Fix typos

statistical-and-mathematical-methods-for-ai/sections/_linear_systems.tex

@@ -54,7 +54,7 @@ has an unique solution iff one of the following conditions is satisfied:
 
 The solution can be algebraically determined as \marginnote{Algebraic solution to linear systems}
 \[ \matr{A}\vec{x} = \vec{b} \iff \vec{x} = \matr{A}^{-1}\vec{b} \]
-However this approach requires to compute the inverse of a matrix, which has a time complexity of $O(n^3)$.
+However, this approach requires to compute the inverse of a matrix, which has a time complexity of $O(n^3)$.
 
 
 
@@ -74,21 +74,23 @@ the matrix $\matr{A} \in \mathbb{R}^{n \times n}$ is factorized into $\matr{A} =
     \item $\matr{U} \in \mathbb{R}^{n \times n}$ is an upper triangular matrix
 \end{itemize}
 %
-As directly solving a system with a triangular matrix has complexity $O(n^2)$ (forward or backward substitutions), 
-the system can be decomposed to:
-\begin{equation}
+The system can be decomposed to:
+\[
     \begin{split}
         \matr{A}\vec{x} = \vec{b} & \iff \matr{LU}\vec{x} = \vec{b} \\
             & \iff \vec{y} = \matr{U}\vec{x} \text{ \& } \matr{L}\vec{y} = \vec{b}
     \end{split}
-\end{equation}
+\]
 To find the solution, it is sufficient to solve in order:
 \begin{enumerate}
     \item $\matr{L}\vec{y} = \vec{b}$ (solved w.r.t. $\vec{y}$)
     \item $\vec{y} = \matr{U}\vec{x}$ (solved w.r.t. $\vec{x}$)
 \end{enumerate}
 
-The overall complexity is $O(\frac{n^3}{3}) + 2 \cdot O(n^2) = O(\frac{n^3}{3})$
+The overall complexity is $O(\frac{n^3}{3}) + 2 \cdot O(n^2) = O(\frac{n^3}{3})$.\\
+$O(\frac{n^3}{3})$ is the time complexity of the LU factorization. 
+$O(n^2)$ is the complexity to directly solving a system with a triangular matrix (forward or backward substitutions).
+
 
 \subsection{Gaussian factorization with pivoting}
 \marginnote{Gaussian factorization with pivoting}
@@ -100,12 +102,12 @@ This is achieved by using a permutation matrix $\matr{P}$, which is obtained as
 
 The permuted system becomes $\matr{P}\matr{A}\vec{x} = \matr{P}\vec{b}$ and the factorization is obtained as $\matr{P}\matr{A} = \matr{L}\matr{U}$.
 The system can be decomposed to:
-\begin{equation}
+\[
     \begin{split}
         \matr{P}\matr{A}\vec{x} = \matr{P}\vec{b} & \iff \matr{L}\matr{U}\vec{x} = \matr{P}\vec{b} \\
             & \iff \vec{y} = \matr{U}\vec{x} \text{ \& } \matr{L}\vec{y} = \matr{P}\vec{b}
     \end{split}
-\end{equation}
+\]
 
 An alternative formulation (which is what \texttt{SciPy} uses) 
 is defined as:
@@ -132,7 +134,7 @@ The two most common families of iterative methods are:
         compute the sequence as:
         \[ \vec{x}_k = \matr{B}\vec{x}_{k-1} + \vec{d} \]
         where $\matr{B}$ is called iteration matrix and $\vec{d}$ is computed from the $\vec{b}$ vector of the system.
-        The time complexity per iteration $O(n^2)$.
+        The time complexity per iteration is $O(n^2)$.
     
     \item[Gradient-like methods] \marginnote{Gradient-like methods}
         have the form:
@@ -142,20 +144,20 @@ The two most common families of iterative methods are:
 
 \subsection{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 or more stopping criteria are needed to determine when to truncate the sequence (as it is theoretically infinite).
 The most common approaches are:
 \begin{descriptionlist}
     \item[Residual based]
         The algorithm is terminated when the current solution is close enough to the exact solution.
         The residual at iteration $k$ is computed as $\vec{r}_k = \vec{b} - \matr{A}\vec{x}_k$.
-        Given a tolerance $\varepsilon$, the algorithm stops when:
+        Given a tolerance $\varepsilon$, the algorithm may stop when:
         \begin{itemize}
-            \item $\Vert \vec{r}_k \Vert \leq \varepsilon$
-            \item $\frac{\Vert \vec{r}_k \Vert}{\Vert \vec{b} \Vert} \leq \varepsilon$
+            \item $\Vert \vec{r}_k \Vert \leq \varepsilon$ (absolute)
+            \item $\frac{\Vert \vec{r}_k \Vert}{\Vert \vec{b} \Vert} \leq \varepsilon$ (relative)
         \end{itemize}
 
     \item[Update based] 
-        The algorithm is terminated when the change between iterations is very small.
+        The algorithm is terminated when the difference between iterations is very small.
         Given a tolerance $\tau$, the algorithm stops when:
         \[ \Vert \vec{x}_{k} - \vec{x}_{k-1} \Vert \leq \tau \]
 \end{descriptionlist}
@@ -183,5 +185,5 @@ Finally, we can define the \textbf{condition number} of a matrix $\matr{A}$ as:
 \[ K(\matr{A}) = \Vert \matr{A} \Vert \cdot \Vert \matr{A}^{-1} \Vert \]
 
 A system is \textbf{ill-conditioned} if $K(\matr{A})$ is large \marginnote{Ill-conditioned}
-(i.e. small perturbation on the input causes large changes in the output).
+(i.e. a small perturbation of the input causes a large change of the output).
 Otherwise it is \textbf{well-conditioned}. \marginnote{Well-conditioned}