[Octave-cvsupdate] SF.net SVN: octave:[5494] trunk/octave-forge/main

[<sim...@us...>]

Revision: 5494
          http://octave.svn.sourceforge.net/octave/?rev=5494&view=rev
Author:   simonepernice
Date:     2008-12-16 20:15:12 +0000 (Tue, 16 Dec 2008)

Log Message:
-----------
First version of SIMP.

Added Paths:
-----------
    trunk/octave-forge/main/simp/
    trunk/octave-forge/main/simp/COPYING
    trunk/octave-forge/main/simp/DESCRIPTION
    trunk/octave-forge/main/simp/INDEX
    trunk/octave-forge/main/simp/doc/
    trunk/octave-forge/main/simp/doc/simp.lyx
    trunk/octave-forge/main/simp/doc/simp.pdf
    trunk/octave-forge/main/simp/inst/
    trunk/octave-forge/main/simp/inst/__add1Int__.m
    trunk/octave-forge/main/simp/inst/__addSingleInt__.m
    trunk/octave-forge/main/simp/inst/__checkInputInt__.m
    trunk/octave-forge/main/simp/inst/__checkInt__.m
    trunk/octave-forge/main/simp/inst/__checkMonotonicity__.m
    trunk/octave-forge/main/simp/inst/__div1Int__.m
    trunk/octave-forge/main/simp/inst/__divSingleInt__.m
    trunk/octave-forge/main/simp/inst/__engSingleFormInt__.m
    trunk/octave-forge/main/simp/inst/__incrementalRatio__.m
    trunk/octave-forge/main/simp/inst/__intSingleToTol100__.m
    trunk/octave-forge/main/simp/inst/__intSingleToTol__.m
    trunk/octave-forge/main/simp/inst/__intSingleToVal__.m
    trunk/octave-forge/main/simp/inst/__intersectSingleInt__.m
    trunk/octave-forge/main/simp/inst/__intervalFunctionMultiInt__.m
    trunk/octave-forge/main/simp/inst/__intervalFunctionNNSingleInt__.m
    trunk/octave-forge/main/simp/inst/__intervalFunctionNSingleInt__.m
    trunk/octave-forge/main/simp/inst/__invertSingleInt__.m
    trunk/octave-forge/main/simp/inst/__mul1Int__.m
    trunk/octave-forge/main/simp/inst/__mulSingleInt__.m
    trunk/octave-forge/main/simp/inst/__negateSingleInt__.m
    trunk/octave-forge/main/simp/inst/__powerSingleInt__.m
    trunk/octave-forge/main/simp/inst/__sqrSingleInt__.m
    trunk/octave-forge/main/simp/inst/__sqrtSingleInt__.m
    trunk/octave-forge/main/simp/inst/__sub1Int__.m
    trunk/octave-forge/main/simp/inst/__subSingleInt__.m
    trunk/octave-forge/main/simp/inst/addInt.m
    trunk/octave-forge/main/simp/inst/dB10ToLinInt.m
    trunk/octave-forge/main/simp/inst/dB20ToLinInt.m
    trunk/octave-forge/main/simp/inst/divInt.m
    trunk/octave-forge/main/simp/inst/engFormInt.m
    trunk/octave-forge/main/simp/inst/errorBarInt.m
    trunk/octave-forge/main/simp/inst/findRootInt.m
    trunk/octave-forge/main/simp/inst/functionInt.m
    trunk/octave-forge/main/simp/inst/intToTol.m
    trunk/octave-forge/main/simp/inst/intToTol100.m
    trunk/octave-forge/main/simp/inst/intToVal.m
    trunk/octave-forge/main/simp/inst/invertAddInt.m
    trunk/octave-forge/main/simp/inst/invertInt.m
    trunk/octave-forge/main/simp/inst/linSpaceTol100Int.m
    trunk/octave-forge/main/simp/inst/linToDB10Int.m
    trunk/octave-forge/main/simp/inst/linToDB20Int.m
    trunk/octave-forge/main/simp/inst/logSpaceTol100Int.m
    trunk/octave-forge/main/simp/inst/monotonicFunctionInt.m
    trunk/octave-forge/main/simp/inst/mulInt.m
    trunk/octave-forge/main/simp/inst/negateInt.m
    trunk/octave-forge/main/simp/inst/plotInt.m
    trunk/octave-forge/main/simp/inst/powerInt.m
    trunk/octave-forge/main/simp/inst/sqrAddInt.m
    trunk/octave-forge/main/simp/inst/sqrInt.m
    trunk/octave-forge/main/simp/inst/sqrtInt.m
    trunk/octave-forge/main/simp/inst/subInt.m
    trunk/octave-forge/main/simp/inst/valtol100ToInt.m

Added: trunk/octave-forge/main/simp/COPYING
===================================================================
--- trunk/octave-forge/main/simp/COPYING	                        (rev 0)
+++ trunk/octave-forge/main/simp/COPYING	2008-12-16 20:15:12 UTC (rev 5494)
@@ -0,0 +1,674 @@
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+    but WITHOUT ANY WARRANTY; without even the implied warranty of
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Added: trunk/octave-forge/main/simp/DESCRIPTION
===================================================================
--- trunk/octave-forge/main/simp/DESCRIPTION	                        (rev 0)
+++ trunk/octave-forge/main/simp/DESCRIPTION	2008-12-16 20:15:12 UTC (rev 5494)
@@ -0,0 +1,12 @@
+Name:  			simp.tar
+Version: 		1.0.0
+Date: 			13/12/2008
+Author: 		Simone Pernice
+Maintainer: 		Simone Pernice
+Title: 			Single Interval Mathematics Package for Octave
+Description: 		This package define the basic operations on intervals. It is useful when some values for a computation are incerte.
+Categories: 		Interval Mathematics Octave
+Url: 			simonepernice.freehostia.com
+Autoload: 		yes
+License: 		GPL version 3
+

Added: trunk/octave-forge/main/simp/INDEX
===================================================================
--- trunk/octave-forge/main/simp/INDEX	                        (rev 0)
+++ trunk/octave-forge/main/simp/INDEX	2008-12-16 20:15:12 UTC (rev 5494)
@@ -0,0 +1,27 @@
+intervals >> Basic Arithmetic
+ addInt
+ negateInt
+ subInt
+ mulInt
+ invertInt
+ invertAddInt
+ divInt
+ powerInt
+ sqrInt
+ sqrtInt
+ addSqrInt
+ engFormInt
+ dB10ToLinInt
+ linToDB10Int
+ dB20ToLinInt
+ linToDB20Int
+ valtol100ToInt
+ intToTol
+ intToTol100
+ intToVal
+ linSpaceTol100Int
+ logSpaceTol100Int
+ monotonicFunctionInt
+ functionInt
+ plotInt
+ errorBarInt

Added: trunk/octave-forge/main/simp/doc/simp.lyx
===================================================================
--- trunk/octave-forge/main/simp/doc/simp.lyx	                        (rev 0)
+++ trunk/octave-forge/main/simp/doc/simp.lyx	2008-12-16 20:15:12 UTC (rev 5494)
@@ -0,0 +1,1137 @@
+#LyX 1.5.6 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass article
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author "" 
+\author "" 
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Single Interval Mathematics Package for Octave
+\end_layout
+
+\begin_layout Standard
+
+\newpage
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset LatexCommand tableofcontents
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\newpage
+
+\end_layout
+
+\begin_layout Section
+Author and version
+\end_layout
+
+\begin_layout Standard
+I am Simone Pernice and I am the writer of the SIMP pachage.
+ If you find any bug or you want to add features, you can contact me at
+ pe...@li....
+ You can find further details on my web site: simonepernice.freehostia.com
+\end_layout
+
+\begin_layout Standard
+List of versions of this document:
+\end_layout
+
+\begin_layout Itemize
+Version 1.0, Turin 14th December 2008, initial draft
+\end_layout
+
+\begin_layout Section
+Preface
+\end_layout
+
+\begin_layout Standard
+I am an engineer, I worked on hardware and software fields.
+ To design an electrical circuit is quite complex because the characteristics
+ of the components involved have huge tolerance some about 5%, while others
+ 80%.
+ The engineer has the target to have the circuit working in the worst case
+ of the components and of the environment.
+ That involves to compute several time the same equation putting inside
+ the worst or best case values.
+ Some time ago I was doing the usual design when I though, what if I use
+ intervals instead of numbers: I may get the results in just one step! Then
+ I went on the Internet looking for Interval Mathematics: I discovered it
+ was invented in the '50s.
+ Eventually I looked for a open source calculator able to manage intervals
+ instead of Numbers.
+ Unfortunately I was not able to find anything.
+ There are just some libraries for C++ language and packages for proprietary
+ software.
+ I was thinking to develop a calculator on my own, but I was not that happy
+ because it would have required a lot of effort.
+ Then I discovered Octave.
+ After a loot at its tutorial I decided to write a package to work on intervals
+ in Octave.
+ It was the right choice, because it tooks just three days while to make
+ a new program from scratch would have taken weeks.
+ The package extends the basic function to single interval mathematics,
+ it is called SIMP.
+ Unfortunately it is not possible to overload Octave operators, therefore
+ new functions working on matrix were added for that purpose.
+\end_layout
+
+\begin_layout Standard
+\begin_inset LatexCommand prettyref
+reference "sec:Introduction"
+
+\end_inset
+
+ shows why numbers are not the best tool when computing real world measures.
+ 
+\begin_inset LatexCommand prettyref
+reference "sec:Intervals"
+
+\end_inset
+
+ explains what is an interval and how a function can be extended to work
+ on intervals.
+ 
+\begin_inset LatexCommand prettyref
+reference "sec:Issues-with-interval"
+
+\end_inset
+
+ explains some issues to be aware when making interval calculations, which
+ are not present in the number computation.
+ 
+\begin_inset LatexCommand prettyref
+reference "sec:Single-Interval-Mathematics"
+
+\end_inset
+
+ shows some examples to explain how the SIMP package can be used.
+ Eventually the 
+\begin_inset LatexCommand prettyref
+reference "sec:Functions"
+
+\end_inset
+
+ provides the detailed list of functions available.
+\end_layout
+
+\begin_layout Section
+Introduction
+\begin_inset LatexCommand label
+name "sec:Introduction"
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+Every day we need to compute the result of a lot of simple mathematical
+ equations.
+ For example the cost of the apples bought at the supermarket is given by
+ the apple cost per kilo times the number of kilos bought: 
+\begin_inset Formula $applePrice=appleCostPerKilo\times kilosOfAppleBought$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+When we need the result of those mathematical expressions, we put the values
+ on the right side of the equation and we got its result on the left side.
+ Well, we usually put wrong numbers on the right side and therefore no doubt
+ we get wrong results.
+ There are a lot of reason why we put wrong values, below some of them follows:
+ 
+\end_layout
+
+\begin_layout Enumerate
+Most of the values are measured, therefore they are known within a given
+ tolerance (looking for accuracy and precision on Wikipedia will provide
+ interesting information); 
+\end_layout
+
+\begin_layout Enumerate
+Some values have an infinite number of digits after the decimal point, like
+ for 
+\begin_inset Formula $\pi$
+\end_inset
+
+ (the ratio between a circumference and its diameter); 
+\end_layout
+
+\begin_layout Enumerate
+Some values change with time or samples or whatever, like the weight of
+ a person (which can change of 5% during a day) or the current gain of a
+ BJT (which can change of 50% from the samples of the same series); 
+\end_layout
+
+\begin_layout Enumerate
+Some value are estimation or guess: something like between a minimum and
+ a maximum.
+ 
+\end_layout
+
+\begin_layout Standard
+For example if a pipe brakes and you want to buy a new one you need its
+ diameter.
+ If you do not have a caliber, you may measure its circumference and divide
+ it by 
+\begin_inset Formula $\pi$
+\end_inset
+
+ (3.1415...):
+\begin_inset Formula $diameter=\frac{circumference}{\pi}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Here there are two errors: the circumference is knows with the tolerance
+ given by your meter, moreover 
+\begin_inset Formula $\pi$
+\end_inset
+
+ has an infinite number of digits while only few of them can be used in
+ the operation.
+ You may think the error is negligible, the result is enough accurate to
+ buy a new pipe in a hardware shop.
+ However the not infinite accuracy of those operation avoid the use of computers
+ as automatic theorem demonstration tools and so on...
+\end_layout
+
+\begin_layout Standard
+This kind of issue is quite common on engineer design.
+ What engineers do is to be sure their design will work in the worst case
+ or in most of the cases (usually more than 99.9%).
+ Here a simple example.
+ Let us say you want to repaint the walls of your living room completely
+ messed up by your children.
+ You need to compute how many paint cans you need to buy.
+ The equation is quite simple:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+paintCans=\frac{2\times\left(roomWidth+roomLength\right)\times roomHeight}{paintLitersPerCan\times paintEfficiency}\label{eq:paintCans}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+where paintEfficiency is how may square meter of surface can be painted
+ with a liter of paint.
+ The problem here is that usually we do not have a meter long enough to
+ measure the room width and length, it is much simpler to count the number
+ of steps to go through it (1 step is about 1m, let us say from 0.9 to 1.1m).
+ Moreover the paint provider usually declare a paint efficiency range.
+ Let us put below the data: 
+\end_layout
+
+\begin_layout Itemize
+roomWidth = 6 
+\end_layout
+
+\begin_layout Itemize
+steps roomLength = 4 
+\end_layout
+
+\begin_layout Itemize
+steps roomHeight = 3 
+\end_layout
+
+\begin_layout Itemize
+meters paintEfficiency = from 0.7 to 1.3 square meters per liter (1 liter
+ per square meter in average) 
+\end_layout
+
+\begin_layout Itemize
+paintLitersPerCan = 40 
+\end_layout
+
+\begin_layout Standard
+To compute the average result just put average values in 
+\begin_inset LatexCommand eqref
+reference "eq:paintCans"
+
+\end_inset
+
+.
+ We get: 
+\begin_inset Formula $paintCans=\frac{2\times\left(6+4\right)\times3}{40\times1}=1.5$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+paint cans, which means two unless you are able to buy just half of a can.
+ 
+\end_layout
+
+\begin_layout Standard
+Are you satisfied with that result? I am not.
+ What if I have underestimated something? As every good engineer I would
+ check what would happen in the worst case, which means 1.1m step and efficiency
+ of just 0.8.
+ Again just substituting those values in 
+\begin_inset LatexCommand eqref
+reference "eq:paintCans"
+
+\end_inset
+
+, we get: 
+\begin_inset Formula $paintCans=\frac{2\times\left(6.6+4.4\right)\times3}{40\times0.7}=2.36$
+\end_inset
+
+.
+ That is really interesting: in the worst case I would miss 0.36 cans, it
+ makes sense to buy three cans to avoid to go back to the hardware shop
+ to buy one more (in the worst case).
+ 
+\end_layout
+
+\begin_layout Standard
+More happy with the result now? I am not completely satisfied, I am asking
+ myself: what if in the best case I need just 1 can? In that case probably
+ I need more accurate data because the result range would be too wide.
+ Eventually from 
+\begin_inset LatexCommand eqref
+reference "eq:paintCans"
+
+\end_inset
+
+ we get: 
+\begin_inset Formula $paintCans=\frac{2\times\left(5.4+3.6\right)\times3}{40\times1.3}=1.04$
+\end_inset
+
+.
+ Which means two cans.
+ I am satisfied, I have to buy at least two cans, but probably I may need
+ one more.
+ In the next paragraph you will see how to do all this stuff in one step
+ using Octave and SIMP.
+ 
+\end_layout
+
+\begin_layout Section
+Intervals
+\begin_inset LatexCommand label
+name "sec:Intervals"
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+As you can see in the example above a lot of computations are required to
+ get an idea of the result.
+ To get the worst (and sometime is required also best) case, you need also
+ to think carefully at the equation because some time you need to put the
+ higher value (for steps) while other times the smallest (for efficiency)
+ to get the worst case and vice versa for the best.
+ There is a much simpler way to work with that issue.
+ You can use intervals instead of number.
+ An interval is:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+[a1,a2]=\left\{ \forall x\in\mathbb{R\mid\mathit{a1}\leqslant\mathit{x}\leqslant\mathit{a2}}\right\} ;a1,a2\in\mathbb{R\cup\left\{ -\infty,+\infty\right\} ;\mathit{a1}\leqslant\mathit{a2};}\label{eq:interval}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In few words, [a, b] is the set of real numbers among a and b.
+ Please note that intervals suit perfectly in all the cases where numbers
+ do not fit (some were showed in the 
+\begin_inset LatexCommand prettyref
+reference "sec:Introduction"
+
+\end_inset
+
+).
+ Moreover if it is correct to use a number, it is possible to use a degenerate
+ interval like [a, a].
+ It is possible also to define functions that work on interval instead of
+ numbers:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+f\left([a1,a2],[b1,b2],...\right)=\left\{ \forall f(a,b,...)\in\mathbb{R}\mid\exists a\in[a1,a2],b\in[b1,b2],...\right\} \label{eq:functionOnInterval}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Please note, with that definition it is possible to extend every function
+ (like addition, multiplication, square, square root, ...) to work on intervals.
+ 
+\end_layout
+
+\begin_layout Standard
+Note also sometime a function may generate several intervals as result,
+ but in this package we will concentrate on single interval function.
+ I will give you an example: 
+\begin_inset Formula $[4,4]/[-2,2]=[-\infty,-2]\cup[2,+\infty]$
+\end_inset
+
+.
+ However the result is a double interval, for that reason if you try to
+ divide for a interval containing 0 in SIMP, you will get an error.
+\end_layout
+
+\begin_layout Standard
+Eventually note that the usually compute the interval result of a function
+ applied to intervals is computationally a long task.
+ In those case we are satisfied by a bigger interval containing the correct
+ interval.
+ However the target is always to get the smaller interval containing the
+ solution.
+ In the example above you would have:
+\end_layout
+
+\begin_layout Itemize
+roomWidth = [5.4, 6.6] meters 
+\end_layout
+
+\begin_layout Itemize
+roomLength = [3.6, 4.4] meters 
+\end_layout
+
+\begin_layout Itemize
+roomHeight = 3 = [3, 3] meters
+\end_layout
+
+\begin_layout Itemize
+paintEfficiency = [0.7, 1.3] square meters per liter 
+\end_layout
+
+\begin_layout Itemize
+paintLitersPerCan = 40 = [40, 40] liters per can
+\end_layout
+
+\begin_layout Standard
+Now you have just to compute the equation once with extended function to
+ get [1.04, 2.36] cans.
+ SIMP, the Octave package I developed, extends all the basic mathematical
+ functions and it provides functions to extend all other available Octave
+ functions.
+ 
+\end_layout
+
+\begin_layout Section
+Issues with interval computation
+\begin_inset LatexCommand label
+name "sec:Issues-with-interval"
+
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+There are few issues you need to know about interval computations.
+ 
+\end_layout
+
+\begin_layout Subsection
+Variables independence 
+\end_layout
+
+\begin_layout Standard
+All the variables you use to make a computation are independent from each
+ other, although they are the same instance of variable for you.
+ For example the perimeter of a rectangle (which was also used in a piece
+ of the equation 
+\begin_inset LatexCommand ref
+reference "eq:paintCans"
+
+\end_inset
+
+) can be written in several different ways.
+ If the dimensions of the rectangle are a end b we can write:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+perimeter=2(a+b)\label{eq:perimetersmall}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+perimeter=a+a+b+b\label{eq:perimeterbig}\end{equation}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In standard mathematics they are the same, while in interval mathematics
+ we have the interval 
+\begin_inset Formula $equation(\ref{eq:perimetersmall})\subseteqq equation(\ref{eq:perimeterbig})$
+\end_inset
+
+.
+ The reason is that every interval that appears in the equation is not dependent
+ from each other, therefore in equation 
+\begin_inset LatexCommand ref
+reference "eq:perimeterbig"
+
+\end_inset
+
+ is like having four different intervals.
+ What is important to remember is that: 
+\emph on
+Every variable should appear just once in the equation if possible in order
+ to get the smallest interval possible as result.
+\end_layout
+
+\begin_layout Subsection
+Hyper-cube Overlapping
+\end_layout
+
+\begin_layout Standard
+The result of a set of equation involving intervals is an Hyper-cube.
+ For example if we have just two equations in the unknowns x and y the result
+ will be a couple of intervals which draw a rectangle on the Cartesian plane.
+ However in general the solution will be smaller, for example just a segment.
+ In that case the rectangle will contain the solution.
+ There is no way to solve that, just remember the solution is usually a
+ super set of the actual one.
+\end_layout
+
+\begin_layout Section
+Introduction on how the package works 
+\begin_inset LatexCommand label
+name "sec:Single-Interval-Mathematics"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Installation
+\end_layout
+
+\begin_layout Standard
+First of all you have to install the SIMP package in Octave, so that Octave
+ will automatically load the new functions at every start.
+ To do that just run Octave from the directory where was downloaded the
+ package and execute the following command:
+\end_layout
+
+\begin_layout Itemize
+pkg install simp.tar.gz
+\end_layout
+
+\begin_layout Subsection
+Some example
+\end_layout
+
+\begin_layout Standard
+SIMP works on matrix, interpreted as intervals.
+ The basic interval is: [min, max], which in Octave is a row vector made
+ by two elements.
+ The added functions work on those matrix.
+ For example to add the intervals [5.4, 6.6] and [3.6, 4.4] it is possible to
+ write:
+\end_layout
+
+\begin_layout Itemize
+addInt ([5.4, 6.6], [3.6, 4.4])
+\end_layout
+
+\begin_layout Description
+> ans = [9, 11]
+\end_layout
+
+\begin_layout Standard
+addInt stands for add intervals.
+ It is possible to use scalar number instead of intervals.
+ Every scalar is computed like: 
+\begin_inset Formula $s=[s,s]$
+\end_inset
+
+.
+ Therefore the following computations give the same result:
+\end_layout
+
+\begin_layout Itemize
+addInt ([5.4, 6.6], 3.6)
+\end_layout
+
+\begin_layout Description
+> ans = [9, 10.2]
+\end_layout
+
+\begin_layout Itemize
+addInt ([5.4, 6.6], [3.6, 3.6])
+\end_layout
+
+\begin_layout Description
+> ans = [9, 10.2]
+\end_layout
+
+\begin_layout Standard
+Octave supports variable declaration, therefore it is possible to write:
+\end_layout
+
+\begin_layout Itemize
+roomWidth = [5.4, 6.6];
+\end_layout
+
+\begin_layout Itemize
+roomLength =[3.6, 4.4];
+\end_layout
+
+\begin_layout Itemize
+addInt (roomWidth, roomLength)
+\end_layout
+
+\begin_layout Description
+> ans = [9, 11]
+\end_layout
+
+\begin_layout Standard
+Now we know enough from Octave to compute the equation 
+\begin_inset LatexCommand ref
+reference "eq:paintCans"
+
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Itemize
+roomHeight = 3;
+\end_layout
+
+\begin_layout Itemize
+paintEfficiency = [0.7, 1.3]; 
+\end_layout
+
+\begin_layout Itemize
+paintLitersPerCan = 40; 
+\end_layout
+
+\begin_layout Itemize
+divInt(mulInt(2, addInt (roomWidth, roomLength), roomHeight), mulInt(paintLiters
+PerCan, paintEfficiency)) 
+\end_layout
+
+\begin_layout Description
+> ans = [1.0385, 2.3571]
+\end_layout
+
+\begin_layout Standard
+Sometime the intervals are expressed in terms of tolerance.
+ The electronic resistors are sold in several tolerances: 1%, 2%, 5%.
+ There is a function to easily make an interval from a tolerance.
+ For example a resistor of 10KOhms with 5% of tolerance can be converted
+ in an interval:
+\end_layout
+
+\begin_layout Itemize
+tollToInt (10000, 5)
+\end_layout
+
+\begin_layout Description
+> ans = [9500, 10500]
+\end_layout
+
+\begin_layout Standard
+If the tolerance is not symmetric, for example +5% and -10%:
+\end_layout
+
+\begin_layout Itemize
+tollToInt (10000, 5, -10)
+\end_layout
+
+\begin_layout Description
+> ans = [9000, 10500]
+\end_layout
+
+\begin_layout Standard
+All the computations are done among intervals, therefore if a result with
+ tolerance is required it is possible to get back the value and tolerance
+ of an interval in the hypothesis that the tolerance is symmetrical:
+\end_layout
+
+\begin_layout Itemize
+engFormInt([9500, 10500])
+\end_layout
+
+\begin_layout Description
+> 10K+-5%
+\end_layout
+
+\begin_layout Standard
+Eventually those functions work on vectors of interv...
 
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