\parskip 8.2pt % sets spacing between paragraphs
%\renewcommand{\baselinestretch}{1.5} % Uncomment for 1.5 spacing between lines
\parindent 0pt % sets leading space for paragraphs
-\linespread{0.8}
+\linespread{1}
\newcommand{\vect}[1]{\boldsymbol{#1}} % Draw a vector
\definecolor{darkred}{rgb}{0.75,0,0}
\definecolor{darkblue}{rgb}{0,0,0.75}
\definecolor{pink}{rgb}{1,0.5,0.5}
-\lstset{language=Java}
+\lstset{language=XML}
\lstset{backgroundcolor=\color{darkgray}}
-\lstset{numbers=left, numberstyle=\tiny, stepnumber=1, numbersep=5pt}
-\lstset{keywordstyle=\color{darkred}\bfseries}
-\lstset{commentstyle=\color{darkblue}}
+%\lstset{numbers=left, numberstyle=\tiny, stepnumber=1, numbersep=5pt}
+%\lstset{keywordstyle=\color{darkred}\bfseries}
+%\lstset{commentstyle=\color{darkblue}}
%\lstset{stringsyle=\color{red}}
-\lstset{showstringspaces=false}
-\lstset{basicstyle=\small}
+%\lstset{showstringspaces=false}
+%\lstset{basicstyle=\small}
\newcommand{\shell}[1]{\texttt{#1}}
\newcommand{\code}[1]{\texttt{#1}}
I'm still not quite sure when this is useful. I haven't been able to find an example for $x$ and $y$ where $x + y \neq \text{Fast2Sum}(x, y)$.
+\section{Handbook of Floating-Point Arithmetic \cite{HFP}}
+This book is amazingly useful and pretty much says everything there is to know about Floating Point numbers.
+It is much easier to read than Goldberg or Priest's papers.
+
+I'm going to start working through it and compile their test programs.
+
+\subsection{A sequence that seems to converge to a wrong limit - pgs 9-10, \cite{HFP}}
+
+\begin{align*}
+ u_n &= \left\{ \begin{array}{c} u_0 = 2 \\ u_1 = -4 \\ u_n = 111 - \frac{1130}{u_{n-1}} + \frac{3000}{u_{n-1}u_{n-2}}\end{array}\right.
+\end{align*}
+
+The limit of the series should be $6$ but when calculated with IEEE floats it is actually $100$
+The authors show that the limit is actually $100$ for different starting values, and the error in floating point arithmetic causes the series to go to that limit instead.
+
+\begin{figure}[H]
+ \centering
+ \includegraphics[width=0.8\textwidth]{figures/handbook1-1.pdf}
+ \caption{Output of Program 1.1 from \emph{Handbook of Floating-Point Arithmetic}\cite{HFP} for various IEEE types}
+ \label{HFP-1-1}
+\end{figure}
+
+\subsection{Mr Gullible and the Chaotic Bank Society pgs 10-11 \cite{HFP}}
+
+This is an example of a sequence involving $e$. Since $e$ cannot be represented exactly with FP, even though the sequence should go to $0$ for $a_0 = e - 1$, the representation of $a_0 \neq e - 1$ so the sequence goes to $\pm \infty$.
+
+To eliminate these types of problems we'd need an \emph{exact} representation of all real numbers.
+For \emph{any} FP representation, regardless of precision (a finite number of digits) there will be numbers that can't be represented exactly hence you could find a similar sequence that would explode.
+
+IE: The more precise the representation, the slower things go wrong, but they still go wrong, {\bf even with errorless operations}.
+
+
+\subsection{Rump's example pg 12 \cite {HFP}}
+
+This is an example where the calculation of a function $f(a,b)$ is not only totally wrong, it gives completely different results depending on the CPU. Despite the CPU conforming to IEEE.
+
+\section{Scalable Vector Graphics (SVG) 1.1 (Second Edition) \cite{svg2011-1.1}}
+
+Scalable Vector Graphics (SVG) is a XML language for describing two dimensional graphics. This document is \url{http://www.w3.org/TR/2011/REC-SVG11-20110816/}, the latest version of the standard at the time of writing.
+
+\subsubsection{Three types of object}
+\begin{enumerate}
+ \item Vector graphics shapes (paths)
+ \item Images (embedded bitmaps)
+ \item Text
+\end{enumerate}
+
+\subsubsection{Rendering Model and Precision}
+
+SVG uses the ``painter's model''. Paint is applied to regions of the page in an order, covering the paint below it according to rules for alpha blending.
+
+``Implementations of SVG are expected to behave as though they implement a rendering (or imaging) model cor-
+responding to the one described in this chapter. A real implementation is not required to implement the model in
+this way, but the result on any device supported by the implementation shall match that described by this model.''
+
+SVG uses {\bf single precision floats}. Unlike PDF and PostScript, the standard specifies this as a {\bf minimum} range from \verb/-3.4e+38F/ to \verb/+3.4e+38F/
+
+``It is recommended that higher precision floating point storage and computation be performed on operations
+such as coordinate system transformations to provide the best possible precision and to prevent round-off errors.''
+
+There is also a ``High Quality SVG Viewers'' requirement to use at least {\bf double} precision floats.
+
+\section{OpenVG Specification 1.1 \cite{rice2008openvg}}
+``OpenVG is an application programming interface (API) for hardware-accelerated two-
+dimensional vector and raster graphics developed under the auspices of the Khronos
+Group \url{www.khronos.org}.''
+
+Specifically, provides the same drawing functionality required by a high-performance SVG document viewer (SVG Tiny 1.2)
+TODO: Look at that $\neq$ SVG 1.1
+
+It is designed to be similar to OpenGL.
+
+\subsubsection{Precision}
+``All floating-point values are specified in standard IEEE 754 format. However,
+implementations may clamp extremely large or small values to a restricted
+range, and internal processing may be performed with lesser precision. At least
+16 bits of mantissa, 6 bits of exponent, and a sign bit must be present, allowing
+values from $\pm 2^{-30}$ to $\pm2^{31}$ to be represented with a fractional precision of at least 1
+in $2^{16}$.''
+
+IEEE but with a non standard number of bits.
+
+Presumably the decreased precision is for increased efficiency the standard states that example applications are for ebooks.
+
+
+\section{Document Object Model --- pugixml 1.4 manual \cite{pugixmlDOM}}
+
+Pugixml is a C++ library for parsing XML documents\cite{kapoulkine2014pugixml}. XML is based on the Document Object Model (DOM); this is explained pretty well by the pugixml manual.
+
+The document is the root node of the tree. Each child node has a type. These may
+\begin{itemize}
+ \item Have attributes
+ \item Have child nodes of their own
+ \item Contain data
+\end{itemize}
+
+In the case of HTML/SVG an XML parser within the browser/viewer creates the DOM tree from the XML (plain text) and then interprets that tree to produce the objects that will be rendered.
+
+Example of XML $\to$ DOM tree given at\cite{pugixmlDOM}.
+Example of XML parsing using pugixml is in \shell{code/src/tests/xml.cpp}
+
+
+\begin{figure}[H]
+ \centering
+\begin{lstlisting}[language=XML,basicstyle=\ttfamily]
+<?xml version="1.0"?>
+<mesh name="mesh_root">
+ <!-- here is a mesh node -->
+ some text
+ <![CDATA[someothertext]]>
+ some more text
+ <node attr1="value1" attr2="value2" />
+ <node attr1="value2">
+ <innernode/>
+ </node>
+</mesh>
+<?include somedata?>
+\end{lstlisting}
+
+ \includegraphics[width=0.6\textwidth]{references/pugixmlDOM-dom_tree.png}
+ \caption{Tree representation of the above listing \cite{pugixmlDOM}}
+\end{figure}
\chapter{General Notes}
+\section{The DOM Model}
+
+A document is modelled as a tree. The root of the tree is the document. This has nodes of varying types. Some nodes may have children, attributes, and data.
+
+\section{Floating-Point Numbers\cite{HFP,goldberg1991whatevery,goldberg1992thedesign,priest1991algorithms}}
+
+A set of FP numbers is characterised by:
+\begin{enumerate}
+ \item Radix (base) $\beta \geq 2$
+ \item Precision %$p \req 2$ ``number of sig digits''
+ \item Two ``extremal`` exponents $e_min < 0 < e_max$ (generally, don't have to have the $0$ in there)
+\end{enumerate}
+
+Numbers are represented by {\bf integers}: $(M, e)$ such that $x = M \times \beta^{e - p + 1}$
+
+Require: $|M| \leq \beta^{p}-1$ and $e_min \leq e \leq e_max$.
+
+Representations are not unique; set of equivelant representations is a cohort.
+
+$\beta^{e-p+1}$ is the quantum, $e-p+1$ is the quantum exponent.
+
+Alternate represetnation: $(s, m, e)$ such that $x = (-1)^s \times m \times \beta^{e}$
+$m$ is the significand, mantissa, or fractional part. Depending on what you read.
+
+
+
+
+
\section{Rounding Errors}
+
They happen. There is ULP and I don't mean a political party.
TODO: Probably say something more insightful. Other than "here is a graph that shows errors and we blame rounding".
-Results of \verb/calculatepi/
+\subsection{Results of calculatepi}
+
+We can calculate pi by numerically solving the integral:
+\begin{align*}
+ \int_0^1 \left(\frac{4}{1+x^2}\right) dx &= \pi
+\end{align*}
+
+Results with Simpson Method:
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{figures/calculatepi.pdf}
git.ucc.asn.au / ipdf / documents.git / blobdiff
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