update readme and questions of project 1

.gitignore

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-.vscode
+.vscode
+.DS_Store

project1/README.md

@@ -6,7 +6,7 @@ conda activate math6601_env
 pip install -r requirements.txt
 ```
 
-### Question 1
+### Run programs
 
 ```bash
 python Project1_Q1.py > Project1_A1.txt

project1/main.tex

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+\documentclass{article}
+
+% Language setting
+\usepackage[english]{babel}
+
+% Set page size and margins
+\usepackage[letterpaper,top=2cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}
+
+% Useful packages
+\usepackage{amsmath}
+\usepackage{graphicx}
+\usepackage{amsfonts}
+\usepackage[colorlinks=true, allcolors=blue]{hyperref}
+\usepackage{array} % For better table column definitions
+
+\title{MATH 6601 - Project 1 Report}
+\author{Zhe Yuan (yuan.1435)}
+\date{Sep 2025}
+
+\begin{document}
+\maketitle
+
+\section{Projections}
+
+\subsection*{(a)}
+The projection of vector $b$ onto the column space of a full-rank matrix $A$ is given by $p=A(A^*A)^{-1}A^*b$. A function \texttt{proj(A, b)} was implemented based on this formula (see \texttt{Project1\_Q1.py}). For $\epsilon = 1$, the projection is:
+\begin{verbatim}
+p = [1.85714286, 1.0, 3.14285714, 1.28571429, 3.85714286]
+\end{verbatim}
+
+\subsection*{(b)}
+As $\epsilon \rightarrow 0$, the normal equation method becomes numerically unstable. A more robust SVD-based method, \texttt{proj\_SVD(A, b)}, was implemented to handle this (see \texttt{Project1\_Q1.py}). The results are summarized in Table~\ref{tab:proj_results}.
+
+\begin{table}[h!]
+\centering
+\small % Use a smaller font for the table
+\begin{tabular}{|c|>{\ttfamily}p{3cm}|>{\ttfamily}p{3cm}|p{3.5cm}|}
+\hline
+\textbf{$\epsilon$} & \textbf{Normal Equation Method \texttt{proj(A,b)}} & \textbf{SVD Method \texttt{proj\_SVD(A,b)}} & \textbf{Difference (Normal Eq - SVD)} \\
+\hline
+\textbf{1.0} & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [ 1.78e-15 -2.22e-15 -8.88e-16  8.88e-16 0.00e+00] \\
+\hline
+\textbf{0.1} & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [ 7.28e-14 -4.44e-16 -2.66e-14 -1.62e-14 -5.28e-14] \\
+\hline
+\textbf{0.01} & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [-5.45e-12 -7.28e-12 -3.45e-13 -4.24e-12 -4.86e-12] \\
+\hline
+\textbf{1e-4} & [1.85714297 1.00000012 3.14285716 1.28571442 3.85714302] & [1.85714286 1. 3.14285714 1.28571429 3.85714286] & [1.11e-07 1.19e-07 1.92e-08 1.36e-07 1.67e-07] \\
+\hline
+\textbf{1e-8} & \textbf{Error:} \texttt{LinAlgError: Singular matrix} & [1.85714286 1. 3.14285714 1.28571428 3.85714286] & Could not compute difference due to previous error. \\
+\hline
+\textbf{1e-16} & \textbf{Error:} \texttt{ValueError: Matrix A must be full rank} & [1.81820151 1. 3.18179849 2.29149804 2.89030045] & Could not compute difference due to previous error. \\
+\hline
+\textbf{1e-32} & \textbf{Error:} \texttt{ValueError: Matrix A must be full rank} & [2. 1. 3. 2.5 2.5] & Could not compute difference due to previous error. \\
+\hline
+\end{tabular}
+\caption{Comparison of results from the normal equation and SVD-based methods.}
+\label{tab:proj_results}
+\end{table}
+
+\noindent Numerical experiments show that as $\epsilon$ becomes small, the difference between the two methods increases. When $\epsilon$ becomes very small (e.g., 1e-8), the normal equation method fails while the SVD method continues to provide a stable solution.
+
+\section{QR Factorisation}
+
+\subsection*{(a, b, c)}
+Four QR factorization algorithms were implemented to find an orthonormal basis $Q$ for the column space of the $n \times n$ Hilbert matrix (for $n=2$ to $20$):
+\begin{enumerate}
+    \item Classical Gram-Schmidt (CGS, \texttt{cgs\_q})
+    \item Modified Gram-Schmidt (MGS, \texttt{mgs\_q})
+    \item Householder QR (\texttt{householder\_q})
+    \item Classical Gram-Schmidt Twice (CGS-Twice, \texttt{cgs\_twice\_q})
+\end{enumerate}
+For implementation details, please see the \texttt{Project1\_Q2.py} file.
+
+\subsection*{(d)}
+The quality of the computed $Q$ from each method was assessed by plotting the orthogonality loss, $\log_{10}(\|I - Q^T Q\|_F)$, versus the matrix size $n$, as shown in Figure~\ref{fig:ortho_loss}.
+
+\begin{figure}[h!]
+    \centering
+    \includegraphics[width=0.8\textwidth]{Project1_A2d_orthogonality_loss.png}
+    \caption{Orthogonality Loss vs. Matrix Size for QR Methods.}
+    \label{fig:ortho_loss}
+\end{figure}
+
+A summary of the loss and computation time for $n = 4, 9, 11$ is provided in Table~\ref{tab:qr_results}.
+
+\begin{table}[h!]
+    \centering
+    \begin{tabular}{|c|l|r|r|}
+        \hline
+        \textbf{n} & \textbf{Method} & \textbf{Loss ($log_{10}$)} & \textbf{Time (s)} \\
+        \hline
+        4 & Classical Gram-Schmidt & -10.311054 & 4.28e-05 \\
+        4 & Modified Gram-Schmidt & -12.374458 & 5.89e-05 \\
+        4 & Householder & -15.351739 & 1.42e-04 \\
+        4 & Classical Gram-Schmidt Twice & -15.570756 & 8.99e-05 \\
+        \hline
+        9 & Classical Gram-Schmidt & 0.389425 & 2.89e-04 \\
+        9 & Modified Gram-Schmidt & -5.257679 & 3.50e-04 \\
+        9 & Householder & -14.715202 & 6.19e-04 \\
+        9 & Classical Gram-Schmidt Twice & -8.398044 & 6.07e-04 \\
+        \hline
+        11 & Classical Gram-Schmidt & 0.609491 & 5.23e-04 \\
+        11 & Modified Gram-Schmidt & -2.005950 & 5.77e-04 \\
+        11 & Householder & -14.697055 & 8.68e-04 \\
+        11 & Classical Gram-Schmidt Twice & -1.745087 & 1.07e-03 \\
+        \hline
+    \end{tabular}
+    \caption{Loss and computation time for selected matrix sizes.}
+    \label{tab:qr_results}
+\end{table}
+
+\subsection*{(e)}
+The speed and accuracy of the four methods were compared based on the plots in Figure~\ref{fig:ortho_loss} and Figure~\ref{fig:time_taken}.
+
+\begin{figure}[h!]
+    \centering
+    \includegraphics[width=0.8\textwidth]{Project1_A2d_time_taken.png}
+    \caption{Computation Time vs. Matrix Size for QR Methods.}
+    \label{fig:time_taken}
+\end{figure}
+
+\begin{itemize}
+    \item \textbf{Accuracy:} Householder is the most accurate and stable. MGS is significantly better than CGS. CGS-Twice improves on CGS, achieving accuracy comparable to MGS. CGS is highly unstable and loses orthogonality quickly.
+    \item \textbf{Speed:} CGS and MGS are the fastest. Householder is slightly slower, and CGS-Twice is the slowest.
+\end{itemize}
+
+\section{Least Square Problem}
+
+\subsection*{(a)}
+The problem is formulated as $Ax = b$, where $A$ is an $(N+1) \times (n+1)$ Vandermonde matrix, $x$ is the $(n+1) \times 1$ vector of unknown polynomial coefficients $[a_0, \dots, a_n]^T$, and $b$ is the $(N+1) \times 1$ vector of known function values $f(t_j)$.
+
+\subsection*{(b)}
+The least squares problem was solved using the implemented Householder QR factorization (\texttt{householder\_lstsq} in \texttt{Project1\_Q3.py}, the same function in Question 2) to find the coefficient vector $x$.
+
+\subsection*{(c)}
+The function $f(t) = 1 / (1 + t^2)$ and its polynomial approximation $p(t)$ were plotted for $N=30$ and $n=5, 15, 30$, as shown in Figure~\ref{fig:ls_approx}.
+
+\begin{figure}[h!]
+    \centering
+    \includegraphics[width=0.8\textwidth]{Project1_A3c_least_squares_approximation.png}
+    \caption{Least squares approximation of $f(t)=1/(1+t^2)$ for varying polynomial degrees $n$.}
+    \label{fig:ls_approx}
+\end{figure}
+
+When $n=15$, the approximation is excellent. When $n=30$, the polynomial interpolates the points but exhibits wild oscillations between them.
+
+\subsection*{(d)}
+No, $p(t)$ will not converge to $f(t)$ if $N = n$ tends to infinity. Polynomial interpolation on equally spaced nodes diverges for this function, as demonstrated by the severe oscillations in the $n=30$ case shown in Figure~\ref{fig:ls_approx}.
+
+\subsection*{(e)}
+The error between the computed and true coefficients was plotted against the polynomial degree $n$, as shown in Figure~\ref{fig:coeff_error}.
+
+\begin{figure}[h!]
+    \centering
+    \includegraphics[width=0.8\textwidth]{Project1_A3e_coefficient_error.png}
+    \caption{Error in computed polynomial coefficients vs. polynomial degree $n$.}
+    \label{fig:coeff_error}
+\end{figure}
+
+The error in the computed coefficients grows significantly as $n$ increases.
+
+\end{document}