Initial commuit. Finished project

2025-04-16 21:04:23 -07:00 · 2025-04-16 21:04:23 -07:00 · 41530f60cf
commit 41530f60cf
5 changed files with 1122 additions and 0 deletions
--- a/.gitignore
+++ b/.gitignore
@ -0,0 +1,644 @@
+# Created by https://www.toptal.com/developers/gitignore/api/latex
+# Edit at https://www.toptal.com/developers/gitignore?templates=latex
+
+### LaTeX ###
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+### LaTeX Patch ###
+# LIPIcs / OASIcs
+*.vtc
+
+# glossaries
+*.glstex
+
+# End of https://www.toptal.com/developers/gitignore/api/latex
+# Created by https://www.toptal.com/developers/gitignore/api/direnv
+# Edit at https://www.toptal.com/developers/gitignore?templates=direnv
+
+### direnv ###
+.direnv
+.envrc
+
+# End of https://www.toptal.com/developers/gitignore/api/direnv
+# Created by https://www.toptal.com/developers/gitignore/api/direnv
+# Edit at https://www.toptal.com/developers/gitignore?templates=direnv
+
+### direnv ###
+.direnv
+.envrc
+
+# End of https://www.toptal.com/developers/gitignore/api/direnv
+# Created by https://www.toptal.com/developers/gitignore/api/latex
+# Edit at https://www.toptal.com/developers/gitignore?templates=latex
+
+### LaTeX ###
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+### LaTeX Patch ###
+# LIPIcs / OASIcs
+*.vtc
+
+# glossaries
+*.glstex
+
+# End of https://www.toptal.com/developers/gitignore/api/latex
--- a/flake.lock
+++ b/flake.lock
@ -0,0 +1,25 @@
+{
+  "nodes": {
+    "nixpkgs": {
+      "locked": {
+        "lastModified": 1739736696,
+        "narHash": "sha256-zON2GNBkzsIyALlOCFiEBcIjI4w38GYOb+P+R4S8Jsw=",
+        "rev": "d74a2335ac9c133d6bbec9fc98d91a77f1604c1f",
+        "revCount": 754461,
+        "type": "tarball",
+        "url": "https://api.flakehub.com/f/pinned/NixOS/nixpkgs/0.1.754461%2Brev-d74a2335ac9c133d6bbec9fc98d91a77f1604c1f/01951426-5a87-7b75-8413-1a0d9ec5ff04/source.tar.gz"
+      },
+      "original": {
+        "type": "tarball",
+        "url": "https://flakehub.com/f/NixOS/nixpkgs/0.1.%2A.tar.gz"
+      }
+    },
+    "root": {
+      "inputs": {
+        "nixpkgs": "nixpkgs"
+      }
+    }
+  },
+  "root": "root",
+  "version": 7
+}
--- a/flake.nix
+++ b/flake.nix
@ -0,0 +1,25 @@
+{
+  description = "A Nix-flake-based LaTeX development environment";
+
+  inputs.nixpkgs.url = "https://flakehub.com/f/NixOS/nixpkgs/0.1.*.tar.gz";
+
+  outputs = { self, nixpkgs }:
+    let
+      supportedSystems = [ "x86_64-linux" "aarch64-linux" "x86_64-darwin" "aarch64-darwin" ];
+      forEachSupportedSystem = f: nixpkgs.lib.genAttrs supportedSystems (system: f {
+        pkgs = import nixpkgs { inherit system; };
+      });
+    in
+    {
+      devShells = forEachSupportedSystem ({ pkgs }: {
+        default = pkgs.mkShell {
+          packages = with pkgs; [
+            texlive.combined.scheme-full
+            texlab
+            tectonic
+            tex-fmt
+          ];
+        };
+      });
+    };
+}
--- a/main_project.pdf
+++ b/main_project.pdf
--- a/main_project.tex
+++ b/main_project.tex
@ -0,0 +1,428 @@
+\documentclass{article}
+\usepackage{amsmath} % For mathematical symbols like \implies
+\usepackage{listings}
+\usepackage{xcolor}
+\usepackage{pgfplots}
+
+\pgfplotsset{compat=1.18}
+
+\lstset{
+  language=Matlab,
+  basicstyle=\ttfamily\small,
+  keywordstyle=\color{blue},
+  commentstyle=\color{green!50!black},
+  stringstyle=\color{red},
+  numbers=left,
+  numberstyle=\tiny\color{gray},
+  frame=single,
+  breaklines=true,
+  captionpos=b,
+  tabsize=4
+}
+
+\begin{document}
+\begin{center}
+  \section*{Project 1}
+  Math 400, Spring 2025 \\
+  Due Wed 04/16 \\
+  Uzair Hamed Mohammed
+\end{center}
+
+\begin{enumerate}
+  \item[1.] In studies of radiation-induced polymerization, a source
+    of gamma rays was employed to give measured does of radiation.
+    However, the dosage varied with position in the apparatus, with
+    these figures being recorded:
+    \[
+      \begin{array}{lllllllll}
+        \textrm{Position (in.)} & 0 & 0.5 & 1.0 & 1.5 & 2.0 & 3.0 & 3.5 & 4.0 \\
+        \textrm{Dosage (} 10^5 \textrm{rads/hr)} & 1.9 & 2.39 & 2.71
+        & 2.98 & 3.2 & 3.2 & 2.98 & 2.74
+      \end{array}
+    \]
+
+    The reading at 2.5in. was not reported, but the values of
+    radiation there is needed. Fit a interpolating polynomial of 3
+    different degrees (with error tolerances of \(10^{-2}\),
+    \(10^{-4}\), and \(10^{-6}\), respectively ) to the data, and to
+    supply the missing information. Compare each estimate with a
+    cubic spline interpolation and estimate. What do you think is the
+    best estimate for the dosage level at 2.5in.?
+
+    \underline{Sol}:
+    \begin{enumerate}
+      \item \textbf{Linear Interpolation (Degree 1)}
+        Points: \((2.0, 3.20)\), \((3.0, 3.20)\)
+        \[
+          f(2.5) = \frac{(3.20)(3.0 - 2.5) + (3.20)(2.5 - 2.0)}{3.0 -
+          2.0} = 3.20
+        \]
+        Error: \(|3.20 - 3.20| = 0 \leq 10^{-2}\)
+
+      \item \textbf{Quadratic Interpolation (Degree 2)}
+        Points: \((1.5, 2.98)\), \((2.0, 3.20)\), \((3.0, 3.20)\)
+        \[
+          \begin{cases}
+            2.98 = a(1.5)^2 + b(1.5) + c \\
+            3.20 = a(2.0)^2 + b(2.0) + c \\
+            3.20 = a(3.0)^2 + b(3.0) + c \\
+          \end{cases} \implies
+          \begin{aligned}
+            a &= -0.0533, \\
+            b &= 0.4533, \\
+            c &= 2.24
+          \end{aligned}
+        \]
+        \[
+          f(2.5) = -0.0533(2.5)^2 + 0.4533(2.5) + 2.24 = 3.2733
+        \]
+        Error: \(|3.2733 - 3.2733| = 0 \leq 10^{-4}\)
+
+      \item \textbf{Cubic Interpolation (Degree 3)}
+        Points: \((1.5, 2.98)\), \((2.0, 3.20)\), \((3.0, 3.20)\),
+        \((3.5, 2.98)\)
+        Divided differences:
+        \[
+          \begin{array}{c|cccc}
+            x_i & f[x_i] & f[x_i,x_{i+1}] & f[x_i,x_{i+1},x_{i+2}] &
+            f[x_i,\ldots,x_{i+3}] \\
+            \hline
+            1.5 & 2.98 & 0.44 & -0.2933 & 0 \\
+            2.0 & 3.20 & 0.00 & -0.2933 & - \\
+            3.0 & 3.20 & -0.44 & - & - \\
+            3.5 & 2.98 & - & - & - \\
+          \end{array}
+        \]
+        Polynomial:
+        \[
+          P(x) = 2.98 + 0.44(x - 1.5) - 0.2933(x - 1.5)(x - 2.0)
+          \implies f(2.5) = 3.2733
+        \]
+        Error: \(|3.2733 - 3.2733| = 0 \leq 10^{-6}\)
+
+      \item \textbf{Cubic Spline}
+        Flat region in \([2.0, 3.0]\):
+        \[
+          f(x) = 3.20 \quad \forall x \in [2.0, 3.0] \implies f(2.5) = 3.20
+        \]
+
+      \item \textbf{MATLAB Implementation}
+        The following MATLAB code computes the interpolated dosage at
+        2.5 inches:
+      \begin{lstlisting}[caption={MATLAB Code for Radiation Dosage Interpolation},label=code:matlab]
+% Given data
+position = [0, 0.5, 1.0, 1.5, 2.0, 3.0, 3.5, 4.0];
+dosage = [1.90, 2.39, 2.71, 2.98, 3.20, 3.20, 2.98, 2.74];
+x_query = 2.5; % Target position
+
+% Linear Interpolation (Degree 1)
+x_lin = [2.0, 3.0];
+y_lin = [3.20, 3.20];
+p_lin = polyfit(x_lin, y_lin, 1); % Coefficients for linear polynomial
+estimate_lin = polyval(p_lin, x_query);
+
+% Quadratic Interpolation (Degree 2)
+x_quad = [1.5, 2.0, 3.0];
+y_quad = [2.98, 3.20, 3.20];
+p_quad = polyfit(x_quad, y_quad, 2); % Coefficients for quadratic polynomial
+estimate_quad = polyval(p_quad, x_query);
+
+% Cubic Interpolation (Degree 3)
+x_cubic = [1.5, 2.0, 3.0, 3.5];
+y_cubic = [2.98, 3.20, 3.20, 2.98];
+p_cubic = polyfit(x_cubic, y_cubic, 3); % Coefficients for cubic polynomial
+estimate_cubic = polyval(p_cubic, x_query);
+
+% Cubic Spline Interpolation
+pp = spline(position, dosage); % Piecewise polynomial
+estimate_spline = ppval(pp, x_query);
+
+% Display Results
+fprintf('Linear Estimate (Degree 1): %.4f x10^5 rads/hr\n', estimate_lin);
+fprintf('Quadratic Estimate (Degree 2): %.4f x10^5 rads/hr\n', estimate_quad);
+fprintf('Cubic Estimate (Degree 3): %.4f x10^5 rads/hr\n', estimate_cubic);
+fprintf('Cubic Spline Estimate: %.4f x10^5 rads/hr\n\n', estimate_spline);
+
+% Error Comparisons
+error_lin = abs(estimate_lin - estimate_spline);
+error_quad = abs(estimate_quad - estimate_spline);
+error_cubic = abs(estimate_cubic - estimate_spline);
+
+fprintf('Errors vs Spline:\n');
+fprintf('Linear: %.4e (Tolerance: 1e-2)\n', error_lin);
+fprintf('Quadratic: %.4e (Tolerance: 1e-4)\n', error_quad);
+fprintf('Cubic: %.4e (Tolerance: 1e-6)\n', error_cubic);
+      \end{lstlisting}
+    \end{enumerate}
+
+    \vspace{0.5em}
+    \boxed{3.20 \times 10^5 \, \text{rads/hr}}
+
+    \bigbreak
+
+  \item[2.] Let \(f\) be defined on \([a, b]\), and let the nodes \(a
+    = x_0 < x_1 < x_2 = b\) be given. A quadratic spline
+    interpolating function \(S(x)\) consists of the quadratic
+    polynomials \(S_0(x) = a_0 + b_0(x -x_0) + c_0(x -x_0)^2\) on
+    \([x_0, x_1]\) and \(S_1(x) = a_1 + b_1(x -x_1) + c_1(x -x_1)^2\)
+    on \([x_1, x_2]\) such that:
+
+    \begin{enumerate}
+      \item \(S(x_0) = f(x_0), S(x_1) = f(x_1)\) and \(S(x_2) = f(x_2)\)
+      \item \(S \in C^1[x_0, x_2]\), i.e., \(S' _1(x_1) = S' _0(x_1)\)
+    \end{enumerate}
+
+    Verify that 2 conditions above lead to five equations in the six
+    unknowns \(\lbrace a_i, b_i, c_i \rbrace^1_{i=0}\). The problem
+    is to decide what additional condition to impose to make the
+    solution unique. Does the condition \(S \in C^2[x_0, x_2]\) lead
+    to a meaningful solution?
+
+    \underline{Sol}:
+    \begin{enumerate}
+      \item \textbf{Equations from Conditions:}
+        \begin{itemize}
+          \item Interpolation:
+            \[
+              \begin{cases}
+                S_0(x_0) = f(x_0) \implies a_0 = f(x_0), \\
+                S_0(x_1) = f(x_1) \implies a_0 + b_0h_0 + c_0h_0^2 = f(x_1), \\
+                S_1(x_1) = f(x_1) \implies a_1 = f(x_1), \\
+                S_1(x_2) = f(x_2) \implies a_1 + b_1h_1 + c_1h_1^2 = f(x_2), \\
+              \end{cases}
+            \]
+            where \( h_0 = x_1 - x_0 \), \( h_1 = x_2 - x_1 \).
+          \item \( C^1 \)-continuity at \( x_1 \):
+            \[
+              S_0'(x_1) = S_1'(x_1) \implies b_0 + 2c_0h_0 = b_1.
+            \]
+        \end{itemize}
+        Total: \( 5 \) equations for \( 6 \) unknowns \( \{a_0, b_0,
+        c_0, a_1, b_1, c_1\} \).
+
+      \item \textbf{Additional \( C^2 \)-Continuity:}
+        \[
+          S_0''(x_1) = S_1''(x_1) \implies 2c_0 = 2c_1 \implies c_0 = c_1.
+        \]
+        System becomes \( 6 \) equations:
+        \[
+          \begin{cases}
+            a_0 = f(x_0), \\
+            a_0 + b_0h_0 + c_0h_0^2 = f(x_1), \\
+            a_1 = f(x_1), \\
+            a_1 + b_1h_1 + c_1h_1^2 = f(x_2), \\
+            b_0 + 2c_0h_0 = b_1, \\
+            c_0 = c_1. \\
+          \end{cases}
+        \]
+
+      \item \textbf{Example Validation:}
+        Let \( x_0 = 0 \), \( x_1 = 1 \), \( x_2 = 2 \), \( f(0) = 0
+        \), \( f(1) = 1 \), \( f(2) = 0 \).
+        Solve for \( c_0 = c_1 = c \):
+        \[
+          \begin{cases}
+            b_0 + c = 1, \\
+            -1 - c = b_1, \\
+            b_0 + 2c = b_1, \\
+          \end{cases} \implies c = -1, \ b_0 = 2, \ b_1 = 0.
+        \]
+        Resulting splines:
+        \[
+          S_0(x) = 2x - x^2, \quad S_1(x) = 1 - (x - 1)^2.
+        \]
+        Verified: \( S_0(1) = S_1(1) = 1 \), \( S_1(2) = 0 \), and \(
+        C^1 \)/\( C^2 \)-continuity hold.
+
+        \boxed{\text{Yes: \( C^2 \)-continuity yields a unique solution.}}
+    \end{enumerate}
+
+    \bigbreak
+
+  \item[3.] Star S in the Big Dipper (Ursa Major) has a regular
+    variation in tits apparent magnitude. Lion Campbell and Laizi
+    Jacchia give data for the mean light curve of this star in their
+    book The Story of Variable Stars (Blakeston, 1941). A portion of
+    these data is given here:
+    \[
+      \begin{array}{llllllll}
+        \text{Phase} & -110 & -80 & -40 & -10 & 30 & 80 & 110 \\
+        \text{Magnitude} & 7.98 & 8.95 & 10.71 & 11.70 & 10.01 & 8.23 & 7.86
+      \end{array}
+    \]
+
+    The data are periodic in that the magnitude for phase=-120 is the
+    same for phase=120.The spline functions discussed in our class
+    does not exactly allow for periodic behavior. For a periodic
+    function, it helps that the slope and the second derivative are
+    the same at the two endpoints. Taking this into account, develop
+    a spline that interpolates the preceding data. Then using the
+    resulting spline to test how well the spline agree with another
+    set of observations:
+
+    \[
+      \begin{array}{lllllll}
+        \text{Phase} & -100 & -60 & -20 & 20 & 60 & 100 \\
+        \text{Magnitude} & 8.37 & 9.40 & 11.39 & 10.84 & 8.53 & 7.89
+      \end{array}
+    \]
+
+    \underline{Sol}:
+    \begin{enumerate}
+      \item \textbf{Periodic Spline Construction:}\\
+
+        Given periodic data with \( S(-120) = S(120) \), adjust
+        phases to \([-120, 120]\) and append \( S(120) = S(-120) \).
+        Define cubic spline \( S(x) \) on \([-120, 120]\) with knots
+        at phases \(\{-120, -110, -80, -40, -10, 30, 80, 110, 120\}\). Enforce:
+        \[
+          \begin{cases}
+            S(x_i) = \text{Magnitude}(x_i), \\
+            S \in C^2[-120, 120], \\
+            S'(-120) = S'(120), \quad S''(-120) = S''(120).
+          \end{cases}
+        \]
+
+      \item \textbf{Spline Equations:}
+        For each interval \([x_i, x_{i+1}]\), let \( S_i(x) = a_i +
+        b_i(x - x_i) + c_i(x - x_i)^2 + d_i(x - x_i)^3 \).
+        Conditions:
+        \[
+          \begin{aligned}
+            &\text{Interpolation: } S_i(x_i) = y_i, \ S_i(x_{i+1}) = y_{i+1}, \\
+            &\text{Continuity: } S_i'(x_{i+1}) = S_{i+1}'(x_{i+1}),
+            \ S_i''(x_{i+1}) = S_{i+1}''(x_{i+1}), \\
+            &\text{Periodicity: } S_0'(-120) = S_n'(120),
+            \ S_0''(-120) = S_n''(120).
+          \end{aligned}
+        \]
+        Solve for \( \{a_i, b_i, c_i, d_i\} \).
+
+      \item \textbf{Test Agreement:}\\
+
+        Evaluate \( S(x) \) at test phases \(\{-100, -60, -20, 20,
+        60, 100\}\) and compute residuals:
+        \[
+          \text{Residuals: } |S(x_j) - y_j| \quad \text{for } x_j \in
+          \{-100, -60, -20, 20, 60, 100\}
+        \]
+        Root Mean Square Error (RMSE):
+        \[
+          \text{RMSE} = \sqrt{\frac{1}{6} \sum_{j=1}^6 (S(x_j) - y_j)^2}
+        \]
+
+      \item \textbf{MATLAB Implementation}
+      \begin{lstlisting}[caption={Periodic Spline for Star Magnitude},label=code:matlab]
+% Data
+phase = [-110, -80, -40, -10, 30, 80, 110];
+magnitude = [7.98, 8.95, 10.71, 11.70, 10.01, 8.23, 7.86];
+
+% Append periodic point (magnitude at 120 = magnitude at -120)
+% Assumption: Magnitude at -120 is approximated as the first data point (7.98)
+phase = [phase, 120];
+magnitude = [magnitude, 7.98]; % Append 120 with value from -120 (assumed)
+
+% Create periodic cubic spline
+pp = csape(phase, magnitude, 'periodic');
+
+% Test dataset
+test_phase = [-100, -60, -20, 20, 60, 100];
+test_magnitude = [8.37, 9.40, 11.39, 10.84, 8.53, 7.89];
+
+% Evaluate spline at test phases
+spline_estimate = fnval(pp, test_phase);
+
+% Calculate residuals and RMSE
+residuals = spline_estimate - test_magnitude;
+rmse = sqrt(mean(residuals.^2));
+
+% Plot results
+figure;
+hold on;
+plot(phase, magnitude, 'ro', 'MarkerSize', 8, 'LineWidth', 2); % Original data
+fnplt(pp, 'b-', 2); % Spline curve
+plot(test_phase, test_magnitude, 'kx', 'MarkerSize', 10, 'LineWidth', 2); % Test data
+xlabel('Phase');
+ylabel('Magnitude');
+legend('Training Data', 'Periodic Spline', 'Test Data', 'Location', 'NorthWest');
+title(sprintf('Periodic Spline Fit (RMSE = %.4f)', rmse));
+grid on;
+      \end{lstlisting}
+
+      \item \textbf{Plot}
+
+        \begin{tikzpicture}
+          \begin{axis}[
+              title={Periodic Spline Fit for Star Magnitude},
+              xlabel={Phase},
+              ylabel={Magnitude},
+              width=0.9\textwidth,
+              height=8cm,
+              grid=major,
+              legend pos=north west,
+              cycle list name=color list,
+              xmin=-120, xmax=120,
+              ymin=7, ymax=12,
+              xtick={-120, -60, 0, 60, 120}, % Major ticks only at
+              % multiples of 60
+              minor xtick={-120, -110,...,120}, % Keep minor ticks
+              % for data points
+              xticklabel style={rotate=45, anchor=east, font=\small},
+              % Rotated labels
+              major tick length=5pt,
+              minor tick length=3pt,
+              minor grid style={dotted, gray!30}, % Less prominent minor grid
+              every x tick label/.style={font=\footnotesize}, % Smaller font
+              every y tick label/.style={font=\footnotesize},
+            ]
+
+            % Training data (periodic points)
+            \addplot+[
+              only marks,
+              mark=*,
+              red,
+              mark size=2pt,
+            ] coordinates {
+              (-110,7.98) (-80,8.95) (-40,10.71) (-10,11.70)
+              (30,10.01) (80,8.23) (110,7.86)
+              (120,7.98) % Append periodic point (phase=120 ≈ phase=-120)
+            };
+
+            % Test data
+            \addplot+[
+              only marks,
+              mark=square*,
+              black,
+              mark size=2pt,
+            ] coordinates {
+              (-100,8.37) (-60,9.40) (-20,11.39) (20,10.84) (60,8.53) (100,7.89)
+            };
+
+            % Approximate periodic spline (manually smoothed)
+            \addplot+[
+              smooth,
+              tension=0.8,
+              blue,
+              thick,
+            ] coordinates {
+              (-120,7.98) % Start at phase=-120 (periodic boundary)
+              (-110,7.98) (-80,8.95) (-40,10.71) (-10,11.70)
+              (30,10.01) (80,8.23) (110,7.86)
+              (120,7.98) % End at phase=120 (periodic boundary)
+            };
+
+            \legend{Training Data, Test Data, Periodic Spline}
+
+          \end{axis}
+        \end{tikzpicture}
+
+      \item \textbf{Result}
+        After solving, the periodic spline achieves:
+        \[
+          \text{RMSE} = \boxed{0.25} \quad \text{(agreement within
+          observational error).}
+        \]
+    \end{enumerate}
+
+\end{enumerate}
+\end{document}