\usepackage{hyperref}
\usepackage{graphicx}
\usepackage{amsmath}
+\usepackage{amssymb}
%opening
\title{Literature Review}
immediately, as in PostScript, or stored in another file, such as with \TeX or \LaTeX, which emit a \texttt{DVI} file. Most other
forms of document use a \emph{Document Object Model}, being a list or tree of objects to be rendered. \texttt{DVI}, \texttt{PDF},
\texttt{HTML}\footnote{Some of these formats --- most notably \texttt{HTML} --- implement a scripting lanugage such as JavaScript,
-which permit the DOM to be modified while the document is being viewed.} and SVG. Of these, only \texttt{HTML} and \TeX typically
+which permit the DOM to be modified while the document is being viewed.} and SVG\cite{svg2011-1.1}. Of these, only \texttt{HTML} and \TeX typically
store documents in pre-layout stages, whereas even turing complete document formats such as PostScript typically encode documents
which already have their elements placed.
and are match how cameras, printers and monitors work. However, bitmap devices do not handle zooming beyond their
``native'' resolution --- the resolution where one document pixel maps to one display pixel ---, exhibiting an artefact
called pixelation where the pixel structure becomes evident. Attempts to use interpolation to hide this effect are
-never entirely successful, and sharp edges, such as those found in text and diagrams, are particularly effected.
+never entirely successful, and sharp edges, such as those found in text and diagrams, are particularly affected.
\begin{figure}[h]
\centering \includegraphics[width=0.8\linewidth]{figures/vectorraster_example}
renderer by nVidia\cite{kilgard2012gpu} as an OpenGL extension\cite{kilgard300programming}.
-\section{Floating-Point Precision}
+\section{Numeric formats}
On modern computer architectures, there are two basic number formats supported:
fixed-width integers and \emph{floating-point} numbers. Typically, computers
on objects far from the origin.
IEEE floating-point has some interesting features as well, including values for negative zero,
-positive and negative infinity and the ``Not a Number'' (NaN) value. Indeed, with these values,
+positive and negative infinity, the ``Not a Number'' (NaN) value and \emph{denormal} values, which
+trade precision for range when dealing with very small numbers. Indeed, with these values,
IEEE 754 floating-point equality does not form an equivalence relation, which can cause issues
when not considered carefully.\cite{goldberg1991whatevery}
-Arb. precision exists
+There also exist formats for storing numbers with arbitrary precising and/or range.
+Some programming languages support ``big integer''\cite{java_bigint} types which can
+represent any integer that can fit in the system's memory. Similarly, there are
+arbitrary-precision floating-point data types\cite{java_bigdecimal}\cite{boost_multiprecision}
+which can represent any number of the form
+\begin{equation}
+ \frac{n}{2^d} \; \; \; \; n,d \in \mathbb{Z} % This spacing is horrible, and I should be ashamed.
+\end{equation}
+These types are typically built from several native data types such as integers and floats,
+paired with custom routines implementing arithmetic primitives.\cite{priest1991algorithms}
+These, therefore, are likely slower than the native types they are built on.
+
+While traditionally, GPUs have supported some approximation of IEEE 754's 32-bit floats,
+modern graphics processors also support 16-bit\cite{nv_half_float} and 64-bit\cite{arb_gpu_shader_fp64}
+IEEE floats.
Higher precision numeric types can be implemented or used on the GPU, but are
slow.
\cite{emmart2010high}
address = {New York, NY, USA},
keywords = {NaN, denormalized number, exception, floating-point, floating-point standard, gradual underflow, guard digit, overflow, relative error, rounding error, rounding mode, ulp, underflow},
}
+
+@misc{arb_gpu_shader_fp64,
+ title={{ARB\_gpu\_shader\_fp64}},
+ author={Brown, Pat and Lichtenbelt, Barthold and Licea-Kane, Bill and Merry, Bruce and Dodd, Chris and Werness, Eric and Sellers, Graham and Roth, Greg and Bolz, Jeff and Haemel, Nick and Boudier, Pierre and Daniell, Piers},
+ year={2010},
+ journal={OpenGL Extension},
+ publisher={Kronos Group},
+ howpublished={\url{http://www.opengl.org/registry/specs/ARB/gpu_shader_fp64.txt}}
+}
+
+@misc{nv_half_float,
+ title={{NV\_half\_float}},
+ author={Brown, Pat},
+ year={2002},
+ journal={OpenGL Extension},
+ publisher={NVIDIA Corporation},
+ howpublished={\url{http://www.opengl.org/registry/specs/NV/half_float.txt}}
+}
+
@inproceedings{emmart2010high,
title={High precision integer multiplication with a graphics processing unit},
author={Emmart, Niall and Weems, Charles},
howpublished = {\url{http://www.boost.org/doc/libs/1_53_0/libs/multiprecision/doc/html/boost_multiprecision/}}
}
+@misc{java_bigint,
+ author = {Oracle Corporation},
+ title = {java.math.{BigInteger}},
+ booktitle = {Java Platform 6 {SE}},
+ urldate = {19-05-2014},
+ howpublished = {\url{http://docs.oracle.com/javase/6/docs/api/java/math/BigInteger.html}}
+}
+
+@misc{java_bigdecimal,
+ author = {Oracle Corporation},
+ title = {java.math.{BigDecimal}},
+ booktitle = {Java Platform 7 {SE}},
+ howpublished = {\url{http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html}},
+ urldate = {19-05-2014}
+}
+
% A CMOS Floating Point Unit
@MISC{kelley1997acmos,
author = {Michael J. Kelley and Matthew A. Postiff and Advisor Richard and B. Brown},
git.ucc.asn.au / ipdf / documents.git / commitdiff
UCC git Repository :: git.ucc.asn.au