X-Git-Url: https://git.ucc.asn.au/?p=ipdf%2Fdocuments.git;a=blobdiff_plain;f=LiteratureNotes.tex;h=a9292dc31dc6eb45fcd034c6c7678fd14116a9bd;hp=2814e4b1e37c3b0cc86e27d61ef7f80ae2639d87;hb=f1069d4e2ac6473ff167bb7385f22b446a91b2dc;hpb=93cb10d1d571c39a1f7b39b88d1fba745f2e31a9

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@@ -578,17 +578,84 @@ There is a version for multiplication.
 
 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.
 
 \chapter{General Notes}
 
+\section{Floating-Point \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}