Uncommitted notes on HFPA and symlink to Sam's Lit Review

[ipdf/documents.git] / LiteratureNotes.tex

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@@ -585,10 +585,65 @@ 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.
 
 
 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}
 
 \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}
 
 \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".
 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".


@@ -610,20 +665,6 @@ Results with Simpson Method:
 Tests with \verb/calculatepi/ show it's not quite as simple as just blindly replacing all your additions with Fast2Sum from Dekker\cite{dekker1971afloating}.
 ie: The graph looks exactly the same for single precision. \verb/calculatepi/ obviously also has multiplication ops in it which I didn't change. Will look at after sleep maybe.
 
 Tests with \verb/calculatepi/ show it's not quite as simple as just blindly replacing all your additions with Fast2Sum from Dekker\cite{dekker1971afloating}.
 ie: The graph looks exactly the same for single precision. \verb/calculatepi/ obviously also has multiplication ops in it which I didn't change. Will look at after sleep maybe.
 

-\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}
-\end{figure}

 
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