]{hypcap} % Links hyperref to object top and not caption
\usepackage[inline]{enumitem} \usepackage[inline]{enumitem}
\usepackage{marginnote} \usepackage{marginnote}
\usepackage[bottom]{footmisc}
\title{Statistical and Mathematical Methods for Artificial Intelligence} \geometry{ margin=3cm, lmargin=2cm, rmargin=4cm, marginparwidth=3cm }
\date{2023 -- 2024} \hypersetup{ colorlinks, citecolor=black, filecolor=black, linkcolor=black, urlcolor=black, linktoc=all }
\hypersetup{ \NewDocumentEnvironment{descriptionlist}{}{%
colorlinks, \begin{description}[labelindent=1em]
citecolor=black, }{
filecolor=black, \end{description}%
linkcolor=black,
urlcolor=black,
linktoc=all
} }
\setlist[description]{labelindent=1em} % Indents `description`
\setlength{\parindent}{0pt} \setlength{\parindent}{0pt}
\renewcommand*{\marginfont}{\color{gray}\footnotesize} \renewcommand*{\marginfont}{\color{gray}\footnotesize}
@ -36,6 +32,10 @@
\newcommand{\matr}[1]{{\bm{#1}}} \newcommand{\matr}[1]{{\bm{#1}}}
\title{Statistical and Mathematical Methods for Artificial Intelligence}
\date{2023 -- 2024}
\begin{document} \begin{document}
\newgeometry{margin=3cm} \newgeometry{margin=3cm}
\makeatletter \makeatletter

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

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statistical-and-mathematical-methods-for-ai/sections/_linear_algebra.tex
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@ -1,7 +1,7 @@
\section{Linear algebra} \chapter{Linear algebra}
\subsection{Vector space} \section{Vector space}
A \textbf{vector space} over $\mathbb{R}$ is a nonempty set $V$, whose elements are called vectors, with two operations: A \textbf{vector space} over $\mathbb{R}$ is a nonempty set $V$, whose elements are called vectors, with two operations:
\marginnote{Vector space} \marginnote{Vector space}
@ -28,7 +28,7 @@ A subset $U \subseteq V$ of a vector space $V$, is a \textbf{subspace} iff $U$ i
\marginnote{Subspace} \marginnote{Subspace}
\subsubsection{Basis} \subsection{Basis}
\marginnote{Basis} \marginnote{Basis}
Let $V$ be a vector space of dimension $n$. Let $V$ be a vector space of dimension $n$.
A basis $\beta = \{ \vec{v}_1, \dots, \vec{v}_n \}$ of $V$ is a set of $n$ linearly independent vectors of $V$.\\ A basis $\beta = \{ \vec{v}_1, \dots, \vec{v}_n \}$ of $V$ is a set of $n$ linearly independent vectors of $V$.\\
@ -41,7 +41,7 @@ The canonical basis of a vector space is a basis where each vector represents a
The canonical basis $\beta$ of $\mathbb{R}^3$ is $\beta = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}$ The canonical basis $\beta$ of $\mathbb{R}^3$ is $\beta = \{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}$
\end{example} \end{example}
\subsubsection{Dot product} \subsection{Dot product}
The dot product of two vectors in $\vec{x}, \vec{y} \in \mathbb{R}^n$ is defined as: \marginnote{Dot product} The dot product of two vectors in $\vec{x}, \vec{y} \in \mathbb{R}^n$ is defined as: \marginnote{Dot product}
\begin{equation*} \begin{equation*}
\left\langle \vec{x}, \vec{y} \right\rangle = \left\langle \vec{x}, \vec{y} \right\rangle =
@ -49,7 +49,7 @@ The dot product of two vectors in $\vec{x}, \vec{y} \in \mathbb{R}^n$ is defined
\end{equation*} \end{equation*}
\subsection{Matrix} \section{Matrix}
This is a {\tiny(very formal definition of)} matrix: \marginnote{Matrix} This is a {\tiny(very formal definition of)} matrix: \marginnote{Matrix}
\begin{equation*} \begin{equation*}
@ -62,14 +62,14 @@ This is a {\tiny(very formal definition of)} matrix: \marginnote{Matrix}
\end{pmatrix} \end{pmatrix}
\end{equation*} \end{equation*}
\subsubsection{Invertible matrix} \subsection{Invertible matrix}
A matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is invertible (non-singular) if: \marginnote{Non-singular matrix} A matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is invertible (non-singular) if: \marginnote{Non-singular matrix}
\begin{equation*} \begin{equation*}
\exists \matr{B} \in \mathbb{R}^{n \times n}: \matr{AB} = \matr{BA} = \matr{I} \exists \matr{B} \in \mathbb{R}^{n \times n}: \matr{AB} = \matr{BA} = \matr{I}
\end{equation*} \end{equation*}
where $\matr{I}$ is the identity matrix. $\matr{B}$ is denoted as $\matr{A}^{-1}$. where $\matr{I}$ is the identity matrix. $\matr{B}$ is denoted as $\matr{A}^{-1}$.
\subsubsection{Kernel} \subsection{Kernel}
The null space (kernel) of a matrix $\matr{A} \in \mathbb{R}^{m \times n}$ is a subspace such that: \marginnote{Kernel} The null space (kernel) of a matrix $\matr{A} \in \mathbb{R}^{m \times n}$ is a subspace such that: \marginnote{Kernel}
\begin{equation*} \begin{equation*}
\text{Ker}(\matr{A}) = \{ \vec{x} \in \mathbb{R}^n : \matr{A}\vec{x} = \nullvec \} \text{Ker}(\matr{A}) = \{ \vec{x} \in \mathbb{R}^n : \matr{A}\vec{x} = \nullvec \}
@ -79,15 +79,15 @@ The null space (kernel) of a matrix $\matr{A} \in \mathbb{R}^{m \times n}$ is a
A square matrix $\matr{A}$ with $\text{\normalfont Ker}(\matr{A}) = \{\nullvec\}$ is non singular. A square matrix $\matr{A}$ with $\text{\normalfont Ker}(\matr{A}) = \{\nullvec\}$ is non singular.
\end{theorem} \end{theorem}
\subsubsection{Similar matrices} \marginnote{Similar matrices} \subsection{Similar matrices} \marginnote{Similar matrices}
Two matrices $\matr{A}$ and $\matr{D}$ are \textbf{similar} if there exists an invertible matrix $\matr{P}$ such that: Two matrices $\matr{A}$ and $\matr{D}$ are \textbf{similar} if there exists an invertible matrix $\matr{P}$ such that:
\[ \matr{D} = \matr{P}^{-1} \matr{A} \matr{P} \] \[ \matr{D} = \matr{P}^{-1} \matr{A} \matr{P} \]
\subsection{Norms} \section{Norms}
\subsubsection{Vector norms} \subsection{Vector norms}
The norm of a vector is a function: \marginnote{Vector norm} The norm of a vector is a function: \marginnote{Vector norm}
\begin{equation*} \begin{equation*}
\Vert \cdot \Vert: \mathbb{R}^n \rightarrow \mathbb{R} \Vert \cdot \Vert: \mathbb{R}^n \rightarrow \mathbb{R}
@ -122,7 +122,7 @@ In some cases, unbalanced results may be given when comparing different norms.
\end{example} \end{example}
\subsubsection{Matrix norms} \subsection{Matrix norms}
The norm of a matrix is a function: \marginnote{Matrix norm} The norm of a matrix is a function: \marginnote{Matrix norm}
\begin{equation*} \begin{equation*}
\Vert \cdot \Vert: \mathbb{R}^{m \times n} \rightarrow \mathbb{R} \Vert \cdot \Vert: \mathbb{R}^{m \times n} \rightarrow \mathbb{R}
@ -148,7 +148,7 @@ Common norms are:
\subsection{Symmetric, positive definite matrices} \section{Symmetric, positive definite matrices}
\begin{description} \begin{description}
\item[Symmetric matrix] \marginnote{Symmetric matrix} \item[Symmetric matrix] \marginnote{Symmetric matrix}
@ -176,7 +176,7 @@ Common norms are:
\subsection{Orthogonality} \section{Orthogonality}
\begin{description} \begin{description}
\item[Angle between vectors] \marginnote{Angle between vectors} \item[Angle between vectors] \marginnote{Angle between vectors}
The angle $\omega$ between two vectors $\vec{x}$ and $\vec{y}$ can be obtained from: The angle $\omega$ between two vectors $\vec{x}$ and $\vec{y}$ can be obtained from:
@ -239,7 +239,7 @@ Common norms are:
\subsection{Projections} \section{Projections}
Projections are methods to map high-dimensional data into a lower-dimensional space Projections are methods to map high-dimensional data into a lower-dimensional space
while minimizing the compression loss.\\ while minimizing the compression loss.\\
\marginnote{Orthogonal projection} \marginnote{Orthogonal projection}
@ -250,7 +250,7 @@ In other words, applying $\pi$ multiple times gives the same result (i.e. idempo
$\pi$ can be expressed as a transformation matrix $\matr{P}_\pi$ such that: $\pi$ can be expressed as a transformation matrix $\matr{P}_\pi$ such that:
\[ \matr{P}_\pi^2 = \matr{P}_\pi \] \[ \matr{P}_\pi^2 = \matr{P}_\pi \]
\subsubsection{Projection onto general subspaces} \marginnote{Projection onto subspace basis} \subsection{Projection onto general subspaces} \marginnote{Projection onto subspace basis}
To project a vector $\vec{x} \in \mathbb{R}^n$ into a lower-dimensional subspace $U \subseteq \mathbb{R}^n$, To project a vector $\vec{x} \in \mathbb{R}^n$ into a lower-dimensional subspace $U \subseteq \mathbb{R}^n$,
it is possible to use the basis of $U$.\\ it is possible to use the basis of $U$.\\
% %
@ -263,7 +263,7 @@ and is found by minimizing the distance between $\pi_U(\vec{x})$ and $\vec{x}$.
\subsection{Eigenvectors and eigenvalues} \section{Eigenvectors and eigenvalues}
Given a square matrix $\matr{A} \in \mathbb{R}^{n \times n}$, Given a square matrix $\matr{A} \in \mathbb{R}^{n \times n}$,
$\lambda \in \mathbb{C}$ is an eigenvalue of $\matr{A}$ \marginnote{Eigenvalue} $\lambda \in \mathbb{C}$ is an eigenvalue of $\matr{A}$ \marginnote{Eigenvalue}
@ -328,7 +328,7 @@ we can prove that $\forall c \in \mathbb{R} \smallsetminus \{0\}:$ $c\vec{x}$ is
\end{theorem} \end{theorem}
\subsubsection{Diagonalizability} \subsection{Diagonalizability}
\marginnote{Diagonalizable matrix} \marginnote{Diagonalizable matrix}
A matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is diagonalizable if it is similar to a diagonal matrix $\matr{D} \in \mathbb{R}^{n \times n}$: A matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is diagonalizable if it is similar to a diagonal matrix $\matr{D} \in \mathbb{R}^{n \times n}$:
\[ \exists \matr{P} \in \mathbb{R}^{n \times n} \text{ s.t. } \matr{P} \text{ invertible and } \matr{D} = \matr{P}^{-1}\matr{A}\matr{P} \] \[ \exists \matr{P} \in \mathbb{R}^{n \times n} \text{ s.t. } \matr{P} \text{ invertible and } \matr{D} = \matr{P}^{-1}\matr{A}\matr{P} \]

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statistical-and-mathematical-methods-for-ai/sections/_linear_systems.tex
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@ -1,4 +1,4 @@
\section{Linear systems} \chapter{Linear systems}
A linear system: A linear system:
\begin{equation*} \begin{equation*}
@ -42,7 +42,7 @@ where:
\subsection{Square linear systems} \section{Square linear systems}
\marginnote{Square linear system} \marginnote{Square linear system}
A square linear system $\matr{A}\vec{x} = \vec{b}$ with $\matr{A} \in \mathbb{R}^{n \times n}$ and $\vec{x}, \vec{b} \in \mathbb{R}^n$ A square linear system $\matr{A}\vec{x} = \vec{b}$ with $\matr{A} \in \mathbb{R}^{n \times n}$ and $\vec{x}, \vec{b} \in \mathbb{R}^n$
has an unique solution iff one of the following conditions is satisfied: has an unique solution iff one of the following conditions is satisfied:
@ -58,14 +58,14 @@ However this approach requires to compute the inverse of a matrix, which has a t
\subsection{Direct methods} \section{Direct methods}
\marginnote{Direct methods} \marginnote{Direct methods}
Direct methods compute the solution of a linear system in a finite number of steps. Direct methods compute the solution of a linear system in a finite number of steps.
Compared to iterative methods, they are more precise but more expensive. Compared to iterative methods, they are more precise but more expensive.
The most common approach consists in factorizing the matrix $\matr{A}$. The most common approach consists in factorizing the matrix $\matr{A}$.
\subsubsection{Gaussian factorization} \subsection{Gaussian factorization}
\marginnote{Gaussian factorization\\(LU decomposition)} \marginnote{Gaussian factorization\\(LU decomposition)}
Given a square linear system $\matr{A}\vec{x} = \vec{b}$, Given a square linear system $\matr{A}\vec{x} = \vec{b}$,
the matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is factorized into $\matr{A} = \matr{L}\matr{U}$ such that: the matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is factorized into $\matr{A} = \matr{L}\matr{U}$ such that:
@ -90,7 +90,7 @@ To find the solution, it is sufficient to solve in order:
The overall complexity is $O(\frac{n^3}{3}) + 2 \cdot O(n^2) = O(\frac{n^3}{3})$ The overall complexity is $O(\frac{n^3}{3}) + 2 \cdot O(n^2) = O(\frac{n^3}{3})$
\subsubsection{Gaussian factorization with pivoting} \subsection{Gaussian factorization with pivoting}
\marginnote{Gaussian factorization with pivoting} \marginnote{Gaussian factorization with pivoting}
During the computation of $\matr{A} = \matr{L}\matr{U}$ During the computation of $\matr{A} = \matr{L}\matr{U}$
(using Gaussian elimination\footnote{\url{https://en.wikipedia.org/wiki/LU\_decomposition\#Using\_Gaussian\_elimination}}), (using Gaussian elimination\footnote{\url{https://en.wikipedia.org/wiki/LU\_decomposition\#Using\_Gaussian\_elimination}}),
@ -115,7 +115,7 @@ The solution to the system ($\matr{P}^T\matr{A}\vec{x} = \matr{P}^T\vec{b}$) can
\subsection{Iterative methods} \section{Iterative methods}
\marginnote{Iterative methods} \marginnote{Iterative methods}
Iterative methods solve a linear system by computing a sequence that converges to the exact solution. Iterative methods solve a linear system by computing a sequence that converges to the exact solution.
Compared to direct methods, they are less precise but computationally faster and more adapt for large systems. Compared to direct methods, they are less precise but computationally faster and more adapt for large systems.
@ -127,7 +127,7 @@ Generally, the first vector $\vec{x}_0$ is given (or guessed). Subsequent vector
as $\vec{x}_k = g(\vec{x}_{k-1})$. as $\vec{x}_k = g(\vec{x}_{k-1})$.
The two most common families of iterative methods are: The two most common families of iterative methods are:
\begin{description} \begin{descriptionlist}
\item[Stationary methods] \marginnote{Stationary methods} \item[Stationary methods] \marginnote{Stationary methods}
compute the sequence as: compute the sequence as:
\[ \vec{x}_k = \matr{B}\vec{x}_{k-1} + \vec{d} \] \[ \vec{x}_k = \matr{B}\vec{x}_{k-1} + \vec{d} \]
@ -138,13 +138,13 @@ The two most common families of iterative methods are:
have the form: have the form:
\[ \vec{x}_k = \vec{x}_{k-1} + \alpha_{k-1}\vec{p}_{k-1} \] \[ \vec{x}_k = \vec{x}_{k-1} + \alpha_{k-1}\vec{p}_{k-1} \]
where $\alpha_{k-1} \in \mathbb{R}$ and the vector $\vec{p}_{k-1}$ is called direction. where $\alpha_{k-1} \in \mathbb{R}$ and the vector $\vec{p}_{k-1}$ is called direction.
\end{description} \end{descriptionlist}
\subsubsection{Stopping criteria} \subsection{Stopping criteria}
\marginnote{Stopping criteria} \marginnote{Stopping criteria}
One ore more stopping criteria are needed to determine when to truncate the sequence (as it is theoretically infinite). One ore more stopping criteria are needed to determine when to truncate the sequence (as it is theoretically infinite).
The most common approaches are: The most common approaches are:
\begin{description} \begin{descriptionlist}
\item[Residual based] \item[Residual based]
The algorithm is terminated when the current solution is close enough to the exact solution. The algorithm is terminated when the current solution is close enough to the exact solution.
The residual at iteration $k$ is computed as $\vec{r}_k = \vec{b} - \matr{A}\vec{x}_k$. The residual at iteration $k$ is computed as $\vec{r}_k = \vec{b} - \matr{A}\vec{x}_k$.
@ -158,12 +158,12 @@ The most common approaches are:
The algorithm is terminated when the change between iterations is very small. The algorithm is terminated when the change between iterations is very small.
Given a tolerance $\tau$, the algorithm stops when: Given a tolerance $\tau$, the algorithm stops when:
\[ \Vert \vec{x}_{k} - \vec{x}_{k-1} \Vert \leq \tau \] \[ \Vert \vec{x}_{k} - \vec{x}_{k-1} \Vert \leq \tau \]
\end{description} \end{descriptionlist}
Obviously, as the sequence is truncated, a truncation error is introduced when using iterative methods. Obviously, as the sequence is truncated, a truncation error is introduced when using iterative methods.
\subsection{Condition number} \section{Condition number}
Inherent error causes inaccuracies during the resolution of a system. Inherent error causes inaccuracies during the resolution of a system.
This problem is independent from the algorithm and is estimated using exact arithmetic. This problem is independent from the algorithm and is estimated using exact arithmetic.