243 lines
6.9 KiB
TeX
243 lines
6.9 KiB
TeX
% Created 2023-09-25 Mon 09:52
|
|
% Intended LaTeX compiler: pdflatex
|
|
\documentclass[11pt]{article}
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage{graphicx}
|
|
\usepackage{longtable}
|
|
\usepackage{wrapfig}
|
|
\usepackage{rotating}
|
|
\usepackage[normalem]{ulem}
|
|
\usepackage{amsmath}
|
|
\usepackage{amssymb}
|
|
\usepackage{capt-of}
|
|
\usepackage{hyperref}
|
|
\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:orga203815}
|
|
Computing \(\epsilon_{\text{mac}}\) for single precision numbers
|
|
|
|
\begin{verbatim}
|
|
(load "../cl/lizfcm.asd")
|
|
(ql:quickload :lizfcm)
|
|
|
|
(let ((domain-values (lizfcm.approx:compute-maceps 1.0
|
|
(lambda (x) x))))
|
|
(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 & 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]
|
|
1.0 & 5.9604645e-08 & 0.0\\[0pt]
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
\(\epsilon_{\text{mac}}\) \(\approx\) 5.9604 \(\cdot\) 10\textsuperscript{-8}
|
|
|
|
\section{Question Two}
|
|
\label{sec:orgdd79be1}
|
|
Computing \(\epsilon_{\text{mac}}\) for double precision numbers:
|
|
|
|
\begin{verbatim}
|
|
(let ((domain-values (lizfcm.approx:compute-maceps 1.0d0
|
|
(lambda (x) x))))
|
|
(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]
|
|
1.0d0 & 1.1102230246251565d-16 & 0.0d0\\[0pt]
|
|
\end{tabular}
|
|
\end{center}
|
|
|
|
Thus, \(\epsilon_{\text{mac}}\) \(\approx\) 1.1102 \(\cdot\) 10\textsuperscript{-16}
|
|
|
|
\section{Question Three - |v|\textsubscript{2}}
|
|
\label{sec:org04608d9}
|
|
\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:orgfb57f3a}
|
|
\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:org7bbdf04}
|
|
\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:orge36996c}
|
|
\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:orgd1577f0}
|
|
\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:org2661676}
|
|
\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} |