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\documentclass[11pt]{article}
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\usepackage{graphicx}
\usepackage{longtable}
\usepackage{wrapfig}
\usepackage{rotating}
\usepackage[normalem]{ulem}
\usepackage{amsmath}
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\notindent \notag  \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
\author{Elizabeth Hunt}
\date{\today}
\title{HW 02}
\hypersetup{
 pdfauthor={Elizabeth Hunt},
 pdftitle={HW 02},
 pdfkeywords={},
 pdfsubject={},
 pdfcreator={Emacs 28.2 (Org mode 9.7-pre)}, 
 pdflang={English}}
\begin{document}

\maketitle
\setlength\parindent{0pt}

\section{Question One}
\label{sec:orga21c813}
Computing \(\epsilon_{\text{mac}}\) for single precision numbers

\begin{verbatim}
(load "../lizfcm.asd")
(ql:quickload :lizfcm)

(let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
                                                   1.0
                                                   1.0)))
  (lizfcm.utils:table (:headers '("a" "h" "err")
                       :domain-order (a h err)
                       :domain-values domain-values)))
\end{verbatim}

(with many rows truncated)

\begin{center}
\begin{tabular}{rrr}
a & h & err\\[0pt]
1.0 & 0.5 & 0.5\\[0pt]
1.0 & 0.25 & 0.25\\[0pt]
1.0 & 0.125 & 0.125\\[0pt]
1.0 & 0.0625 & 0.0625\\[0pt]
1.0 & 0.03125 & 0.03125\\[0pt]
1.0 & 1.9073486e-06 & 1.9073486e-06\\[0pt]
1.0 & 9.536743e-07 & 9.536743e-07\\[0pt]
1.0 & 4.7683716e-07 & 4.7683716e-07\\[0pt]
1.0 & 2.3841858e-07 & 2.3841858e-07\\[0pt]
1.0 & 1.1920929e-07 & 1.1920929e-07\\[0pt]
\end{tabular}
\end{center}

\(\epsilon_{\text{mac single precision}}\) \(\approx\) 1.192(10\textsuperscript{-7})

\section{Question Two}
\label{sec:org06c4a23}
Computing \(\epsilon_{\text{mac}}\) for double precision numbers:

\begin{verbatim}
(let ((domain-values (lizfcm.approx:compute-maceps (lambda (x) x)
                                                   1.0d0
                                                   1.0d0)))
  (lizfcm.utils:table (:headers '("a" "h" "err")
                       :domain-order (a h err)
                       :domain-values domain-values)))
\end{verbatim}

(with many rows truncated)
\begin{center}
\begin{tabular}{rrr}
a & h & err\\[0pt]
1.0d0 & 0.5d0 & 0.5d0\\[0pt]
1.0d0 & 0.25d0 & 0.25d0\\[0pt]
1.0d0 & 0.125d0 & 0.125d0\\[0pt]
1.0d0 & 0.0625d0 & 0.0625d0\\[0pt]
1.0d0 & 0.03125d0 & 0.03125d0\\[0pt]
1.0d0 & 0.015625d0 & 0.015625d0\\[0pt]
1.0d0 & 0.0078125d0 & 0.0078125d0\\[0pt]
1.0d0 & 0.00390625d0 & 0.00390625d0\\[0pt]
1.0d0 & 0.001953125d0 & 0.001953125d0\\[0pt]
1.0d0 & 7.105427357601002d-15 & 7.105427357601002d-15\\[0pt]
1.0d0 & 3.552713678800501d-15 & 3.552713678800501d-15\\[0pt]
1.0d0 & 1.7763568394002505d-15 & 1.7763568394002505d-15\\[0pt]
1.0d0 & 8.881784197001252d-16 & 8.881784197001252d-16\\[0pt]
1.0d0 & 4.440892098500626d-16 & 4.440892098500626d-16\\[0pt]
1.0d0 & 2.220446049250313d-16 & 2.220446049250313d-16\\[0pt]
\end{tabular}
\end{center}

Thus, \(\epsilon_{\text{mac double precision}}\) \(\approx\) 2.220 \(\cdot\) 10\textsuperscript{-16}

\section{Question Three - |v|\textsubscript{2}}
\label{sec:orgf181ba7}
\begin{verbatim}
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
      (2-norm (lizfcm.vector:p-norm 2)))
  (lizfcm.utils:table (:headers '("x" "y" "2norm")
                       :domain-order (x y)
                       :domain-values vs)
    (funcall 2-norm (list x y))))
\end{verbatim}


\begin{center}
\begin{tabular}{rrr}
x & y & 2norm\\[0pt]
1 & 1 & 1.4142135\\[0pt]
2 & 3 & 3.6055512\\[0pt]
4 & 5 & 6.4031243\\[0pt]
-1 & 2 & 2.236068\\[0pt]
\end{tabular}
\end{center}

\section{Question Four - |v|\textsubscript{1}}
\label{sec:org2196087}
\begin{verbatim}
(let ((vs '((1 1) (2 3) (4 5) (-1 2)))
      (1-norm (lizfcm.vector:p-norm 1)))
  (lizfcm.utils:table (:headers '("x" "y" "1norm")
                       :domain-order (x y)
                       :domain-values vs)
    (funcall 1-norm (list x y))))
\end{verbatim}


\begin{center}
\begin{tabular}{rrr}
x & y & 1norm\\[0pt]
1 & 1 & 2\\[0pt]
2 & 3 & 5\\[0pt]
4 & 5 & 9\\[0pt]
-1 & 2 & 3\\[0pt]
\end{tabular}
\end{center}

\section{Question Five - |v|\textsubscript{\(\infty\)}}
\label{sec:org11b8894}
\begin{verbatim}
(let ((vs '((1 1) (2 3) (4 5) (-1 2))))
  (lizfcm.utils:table (:headers '("x" "y" "max-norm")
                       :domain-order (x y)
                       :domain-values vs)
    (lizfcm.vector:max-norm (list x y))))
\end{verbatim}


\begin{center}
\begin{tabular}{rrr}
x & y & infty-norm\\[0pt]
1 & 1 & 1\\[0pt]
2 & 3 & 3\\[0pt]
4 & 5 & 5\\[0pt]
-1 & 2 & 2\\[0pt]
\end{tabular}
\end{center}

\section{Question Six - ||v - u|| via |v|\textsubscript{2}}
\label{sec:orga2324b2}
\begin{verbatim}
(let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
       (vs2 '((7 9) (2 2) (8 -1) (4 4)))
       (2-norm (lizfcm.vector:p-norm 2)))
  (lizfcm.utils:table (:headers '("v1" "v2" "2-norm-d")
                       :domain-order (v1 v2)
                       :domain-values (mapcar (lambda (v1 v2)
                                                (list v1 v2))
                                              vs
                                              vs2))
    (lizfcm.vector:distance v1 v2 2-norm)))
\end{verbatim}


\begin{center}
\begin{tabular}{llr}
v1 & v2 & 2-norm\\[0pt]
(1 1) & (7 9) & 10.0\\[0pt]
(2 3) & (2 2) & 1.0\\[0pt]
(4 5) & (8 -1) & 7.2111025\\[0pt]
(-1 2) & (4 4) & 5.3851647\\[0pt]
\end{tabular}
\end{center}

\section{Question Seven - ||v - u|| via |v|\textsubscript{1}}
\label{sec:org388fbc7}
\begin{verbatim}
(let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
       (vs2 '((7 9) (2 2) (8 -1) (4 4)))
       (1-norm (lizfcm.vector:p-norm 1)))
  (lizfcm.utils:table (:headers '("v1" "v2" "1-norm-d")
                       :domain-order (v1 v2)
                       :domain-values (mapcar (lambda (v1 v2)
                                                (list v1 v2))
                                              vs
                                              vs2))
    (lizfcm.vector:distance v1 v2 1-norm)))
\end{verbatim}


\begin{center}
\begin{tabular}{llr}
v1 & v2 & 1-norm-d\\[0pt]
(1 1) & (7 9) & 14\\[0pt]
(2 3) & (2 2) & 1\\[0pt]
(4 5) & (8 -1) & 10\\[0pt]
(-1 2) & (4 4) & 7\\[0pt]
\end{tabular}
\end{center}

\section{Question Eight - ||v - u|| via |v|\textsubscript{\(\infty\)}}
\label{sec:org6e77f76}
\begin{verbatim}
(let* ((vs '((1 1) (2 3) (4 5) (-1 2)))
       (vs2 '((7 9) (2 2) (8 -1) (4 4))))
  (lizfcm.utils:table (:headers '("v1" "v2" "max-norm-d")
                       :domain-order (v1 v2)
                       :domain-values (mapcar (lambda (v1 v2)
                                                (list v1 v2))
                                              vs
                                              vs2))
    (lizfcm.vector:distance v1 v2 'lizfcm.vector:max-norm)))
\end{verbatim}

\begin{center}
\begin{tabular}{llr}
v1 & v2 & max-norm-d\\[0pt]
(1 1) & (7 9) & -6\\[0pt]
(2 3) & (2 2) & 1\\[0pt]
(4 5) & (8 -1) & 6\\[0pt]
(-1 2) & (4 4) & -2\\[0pt]
\end{tabular}
\end{center}
\end{document}