Added tex files to book/

5 files changed, 489 insertions, 0 deletions

diff --git a/book/module1/installing_anaconda.tex b/book/module1/installing_anaconda.tex
new file mode 100644
index 0000000..0ff3d6f
--- /dev/null
+++ b/book/module1/installing_anaconda.tex
@@ -0,0 +1,158 @@
+\section{Installing Anaconda}\label{installing-anaconda}
+
+This tutorial will cover the steps on how to install Anaconda.
+
+\emph{Note for Advanced users: For those who wish to have a lightweight
+installation, can install miniconda or miniForge, however for this
+course we will show you how to use Anaconda Navigator. If you've never
+used the programs before then using Anaconda is recommended.}
+
+\subsubsection{What is Anaconda?}\label{what-is-anaconda}
+
+Anaconda Distribution is a popular open-source Python distribution
+specifically designed for scientific computing, data science, machine
+learning, and artificial intelligence applications. It simplifies the
+set up and use of Python for data science, machine learning, and
+scientific computing. It comes with all the important tools you need,
+like NumPy, Pandas, and JupyterLab, so you don't have to install
+everything separately. The Conda package manager helps you install and
+update software without worrying about breaking other programs. It also
+lets you create separate environments, so different projects don't
+interfere with each other. Additionally, Anaconda includes programs like
+JupyterLab for interactive coding, and Spyer a MATLAB-like IDE.
+
+\subsection{Instructions}\label{instructions}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\item
+  Find the latest version of Navigator from the official Anaconda
+  Inc.~website: \href{https://www.anaconda.com/download}{Download
+  Anaconda}
+\item
+  Press the \emph{Download Now} button.
+\item
+  Press the \emph{Skip registration} button below the submit button,
+  otherwise submit your email address to subscribe to the Anaconda email
+  list.
+\item
+  Under Anaconda Installers press \emph{Download} or find the
+  appropriate version for your operating system below.
+\end{enumerate}
+
+Proceed to next section for your respective operating system.
+
+\subsubsection{Windows}\label{windows}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\setcounter{enumi}{4}
+\tightlist
+\item
+  Once the download is complete, double click the executable (.exe) file
+  to start the installer. Proceed with the installation instructions.
+\end{enumerate}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_01_welcome.png}
+\caption{Welcome screen}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_02_terms.png}
+\caption{Terms and conditions}
+\end{figure}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\setcounter{enumi}{5}
+\tightlist
+\item
+  Select the \emph{Just Me} recommended option.
+\end{enumerate}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_03_for.png}
+\caption{Install for}
+\end{figure}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\setcounter{enumi}{6}
+\tightlist
+\item
+  You can leave the destination folder as is, just make sure you have a
+  minimum of \textasciitilde5 GB available storage space. Press
+  \emph{Next} to proceed.
+\end{enumerate}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_04_destination.png}
+\caption{Installation destination}
+\end{figure}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\setcounter{enumi}{7}
+\tightlist
+\item
+  It is recommended to register Anaconda3 as the default python version
+  if you already have an instance of python installed. Otherwise, you
+  can leave the checkboxes as defaults.
+\end{enumerate}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_05_advanced.png}
+\caption{Avanced Options}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_06_installing.png}
+\caption{Installing}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_07_installing2.png}
+\caption{Installing 2}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_08_installing_complete.png}
+\caption{Complete}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_09_cloud.png}
+\caption{Cloud}
+\end{figure}
+
+\begin{figure}
+\centering
+\includegraphics{figures/installingAnaconda_windows_installer_10_finish.png}
+\caption{Finish}
+\end{figure}
+
+\begin{enumerate}
+\def\labelenumi{\arabic{enumi}.}
+\setcounter{enumi}{8}
+\tightlist
+\item
+  You made it! Anaconda is now installed, you are ready for launch.
+  Assuming that you didn't add Anaconda to PATH environment variable you
+  will need to start navigator from the start menu.
+\end{enumerate}
+
+\subsubsection{Mac/Linux}\label{maclinux}
+
+Anaconda provides installation documentation for Mac and Linux users,
+please refer to the
+\href{https://docs.anaconda.com/anaconda/install/}{documentation page}.
diff --git a/book/module2/documentation.tex b/book/module2/documentation.tex
new file mode 100644
index 0000000..ebd9d87
--- /dev/null
+++ b/book/module2/documentation.tex
@@ -0,0 +1,64 @@
+\section{Documentation of your code}\label{documentation-of-your-code}
+
+Code documentation is essential for maintaining and scaling software
+projects. Whether it's an open-source project or your own private code.
+It ensures that you can understand, use, troubleshoot and build upon the
+code in the future.
+
+\subsection{Keep Detailed and Accurate
+Notes}\label{keep-detailed-and-accurate-notes}
+
+Just as a recipe requires clear instructions, your code should be
+accompanied by comprehensive notes. Document your process thoroughly to
+ensure that others (and future you) can follow along without confusion.
+
+When documenting a project, it's essential to include detailed notes
+that capture not just what the code does, but how it was developed. This
+includes recording libraries used, citing any external code snippets
+along with their sources, and outlining the sequence of steps taken
+throughout the coding process. Such comprehensive documentation enables
+others---and your future self---to understand, recreate, and maintain
+the project more effectively, reducing confusion and improving long-term
+usability.
+
+\subsection{Explain Your Decisions}\label{explain-your-decisions}
+
+In programming, there are often several valid approaches to solving a
+problem. When documenting your code, it's important to clarify why you
+chose a particular method---especially if it deviates from common
+practices. Anticipating potential questions and addressing them directly
+in your documentation helps others follow your reasoning and builds
+trust in your solution. \includegraphics{figures/rubberDuck.png} A
+useful strategy for articulating these decisions is the ``rubber duck''
+technique---explaining your code as if you're teaching it to someone
+else. Whether spoken aloud or written down, this practice helps you
+clarify your logic and communicate the reasoning behind your choices,
+providing valuable context for future collaborators or
+reviewers.\hspace{0pt}
+
+\subsection{Include a README}\label{include-a-readme}
+
+A README file serves as the introduction to your project. It should be
+placed in the top-level directory and provide essential
+information.\hspace{0pt} A good readme file may include: - Project title
+and description - Installation instructions - Usage examples -
+Contribution guidelines (if applicable) - License information (if
+applicable)
+
+This file acts as a roadmap for anyone interacting with your project.
+\hspace{0pt}
+
+\subsection{In-line Comments}\label{in-line-comments}
+
+While external documentation is vital, in-code comments provide
+immediate context. Use them to explain complex logic or important
+sections within your code.\hspace{0pt}Here are some guidelines to
+follow: Keep comments concise and relevant. Avoid stating the obvious;
+focus on the ``why'' rather than the ``what.''.
+
+\subsection{Maintain and Update
+Documentation}\label{maintain-and-update-documentation}
+
+Similarly to your code, the documentation should evolve alongside your
+code. Regularly review and update it to reflect changes, ensuring
+accuracy and relevance.\hspace{0pt}
diff --git a/book/module2/error.tex b/book/module2/error.tex
new file mode 100644
index 0000000..2646dc8
--- /dev/null
+++ b/book/module2/error.tex
@@ -0,0 +1,97 @@
+\section{Errors in Numerical
+Computations}\label{errors-in-numerical-computations}
+
+In any numerical method, \textbf{error} is inevitable. Understanding
+\textbf{what kinds of errors occur} and \textbf{why} is essential to
+building reliable and accurate computations.
+
+We mainly classify errors into two major types: - Truncation Error -
+Round-off Error
+
+\subsection{What is Error?}\label{what-is-error}
+
+Let's remind ourselves what error is: \[
+\text{Error} = \text{True Value} - \text{Approximate Value}
+\] However, often the \textbf{true value} is unknown, so we focus on
+\textbf{reducing} and \textbf{analyzing} different types of errors
+instead of eliminating them completely. This can be done by using
+relative error when using iterative methods and is calculated as
+follows: \[
+\text{Relative Error} = \frac{\text{Best} - \text{Second to best}}{Best}
+\]
+
+\subsection{Truncation Error}\label{truncation-error}
+
+Truncation error occurs \textbf{when an infinite process is approximated
+by a finite process}.\\
+In simple terms, it happens \textbf{when you cut off or ``truncate''
+part of the computation}. An example of this could be using a finite
+number of terms from a Taylor Series expansion to approximate a
+function.
+
+Approximating \(e^x\) by the first few terms of its Taylor series:
+
+\[e^x \approx 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!}​\]
+
+The error comes from \textbf{neglecting} all the higher order terms
+(\(\frac{x^4}{4!}, \frac{x^5}{5!}\), \ldots).
+
+Truncation error occurs when using numerical methods such as
+approximating and calculating derivatives and integrals. A
+representation of the truncation error is show in the figure below.
+Using our numerical methods we are left if some degree of error.
+
+\begin{figure}
+\centering
+\includegraphics{figures/truncationError.png}
+\caption{Representation of truncation error under a curve}
+\end{figure}
+
+In order to reduce truncation error there are a few things we can do: -
+Include more terms (higher-order methods) - Decrease step sizes (e.g.,
+smaller \(\Delta x\) in approximations) - Use better approximation
+algorithms.
+
+\subsection{Round-off Error}\label{round-off-error}
+
+Round-off error is caused by \textbf{the limited precision} with which
+computers represent numbers. Since computers cannot store an infinite
+number of digits, \textbf{they round off} after a certain number of
+decimal or binary places. For example, instead of representing π with
+infinite decimal places it may be rounded off to approximately 16 digits
+depending on number of bits and the representation of the bits.
+
+In other words, round-off error happens because of how computers store
+numbers. For a double-floating point, the number is stored using
+64-bits. The more bits we use, the more precise of a number we can
+store. However, it makes it costs us more memory making it more
+computational expensive.
+
+While individual round-off errors may seem negligible, their effects can
+\textbf{accumulate over repeated computations}, leading to significant
+inaccuracies. This is particularly problematic in operations such as
+\textbf{subtracting two nearly equal numbers}, where \textbf{loss of
+significance} can occur, severely reducing numerical precision and
+amplifying the impact of round-off error.
+
+\subsubsection{How to Reduce Round-off
+Error:}\label{how-to-reduce-round-off-error}
+
+To reduce round-off error, use higher-precision data types when storing
+numerical values. Additionally, code and algorithms should be structured
+to \textbf{avoid subtracting nearly equal numbers}, a common source of
+significant error. Finally, employing \textbf{numerically stable
+algorithms} is essential for minimizing the accumulation of round-off
+errors during computation.
+
+\subsection{Total Error}\label{total-error}
+
+Truncation and round-off error are inversely proportional, meaning that
+if we decrease one, the other increases. If we want to minimize total
+error we must find the optimal point between step size and error.
+
+\begin{figure}
+\centering
+\includegraphics{figures/totalError.png}
+\caption{Total Error}
+\end{figure}
diff --git a/book/module2/non_linear_eqn_solver.tex b/book/module2/non_linear_eqn_solver.tex
new file mode 100644
index 0000000..5550d12
--- /dev/null
+++ b/book/module2/non_linear_eqn_solver.tex
@@ -0,0 +1,129 @@
+\section{Solving non-linear
+equations}\label{solving-non-linear-equations}
+
+\subsection{Introduction}\label{introduction}
+
+\subsection{Prerequisites}\label{prerequisites}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{import}\NormalTok{ numpy}
+\ImportTok{import}\NormalTok{ scipy}
+\ImportTok{import}\NormalTok{ sympy}
+\end{Highlighting}
+\end{Shaded}
+
+\subsection{fsolve from SciPy}\label{fsolve-from-scipy}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{from}\NormalTok{ scipy.optimize }\ImportTok{import}\NormalTok{ fsolve}
+
+\KeywordTok{def}\NormalTok{ equations(}\BuiltInTok{vars}\NormalTok{):}
+\NormalTok{    x, y }\OperatorTok{=} \BuiltInTok{vars}
+\NormalTok{    eq1 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ y}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}} \DecValTok{25}
+\NormalTok{    eq2 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}}\NormalTok{ y}
+    \ControlFlowTok{return}\NormalTok{ [eq1, eq2]}
+
+\NormalTok{initial\_guess }\OperatorTok{=}\NormalTok{ [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]}
+\NormalTok{solution }\OperatorTok{=}\NormalTok{ fsolve(equations, initial\_guess)}
+\BuiltInTok{print}\NormalTok{(}\StringTok{"Solution:"}\NormalTok{, solution)}
+\end{Highlighting}
+\end{Shaded}
+
+\subsection{root from SciPy}\label{root-from-scipy}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{from}\NormalTok{ scipy.optimize }\ImportTok{import}\NormalTok{ root}
+
+\KeywordTok{def}\NormalTok{ equations(}\BuiltInTok{vars}\NormalTok{):}
+\NormalTok{    x, y }\OperatorTok{=} \BuiltInTok{vars}
+\NormalTok{    eq1 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ y}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}} \DecValTok{25}
+\NormalTok{    eq2 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}}\NormalTok{ y}
+    \ControlFlowTok{return}\NormalTok{ [eq1, eq2]}
+
+\NormalTok{initial\_guess }\OperatorTok{=}\NormalTok{ [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]}
+\NormalTok{solution }\OperatorTok{=}\NormalTok{ root(equations, initial\_guess)}
+\BuiltInTok{print}\NormalTok{(}\StringTok{"Solution:"}\NormalTok{, solution.x)}
+\end{Highlighting}
+\end{Shaded}
+
+\subsection{minimize from SciPy}\label{minimize-from-scipy}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{from}\NormalTok{ scipy.optimize }\ImportTok{import}\NormalTok{ minimize}
+
+\CommentTok{\# Define the equations}
+\KeywordTok{def}\NormalTok{ equation1(x, y):}
+    \ControlFlowTok{return}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ y}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}} \DecValTok{25}
+
+\KeywordTok{def}\NormalTok{ equation2(x, y):}
+    \ControlFlowTok{return}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}}\NormalTok{ y}
+
+\CommentTok{\# Define the objective function for optimization}
+\KeywordTok{def}\NormalTok{ objective(xy):}
+\NormalTok{    x, y }\OperatorTok{=}\NormalTok{ xy}
+    \ControlFlowTok{return}\NormalTok{ equation1(x, y)}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ equation2(x, y)}\OperatorTok{**}\DecValTok{2}
+
+\CommentTok{\# Initial guess}
+\NormalTok{initial\_guess }\OperatorTok{=}\NormalTok{ [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]}
+
+\CommentTok{\# Perform optimization}
+\NormalTok{result }\OperatorTok{=}\NormalTok{ minimize(objective, initial\_guess)}
+\NormalTok{solution\_optimization }\OperatorTok{=}\NormalTok{ result.x}
+
+\BuiltInTok{print}\NormalTok{(}\StringTok{"Optimization Method Solution:"}\NormalTok{, solution\_optimization)}
+\end{Highlighting}
+\end{Shaded}
+
+\subsection{nsolve from SymPy}\label{nsolve-from-sympy}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{from}\NormalTok{ sympy }\ImportTok{import}\NormalTok{ symbols, Eq, nsolve}
+
+\CommentTok{\# Define the variables}
+\NormalTok{x, y }\OperatorTok{=}\NormalTok{ symbols(}\StringTok{\textquotesingle{}x y\textquotesingle{}}\NormalTok{)}
+
+\CommentTok{\# Define the equations}
+\NormalTok{eq1 }\OperatorTok{=}\NormalTok{ Eq(x}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ y}\OperatorTok{**}\DecValTok{2}\NormalTok{, }\DecValTok{25}\NormalTok{)}
+\NormalTok{eq2 }\OperatorTok{=}\NormalTok{ Eq(x }\OperatorTok{{-}}\NormalTok{ y, }\DecValTok{0}\NormalTok{)}
+
+\CommentTok{\# Initial guess for the solution}
+\NormalTok{initial\_guess }\OperatorTok{=}\NormalTok{ [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]}
+
+\CommentTok{\# Use nsolve to find the solution}
+\NormalTok{solution }\OperatorTok{=}\NormalTok{ nsolve([eq1, eq2], [x, y], initial\_guess)}
+\BuiltInTok{print}\NormalTok{(}\StringTok{"Solution:"}\NormalTok{, solution)}
+\end{Highlighting}
+\end{Shaded}
+
+\subsection{newton\_method from NumPy}\label{newton_method-from-numpy}
+
+\begin{Shaded}
+\begin{Highlighting}[]
+\ImportTok{import}\NormalTok{ numpy }\ImportTok{as}\NormalTok{ np}
+
+\KeywordTok{def}\NormalTok{ equations(}\BuiltInTok{vars}\NormalTok{):}
+\NormalTok{    x, y }\OperatorTok{=} \BuiltInTok{vars}
+\NormalTok{    eq1 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{+}\NormalTok{ y}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}} \DecValTok{25}
+\NormalTok{    eq2 }\OperatorTok{=}\NormalTok{ x}\OperatorTok{**}\DecValTok{2} \OperatorTok{{-}}\NormalTok{ y}
+    \ControlFlowTok{return}\NormalTok{ np.array([eq1, eq2])}
+
+\KeywordTok{def}\NormalTok{ newton\_method(initial\_guess, tolerance}\OperatorTok{=}\FloatTok{1e{-}6}\NormalTok{, max\_iter}\OperatorTok{=}\DecValTok{100}\NormalTok{):}
+    \BuiltInTok{vars} \OperatorTok{=}\NormalTok{ np.array(initial\_guess, dtype}\OperatorTok{=}\BuiltInTok{float}\NormalTok{)}
+    \ControlFlowTok{for}\NormalTok{ \_ }\KeywordTok{in} \BuiltInTok{range}\NormalTok{(max\_iter):}
+\NormalTok{        J }\OperatorTok{=}\NormalTok{ np.array([[}\DecValTok{2} \OperatorTok{*} \BuiltInTok{vars}\NormalTok{[}\DecValTok{0}\NormalTok{], }\DecValTok{2} \OperatorTok{*} \BuiltInTok{vars}\NormalTok{[}\DecValTok{1}\NormalTok{]], [}\DecValTok{2} \OperatorTok{*} \BuiltInTok{vars}\NormalTok{[}\DecValTok{0}\NormalTok{], }\OperatorTok{{-}}\DecValTok{1}\NormalTok{]])}
+\NormalTok{        F }\OperatorTok{=}\NormalTok{ equations(}\BuiltInTok{vars}\NormalTok{)}
+\NormalTok{        delta }\OperatorTok{=}\NormalTok{ np.linalg.solve(J, }\OperatorTok{{-}}\NormalTok{F)}
+        \BuiltInTok{vars} \OperatorTok{+=}\NormalTok{ delta}
+        \ControlFlowTok{if}\NormalTok{ np.linalg.norm(delta) }\OperatorTok{\textless{}}\NormalTok{ tolerance:}
+            \ControlFlowTok{return} \BuiltInTok{vars}
+
+\NormalTok{initial\_guess }\OperatorTok{=}\NormalTok{ [}\DecValTok{1}\NormalTok{, }\DecValTok{1}\NormalTok{]}
+\NormalTok{solution }\OperatorTok{=}\NormalTok{ newton\_method(initial\_guess)}
+\BuiltInTok{print}\NormalTok{(}\StringTok{"Solution:"}\NormalTok{, solution)}
+\end{Highlighting}
+\end{Shaded}
diff --git a/book/module3/numerical_methods.tex b/book/module3/numerical_methods.tex
new file mode 100644
index 0000000..737d69d
--- /dev/null
+++ b/book/module3/numerical_methods.tex
@@ -0,0 +1,41 @@
+\section{Numerical Methods}\label{numerical-methods}
+
+Engineering
+
+\subsection{What is a numerical
+method?}\label{what-is-a-numerical-method}
+
+Numerical methods are techniques that transform mathematical problems
+into forms that can be solved using arithmetic and logical operations.
+Because digital computers excel at these computations, numerical methods
+are often referred to as computer mathematics.
+
+\subsection{Numerical Differentiation}\label{numerical-differentiation}
+
+Forwards difference Backwards difference
+
+\href{https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter20.00-Numerical-Differentiation.html}{Read
+More}
+
+\subsection{Roots and Optimization}\label{roots-and-optimization}
+
+Incremental Search Bisection Modified Secant Newton-Raphson
+
+\subsection{Numerical Integration}\label{numerical-integration}
+
+Trapezoidal
+
+Simpson's Rule
+
+\href{https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter21.00-Numerical-Integration.html}{Read
+More}
+
+\subsection{Numerical Solutions of Ordinary Differential
+Equations}\label{numerical-solutions-of-ordinary-differential-equations}
+
+Euler's Method - Forward - Backwards
+
+Runge-Kutta Method
+
+\href{https://pythonnumericalmethods.studentorg.berkeley.edu/notebooks/chapter22.00-ODE-Initial-Value-Problems.html}{Read
+More}