diff --git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet
index 4a1cd0c..e63cdd5 100644
--- a/books/bookvol10.1.pamphlet
+++ b/books/bookvol10.1.pamphlet
@@ -3,6 +3,260 @@
 \input{bookheader.tex}
 \mainmatter
 \setcounter{chapter}{0} % Chapter 1
+\chapter{Interval Arithmetic}
+Lambov \cite{Lambov06} defines a set of useful formulas for 
+computing intervals using the IEEE-754 floating-point standard.
+
+The first thing to note is that IEEE floating point defaults to 
+{\bf round-to-nearest}. However, Lambov sets the rounding mode
+to {\bf round to $-\infty$}. Computing lower bounds directly uses
+the hardware floating point operations but computing upper bounds
+he uses the identity
+\[\Delta(x) = -\nabla(-x)\]
+so that the upper bound of the pair of bounds is always negated.
+That is,
+\[ x = \left[\underline{x},\overline{x}\right] = 
+\left<\underline{x},-\overline{x}\right> \]
+
+Given that convention
+\begin{itemize}
+\item the sum of $x$ and $y$ is evaluated by
+\[\left<\nabla(\underline{x} + \underline{y}),
+  -\nabla(-\overline{x} - \overline{y})\right> \]
+\item changing the sign of an interval $x$ is achieved by swapping
+the two bounds, that is $\left<-\overline{x},\underline{x}\right>$
+\item joining two intervals (that is, finding an interval containing
+all numbers in both, or finding the minimum of the lower bounds
+and the maximum of the higher bounds) is performed as
+\[\left<min(\underline{x},\underline{y}),
+  -min((-\overline{x}),(-\overline{y}))\right>\]
+\end{itemize}
+
+Lambov defines operations which, under the given rounding condition,
+give the tightest bounds.
+
+\section{Addition}
+\[ x + y = \left[\underline{x}+\underline{y},\overline{x}+\overline{y}\right]
+\subseteq \left<\nabla(\underline{x}+\underline{y})
+-\nabla((-\overline{x})+(-\overline{y}))\right>\]
+The negated sign of the higher bound ensures the proper direction
+of the rounding.
+
+\section{Sign Change}
+\[ -x = \left[-\overline{x},-\underline{x}\right] =
+\left<-\overline{x},\underline{x}\right> \]
+This is a single swap of the two values. No rounding is performed.
+
+\section{Subtraction}
+\[ x-y = \left[\underline{x}-\overline{y},\overline{x}-\underline{y}\right]
+\subseteq \left<\nabla(\underline{x}+(-\overline{y})),
+-\nabla((-\overline{x})+\underline{y})\right>\]
+Subtraction is implemented as $x+(-y)$.
+
+\section{Multiplication}
+\[ xy = \left[min(\underline{x}\underline{y},
+                  \underline{x}\overline{y},
+                  \overline{x}\underline{y},
+                  \overline{x}\overline{y}),
+              max(\underline{x}\underline{y},
+                  \underline{x}\overline{y},
+                  \overline{x}\underline{y},
+                  \overline{x}\overline{y})\right]
+\]
+The rounding steps are part of the operation so all 8 multiplications
+are required. Lambov notes that since
+\[\Delta(\nabla(r)+\epsilon) \ge \Delta(r) \]
+for $\epsilon$ being the smallest representable positive number, one
+can do with 4 multiplications at the expense of some accuracy.
+
+In Lambov's case he makes the observation that
+\[
+xy=\left\{
+\begin{array}{l}
+\left[min(\underline{x}\underline{y},\overline{x}\underline{y}),
+      max(\overline{x}\overline{y},\underline{x}\overline{y})\right],
+      {\rm\ if\ }0 \le \underline{x} \le \overline{x}\\
+\left[min(\underline{x}\overline{y},\overline{x}\underline{y}),
+      max(\overline{x}\overline{y},\underline{x}\underline{y})\right],
+      {\rm\ if\ }\underline{x} < 0 \le \overline{x}\\
+\left[min(\underline{x}\overline{y},\overline{x}\overline{y}),
+      max(\overline{x}\underline{y},\underline{x}\underline{y})\right],
+      {\rm\ if\ }\underline{x} \le \overline{x} < 0
+\end{array}
+\right.
+\]
+from which he derives the formula actually used
+\[xy \subseteq \left<min(\nabla(a\underline{x}),\nabla(b(-\overline{x}))),
+-min(\nabla(c(-\overline{x})),\nabla(d\underline{x}))\right>
+\]
+where 
+\[
+\begin{array}{rcl}
+a & = & \left\{ 
+\begin{array}{rl}
+\underline{y} & {\rm if\ } 0 \le \underline{x}\\
+-(-\overline{y}) & {\rm otherwise}
+\end{array}\right.\\
+b & = & \left\{ 
+\begin{array}{rl}
+-\underline{y} & {\rm if\ } (-\overline{x}) \le 0\\
+(-\overline{y}) & {\rm otherwise}
+\end{array}\right.\\
+c & = & \left\{
+\begin{array}{rl}
+-(-\overline{y}) & {\rm if\ } (-\overline{x}) \le 0\\
+\underline{y} & {\rm otherwise}
+\end{array}\right.\\
+d & = & \left\{
+\begin{array}{rl}
+(-\overline{y}) & {\rm if\ } 0 \le \underline{x}\\
+-\underline{y} & {\rm otherwise}
+\end{array}\right.
+\end{array}
+\]
+which computes the rounded results of the original multiplication
+formula but achieves better performance.
+
+\section{Multiplication by a positive number}
+If one of the numbers is known to be positive (e.g. a constant) then 
+\[ {\rm if\ }x > 0 {\rm\ then\ } xy \equiv
+\left[min(\underline{x}\underline{y},\overline{x}\underline{y}),
+       max(\overline{x}\overline{y},\underline{x}\overline{y})
+\right]
+\]
+This formula is faster than the general multiplication formula.
+
+\section{Multiplication of Two Positive Numbers}
+
+If both multiples are positive simply change the sign of the 
+higher bound on one of the arguments prior to multiplication.
+If one of the numbers is a constant this can be arranged to
+skip the sign change.
+
+\section{Division}
+Division is an expensive operation.
+\[\frac{x}{y} =
+\left[
+min\left(\frac{\underline{x}}{\underline{y}},
+         \frac{\underline{y}}{\overline{y}},
+         \frac{\overline{x}}{\underline{y}},
+         \frac{\overline{x}}{\overline{y}}\right),
+max\left(\frac{\underline{x}}{\underline{y}},
+         \frac{\underline{x}}{\overline{y}},
+         \frac{\overline{x}}{\underline{y}},
+         \frac{\overline{x}}{\overline{y}}\right)
+\right]\]
+which is undefined if $0 \in y$. To speed up the computation Lambov
+uses the identity
+\[ \frac{x}{y} = x\frac{1}{y} \]
+
+Lambov does a similar analysis to improve the overall efficiency.
+\[\frac{x}{y} = \left\{
+\begin{array}{rl}
+\left[min(\frac{\underline{x}}{\underline{y}},
+          \frac{\underline{x}}{\overline{y}}),
+      max(\frac{\overline{x}}{\underline{y}},
+          \frac{\overline{x}}{\overline{y}})\right], &
+{\rm if\ }0 < \underline{y} \le \overline{y}\\
+{\rm exception} & 
+{\rm if\ }\underline{y} \le 0 \le \overline{y}\\
+\left[min(\frac{\overline{x}}{\underline{y}},
+          \frac{\overline{x}}{\overline{y}}),
+      max(\frac{\underline{x}}{\underline{y}},
+          \frac{\underline{x}}{\overline{y}})\right], &
+{\rm if\ }\underline{y} \le \overline{y} \le 0
+\end{array}
+\right.
+\]
+
+The formula he uses is
+\[\frac{x}{y} \subseteq
+\left<min\left(\nabla\left(\frac{a}{\underline{y}}\right),
+               \nabla\left(\frac{-a}{(-\overline{y})}\right)\right),
+     -min\left(\nabla\left(\frac{-b}{(-\overline{y})}\right),
+               \nabla\left(\frac{b}{\underline{y}}\right)\right)
+\right>
+\]
+where
+\[
+\begin{array}{rcl}
+a & = & \left\{
+\begin{array}{rl}
+\underline{x} & {\rm if\ }(-\overline{y}) \le 0\\
+-(-\overline{x}) & {\rm otherwise}
+\end{array}\right.\\
+b & = & \left\{
+\begin{array}{rl}
+(-\overline{x}) & {\rm if\ }0 \le \underline{y}\\
+-\underline{x} & {\rm otherwise}
+\end{array}\right.
+\end{array}
+\]
+
+\section{Reciprocal}
+\[\frac{1}{x} = \left[\frac{1}{\overline{x}},\frac{1}{\underline{x}}\right]
+\subseteq 
+\left<\nabla\left(\frac{-1}{(-\overline{x})}\right),
+      \nabla\left(\frac{-1}{\underline{x}}\right)
+\right>
+\]
+which is undefined if $0 \in x$. Lambov implements this by checking for
+zero, followed by division of $-1$ by the argument and swapping the
+two components.   
+
+\section{Absolute Value}
+
+\[\abs{x} = 
+\left[max(\underline{x},-\overline{x},0),
+      max(-\underline{x},\overline{x})\right] =
+\left<max(0,\underline{x},(-\overline{x})),
+     -min(\underline{x},(-\overline{x}))\right>
+\]
+
+\section{Square}
+\[x^2 = \abs{x}\abs{x}\]
+using multiplication by positive numbers, mentioned above.
+
+\section{Square Root}
+
+\[\sqrt{x} = \left[\sqrt{\underline{x}},\sqrt{\overline{x}}\right]\]
+which is defined if $0 \le \underline{x}$
+
+Lambov notes that this formula has a rounding issue. He notes that
+since
+\[\Delta(r) \le -\nabla(-\epsilon - \nabla(r))\]
+he uses the formula
+\[\sqrt{x} \subseteq
+\left\{
+\begin{array}{l}
+\left<\nabla\left(\sqrt{\underline{x}}\right),
+     -\nabla\left(\sqrt{-(-\overline{x})}\right)\right>,
+{\rm\ if\ }\nabla\left(\nabla\left(\sqrt{-(-\overline{x})}\right)\right)^2
+= -(-\overline{x})\\
+\left<\nabla\left(\sqrt{\underline{x}}\right),
+      \nabla\left(\nabla\left(-\epsilon-\sqrt{-(-\overline{x})}\right)
+      \right)\right>,
+{\rm\ otherwise}
+\end{array}
+\right.
+\]
+where $\epsilon$ is the smallest representable positive number.
+
+The first branch of this formula is only satisfied if the result of
+$\sqrt{-(-\overline{x})}$ is exactly representable, in which case
+\[\nabla\left(\sqrt{-(-\overline{x})}\right) =
+  \nabla\left(\sqrt{-(-\overline{x})}\right)
+\]
+otherwise the second branch of the formula adjusts the high bound
+to the next representable number. If tight bounds are not required
+the second branch is always sufficient.
+
+If the argument is entirely negative, the implementation will raise
+an exception. If it contains a negative part, the implementation will
+crop it to only its non-negative part to allow that computations
+such as $\sqrt{0}$ ca be carried out in exact real arithmetic.
+
+
 \chapter{Integration}
 An {\sl elementary function}
 \index{elementary function}
@@ -8355,6 +8609,10 @@ PhD thesis, MIT, Computer Science, 1984
 M. van Hoeij. ``An algorithm for computing an integral
 basis in an algebraic function field'' {\sl J. Symbolic Computation}
 18(4):353-364, October 1994
+\bibitem [Lambov 06]{Lambov06} Lambov, Branimir\\
+``Interval Arithmetic Using SSE-2\\
+in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0
+(2006) pp102-113
 \bibitem[Wa03]{Wa03}
 Watt, Stephen, ``Aldor'', \verb|www.aldor.org|
 \bibitem[We71]{We71}
diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet
index 0be4854..2b24ac7 100644
--- a/books/bookvolbib.pamphlet
+++ b/books/bookvolbib.pamphlet
@@ -3194,5 +3194,12 @@ Greve, David A.; McClurg, Jedidiah R.\\
 ``Development of a Translator from LLVM to ACL2''\\
 \verb|arxiv.org/pdf/1406.1566|
 
+\subsection{Interval Arithmetic} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\bibitem [Lambov 06]{Lambov06} Lambov, Branimir\\
+``Interval Arithmetic Using SSE-2\\
+in Lecture Notes in Computer Science, Springer ISBN 978-3-540-85520-0
+(2006) pp102-113
+
 \end{thebibliography}
 \end{document}
diff --git a/changelog b/changelog
index fa1c2dc..263703c 100644
--- a/changelog
+++ b/changelog
@@ -1,3 +1,6 @@
+20140617 tpd src/axiom-website/patches.html 20140617.01.tpd.patch
+20140617 tpd books/bookvol10.1 add interval arithmetic
+20140617 tpd books/bookvolbib add interval arithmetic
 20140616 tpd src/axiom-website/patches.html 20140616.01.tpd.patch
 20140616 tpd src/input/Makefile add polygamma
 20140616 tpd src/input/polygamma.input add a polygamma example
diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html
index 263a145..b76c633 100644
--- a/src/axiom-website/patches.html
+++ b/src/axiom-website/patches.html
@@ -4474,6 +4474,8 @@ buglist bug 7258: acosh(0.0) invalid argument to acosh
 buglist wish 1011: sum(1/(k+a), k=1..n) by Gosper's method
 <a href="patches/20140616.01.tpd.patch">20140616.01.tpd.patch</a>
 src/input/polygamma.input add a polygamma example
+<a href="patches/20140617.01.tpd.patch">20140617.01.tpd.patch</a>
+books/bookvol10.1, bookvolbib add interval arithmetic
  </body>
 </html>