From: rbk5287@usl.edu (Kearfott Ralph B)
Subject: Update -- Interval Analysis conference
Organization: University of Southwestern Louisiana
Date: Sun, 23 Aug 1992 22:44:49 GMT

Since our last posting, we have received 57 abstracts from 16 countries
for our conference on Numerical Analysis with Automatic Result
Verification, in the areas of Linear Systems, Applications, Arithmetic,
Computing Ranges, Nonlinear Systems, Global Optimization, Automatic
Differentiation, ODE's, PDE's, and others.  Additionally, Vladik
Kreinovich has been assembling abstracts for an associated workshop on
Interval Methods in Artificial Intelligence, to be held as an integral
part of the conference.

We have decided to extend the deadline for abstracts (originally August
first), to allow wider participation.  In particular, we encourage more
participation from researchers in the western hemisphere.  This meeting
will be a chance for colleagues throughout the world to exchange ideas
and views.

The announcement is appended.

Baker Kearfott
rbk@usl.edu (Internet)

P. S. From my last USENET posting, a reader requested an elementary
explanation of interval arithmetic.  I have appended an elementary
explanation, with references, in AMS-TeX, below the conference
announcement.  For those who cannot process AMS-TeX, the references may
still be legible in their unprocessed form.  Please feel free to contact
me if you require further assistance from me.

An excellent reference which has appeared since I wrote the appended
class notes is:

E. Hansen, GLOBAL OPTIMIZATION USING INTERVAL ANALYSIS, Marcel Dekker,
New York, 1992.

========================================================================
========================================================================


                            CALL FOR PAPERS

                           REGISTRATION FORMS

                                  and

                          FURTHER INFORMATION

                                  for

                     An International Conference on

                        NUMERICAL ANALYSIS WITH
                     AUTOMATIC RESULT VERIFICATION

                Mathematics, Applications, and Software

                   February 25 through March 1, 1993
                          Lafayette, Louisiana


General Information
------- -----------

Interval analysis is applicable in scientific computations in which
reliability or automatic verification, or mathematical rigor in
computational results is desirable.

This conference has the following goals.

  *  To provide an accessible forum for researchers in the field to
     exchange the most recent results in interval computations.

  *  To further delineate the role of interval computations in practical
     (applied and industrial) problems, and to identify tasks which must
     be completed to facilitate its optimal use in such settings.

  *  To highlight the role of interval mathematics in more purely
     academic pursuits, such as automatic theorem proving.

  *  To stimulate interest and creative research in the field.

Theory, software, computational results, etc. will be presented.

Topics covered include, but are not limited to

  *  Arithmetic
  *  Programming languages and general software tools
  *  Nonlinear systems of equations
  *  Nonlinear optimization
  *  Quadrature
  *  Ordinary differential equations
  *  Partial differential equations
  *  Sensitivity analysis
  *  Linear algebra and linear operators
  *  Industrial and scientific applications

We plan to publish a refereed proceedings.


Scientific Committee
---------- ---------

  *  G. Alefeld (University of Karlsruhe)
  *  G. Corliss (Marquette University)
  *  B. Kearfott (University of Southwestern Louisiana)
  *  U. Kulisch (University of Karlsruhe)
  *  H. Stetter (Technical University of Vienna)


Local Arrangements Committee
----- ------------ ---------

  *  W. Andrepont     *  R. Sidman
  *  B. Kearfott      *  R. Waggoner
  *  L. Roeling       *  T. West


Some of the Highlighted Speakers
---- -- --- ----------- --------

The following is a partial list of highlighted speakers (in plenary
sessions).

  *  Prof. Fernando Alvarado (Wisconsin)
  *  Dr. Eldon Hansen (Los Altos, California)
  *  Prof. Arnold Neumaier (Universitaet Freiburg)
  *  Priv.-Doz. Dr. Michael Plum (Universitaet Koeln)
  *  Prof. Dr. Siegfried Rump (Universitaet Hamburg)

Additional highlighted speakers will be chosen based on the abstracts.
For this reason, we are urging participants to submit their abstracts
early.


Call for papers
---- --- ------

Papers and additional speakers are welcome.  Contributors should send a
one page abstract to

                Interval Methods Conference
                C/O R. Baker Kearfott
                Department of Mathematics
                University of Southwestern Louisiana
                U.S.L. Box 4-1010
                Lafayette, LA 70504-1010

                Office phone: (318) 231-5270
                Home phone:   (318) 981-9744
                email:  confmath@usl.edu  (Internet)

We are planning facilities for demonstration of software, and will have
a poster session if there is sufficient interest.  Please state your
needs if you wish to use such a format.

The abstract may be sent via electronic mail if it is in some version of
TeX or in an ASCII format.

Abstracts are due by November 1, 1992.


Conference Proceedings
---------- -----------

Papers for the conference proceedings should be prepared by the time of
the conference or shortly thereafter.  Details will be forthcoming.


Workshop on Interval Methods in Artificial Intelligence
-------- -- -------- ------- -- ---------- ------------

A workshop on Interval Methods in Artificial Intelligence will be held
during the conference.  The organizing committee for this set of talks
includes, but is not limited to

  *  Bart Kosko (University of Southern California)
  *  Vladik Kreinovich (University of Texas at El Paso)
  *  Raymond Ng (University of Maryland)
  *  V. S. Subrahmanian (University of Maryland)
  *  Patrick Suppes (Stanford University)
  *  Lotfi Zadeh (University of California at Berkeley)

Abstracts and papers may be submitted specifically to this workshop.
For more information on the workshop, contact:

                Vladik Kreinovich
                Computer Science Department
                University of Texas at El Paso

                Office phone: (915) 747-5470
                Fax: (915) 747-5616
                email: vladik@cs.ep.utexas.edu


Conference Location
---------- --------

The conference will be in Lafayette, Louisiana, and will be partially
supported by the University of Southwestern Louisiana.

Lafayette, a town of approximately 100,000 with no heavy industry, is
the site of the University of Southwestern Louisiana, the second largest
public university in Louisiana, serving 16,000 students.  The
surrounding region a rich subtropical agricultural delta of the
Mississippi River and home of the Cajuns, a French-speaking population,
descendants of the uprooted Acadians of Canada.

The conference begins two days after Mardi Gras. traditional (Cajun)
Mardi Gras celebrations from the seventeenth century take place near
Lafayette during this time.  Also, weeks of elaborate Mardi Gras
(carnival) celebrations occur in New Orleans, which is 2.5 hours by car
from Lafayette.

Outdoor activities include boating and fishing, with nearby state parks,
botanical gardens, and the Atchafalaya swamp.  There should be floral
blooms during the conference.  There are two operating replicas of
traditional Cajun villages near Lafayette.

The average high temperature in late February is around 68 degrees
Fahrenheit and the average low is around 45 degrees Fahrenheit, but with
substantial variation.


Social Events
------ ------

Friday afternoon, conference participants will sample Cajun food,
dancing, and history at Vermilionville, a living museum.  Saturday
evening, there will be a crawfish boil with a Cajun band.  These events
are included in the registration fee.

Additional activities, such as boat tours of the Atchafalaya swamp,
visits to botanical gardens, and visits to renowned local restaurants,
may also be arranged.  Participants also have the option of arriving
early to take part in Mardi Gras festivities.  These occur both around
Lafayette and in New Orleans on and before February 23 (two days before
the conference).


Accommodations
--------------

The conference will be held at the Lafayette Holidome (Holiday Inn
Central), a full-service hotel with various amenities.  Seventy-five
rooms have been reserved, at the rate of $53.00 US for single occupancy,
and a rate of $56.00 US total per room for double, triple, or quadruple
occupancy, not including a 10.5% lodging tax.  Reservations are to be
made directly with the hotel.  This can be done by returning the
enclosed hotel registration card.  Information may be obtained toll free
within the United States by calling 1-(800) 465-4329 (1-(800) HOLIDAY).
The hotel may also be called at (318) 233-6815.

The Holidome has full dining facilities, and there is a selection of
alternate eating establishments within walking distance.  There are
other popular restaurants in and around Lafayette.

Please contact Baker Kearfott if further information or alternate
arrangements are required.


Registration Form
------------ ----

To register, send the following form, along with payment, to:

                Interval Methods Conference
                C/O R. Baker Kearfott
                Department of Mathematics
                University of Southwestern Louisiana
                U.S.L. Box 4-1010
                Lafayette, LA 70504-1010

                Office phone: (318) 231-5270
                Home phone:   (318) 981-9744
                email:  confmath@usl.edu  (Internet)

---------------------------------------------------------------------

         REGISTRATION FOR THE 1993 INTERVAL METHODS CONFERENCE


             Name: ____________________________________________

          Address: ____________________________________________

                   ____________________________________________

                   ____________________________________________

                   ____________________________________________

             City: ____________________________________________

State, Province, or Country: __________________________________

         Zip or Postal Code: __________________________________

                  Telephone: __________________________________

              Fax: __________________________________ (optional)

            email: __________________________________ (optional)

____  Registration fee enclosed

____  I do not wish to register now, but please send more information.

Registration fees will be:

Before the conference -- $ 80 US
    At the conference -- $105 US

The registration fees will include the two social events mentioned
above, as well as coffee and snacks and miscellaneous expenses
associated with the meeting.

For pre-registration, make a check or money order payable in U.S. funds
to R. Baker Kearfott and send to the above address by October 1, 1992.
=========================================================================
=========================================================================


\magnification=\magstep1
%ACHTUNG!!! This paper uses AMS-Tex in with features of the Plain
%format which are consistent with it.
%
%Uncomment the following lines if you need to input the plain and
%AMS-TeX formats from provided TeX input files.
%\input plain
%\input amstex

% This set of AMS-TeX macros is appropriate for papers dealing with
% the interval solution of nonlinear systems of equations.
\def\sclvec #1 #2{(#1_1,#1_2,\dots,#1_#2)}
\def\ivlvec#1#2{(\bold#1_1,\bold#1_2,\dots,\bold#1-$2)}
\def\ivl #1{{\bold #1}}
\def\cspace{\Cal C}
\def\cpx #1{\tilde #1}
\def\cpxivl #1{{\tilde{\bold #1}}}

% This set of macros is appropriate for most papers.
\def \cf{cf.}
\def \eg{eg.}
\def \ibid{(ibid.)}

% This set of macros is appropriate for this paper.
\define\ftext{\foldedtext\foldedwidth{3in}}

\nopagenumbers
\centerline{\bf How Interval Arithmetic Works}
\medskip
Refer to [1] or [6] for a thorough
introduction to interval mathematics.  Further references
appear in [5], and a list of about 2000 papers on the
subject appears in [3] and [4].
The recent proceedings [7] contains assessments of the
role of interval mathematics in scientific computation.
Here, we will give an elementary explanation of some of
the most important concepts.

        We will denote interval quantities throughout with
boldface.

        Interval arithmetic is based on defining the four
elementary arithmetic operations on intervals.  Let
$\ivl a = [a_l,a_u]$
and $\ivl b = [b_l,b_u]$ be intervals.
Then, if $\text{\it op} \in \{+,-,*,/\}$, we define
$$ \ivl a \text{\ \it op \ }  \ivl b
= \left\{ x \text{\ \it op \ } y \mid x \in \ivl a \text{\ and\ }
y \in \ivl b \right\}.
\tag{2.1}$$
For example,
$ \ivl a + \ivl b = [ a_l + b_l, a_u + b_u ]$.
In fact, all four operations can be defined in terms of addition,
subtraction, multiplication, and division of the endpoints of the
intervals, although multiplication and division may require
comparison of several results.  The result of these operations is
an interval except when we compute $\ivl a / \ivl b
\text{\ and \ } 0 \in \ivl b$ . (See \eg\  [6],
pp. 66-68 for a discussion of the latter case.)

        A large part of interval mathematics' power lies in the
ability to compute {\it inclusion monotonic interval extensions} of
functions.  If f is a continuous function of a real variable,
then an inclusion monotonic interval extension $\ivl f$ is defined to be
a function from the set of intervals to the set of intervals,
such that, if $\ivl x$ is an interval in the domain of $\ivl f$,
$$\left\{ f(x) \mid x \in \ivl x \right\} \subset \ivl f(\ivl x)$$
and such that
$$ \ivl x \subset \ivl y \implies \ivl f(\ivl x) \subset
\ivl f(\ivl y).$$

        Inclusion monotonic interval extensions of a polynomial
may be obtained by simply replacing the dependent variable by an
interval and by replacing the additions and multiplications by
the corresponding interval operations.  For example, if
$p(x) = x^2 - 4$, then $\ivl p([1,2])$ may be defined by
$$\ivl p([1,2]) = ([1,2])^2 - 4  =  ([1,4]) - [4,4] = [-3,0].$$

The result of an elementary
interval operation is precisely the range of values that the
usual result attains as we let the operands range over the two
intervals.  However, the value of an interval extension of a
function is not precisely the range of the function over its
interval operand, but only contains this range, and different
(mathematically equivalent) forms of the function give rise
to different interval extensions.  For example, if we write $p$ above
as $p(x)=(x - 2)(x + 2)$, then the corresponding interval extension
gives
$$\align
\ivl p([1,2]) &=  ([1,2] - 2) ([1,2] + 2) =  [-1,0] [3,4] \\
                 &=  [-4,0],
\endalign$$
which is not as good as the previous extension.

        We may use the mean value theorem or Taylor's theorem
with remainder formula to obtain interval extensions of
transcendental functions.  For example, suppose $\ivl x$ is an interval
and $a \in \ivl x$.  Then, for any $y \in \ivl x$, we have
$$\sin(y) = \sin(a) + (y - a) \cos(a) - (y - a)^2/2 \sin(c)$$
for some $c$ between $a$ and $y$.  If $a$ and $y$ are both within a range
where the sine function is non-negative, then we obtain
$$\sin(y) \in  \sin(a) + (\ivl x -a) \cos(a) -
{(\ivl x - a)^2\over 2}.$$
The right side of this relationship gives the value of an
interval extension of $\sin(x)$, albeit a somewhat crude one.

See [9] for more techniques of
producing interval extensions.

        Mathematically rigorous interval extensions can be
computed in finite precision arithmetic via the use of {\it directed
roundings}.  If $x$ and $y$ are machine-representable numbers and
{\it op} is one of the four elementary operations $+$, $-$, $*$, or $/$,
then, $x$ {\it op} $y$ is not normally representable in the machine's
memory.  In interval arithmetic with directed rounding, if
$\ivl x \text{\ \it op\ } \ivl y = [c,d]$,
then we always round the computed value for $c$ down to a machine
number less than the actual value of $c$, and
and we always similarly round
the value for $d$ up.  To be completely rigorous, we also first apply
directed rounding to the initial data while storing it.

If interval arithmetic with directed rounding is used to
compute an interval extension $\ivl f$ of $f$, if
$[c,d] = \ivl f([a,b])$,
and $[c,d]$ does not contain zero, then this is a rigorous proof
(regardless of the machine wordlength, etc.) that there is no
root of $f$ in $[a,b]$.  This concept is also valid if $\ivl F$
is an interval vector-valued function of an interval vector $\ivl X$,
and complex interval arithmetic can also be defined.  These
facts should enable us to computationally but rigorously check
the stability of systems.  (See Section 4 below.)

If available, the language
Fortran-SC is a convenient way of programming interval arithmetic
computations.  This precompiler is available on IBM mainframe
equipment, and requires the ACRITH subroutine package;  see
[2] or [10].  There is also Pascal-SC for personal computers;
see [8].  For a discussion of other programming
languages and packages for interval arithmetic, see [13].
\bigskip
\smallskip


\noindent{\bf References}
\medskip

\item{[1]} G. ALEFELD and J. HERZBERGER,
{\sl Introduction to Interval Computations\/},
Academic Press,
New York, etc.,
1983.
\bigskip
\item{[2]}
J. H. BLEHER, S. M. RUMP, U. KULISCH, M. METZGER, and W. WALTER,
``Fortran-SC -- A Study of a Fortran Extension
for Engineering Scientific Computations with Access to
ACRITH",
{\sl Computing\/},
v. 39, 1987, pp. 93--110.
\bigskip
\item{[3]}
J. GARLOFF,
``Interval Mathematics: A Bibliography",
preprint,
Institut f\"ur Angewandte Mathema\-tik der Universit\"at Frei\-burg,
{\sl Frei\-burg\-er Inter\-vall-Berichte\/},
v. 85, 1985, pp. 1--222.
\bigskip
\item{[4]}
J. GARLOFF,
``Bibliography on Interval Mathematics, Continuation"
pre\-print,
Institut f\"ur An\-gewand\-te Mathema\-tik der Universit\"at Frei\-burg,
{\sl Frei\-burg\-er Intervall-Berichte\/},
v. 87, 1987, pp. 1--50.
\bigskip
\item{[5]}
R. B. KEARFOTT,
``Interval Arithmetic Techniques in the
        Computational Solution of Nonlinear Systems of Equations:
        Introduction, Examples, and Comparisons",
to appear in the
proceedings of the 1988 AMS-SIAM Summer Seminar in Applied
        Mathematics,
Colorado State University, July 18-29, 1988.
\bigskip
\item{[6]}
R. E. MOORE,
{\sl Methods and Applications of Interval Analysis\/},
SIAM, Philadelphia, 1979.
\bigskip
\item{[7]}
R. E. MOORE, ed.,
{\sl Reliability in Computing\/},
Academic Press, New York, etc., 1988.
\bigskip
\item{[8]}
L. B. RALL,
``An Introduction to the Scientific Computing Language Pascal-SC",
{\sl Comput. Math. Appl.\/},
v. 14,
1987,
pp. 53--69.
\bigskip
\item{[9]}
H. RATSCHEK and J. G. ROKNE,
{\sl Computer Methods for the Range of Functions\/},
Horwood,
Chichester, England,
1984.
\bigskip
\item{[10]}
W. WALTER and M. METZGER,
``Fortran-SC, A Fortran
         Extension for Engineering/Scientific Computation with
        Access to ACRITH",
in {\sl Reliability in Computing\/},
R. E. Moore, ed.,
Academic Press, New York, etc.,
1988.

\bye
Newsgroups: sci.math