krj@csri.toronto.edu (Ken Jackson) (06/13/88)
NA Digest Monday, June 13, 1988 Volume 88 : Issue 23
Today's Editor: Cleve Moler
Today's Topics:
Conference on Iterative Methods for Large Linear Systems
First SIAG/LA Linear Algebra Prize
Source for DEROOT and ODERT
Winnipeg Numerical Analysis Meeting
NATO Institute on Orthogonal Polynomials
Contributions to Floating-Point Indoctrination
AMS Reviews in Partial Differential Equations
Two Tenured Positions at University of Umea
Minnesota IMA Summer Program on Signal Processing
Conference on Applications of the Connection Machine
Rational Approximation of Floating Point Numbers
-------------------------------------------------------
From: Bob Ward <ward@msr.epm.ornl.gov>
Date: Mon, 6 Jun 88 16:52:21 EDT
Subject: Conference on Iterative Methods for Large Linear Systems
Second Announcement and Registration Form
CONFERENCE ON ITERATIVE METHODS FOR LARGE LINEAR SYSTEMS
October 19, 20 and 21, 1988
Celebrating the 65th birthday of David M. Young, Jr.
OBJECTIVE: This conference will be dedicated to providing an overview of the
state of the art in the use of iterative methods for solving sparse linear
systems with an eye to contributions of the past, present, and future. The
emphasis will be placed upon identifying current and future research
directions in the mainstream of modern scientific computing. Recently, the
use of iterative methods for solving linear systems has experienced a
resurgence of activity as scientists attack extremely complicated three
dimensional problems using vector and parallel supercomputers. Many
research advances in the development of iterative methods for high-speed
computers over the past forty years will be reviewed as well as focusing on
current research.
INVITED SPEAKERS:
L. Adams (Washington University)
O. Axelsson (University of Nijmegen, Netherlands)
G. Birkhoff (Harvard University)
P. Concus (Lawrence Berkeley Laboratory)
L. Ehrlich (John Hopkins University)
H. Elman (University of Maryland)
G. Golub (Stanford University)
L. Hageman (Bettis Atomic Power Laboratory)
R. Lynch (Purdue University)
T. Manteuffel (Los Alamos National Laboratory)
J. Ortega (University of Virginia)
J. Rice (Purdue University)
P. Saylor (University of Illinois)
R. Varga (Kent State University)
W. Wachspress (University of Tennessee)
M. Wheeler (Rice University)
O. Widlund (Courant Institute)
D. Young (University of Texas)
ORGANIZATION: The host organization is the Center for Numerical Analysis
(CNA) of The University of Texas at Austin. The members of the local
organizing committee are D. Kincaid, L. Hayes, G. Carey and W. Cheney. The
conference is being co-sponsored by the Special Interest Groups for Linear
Algebra and Supercomputing of the Society for Industrial and Applied
Mathematics.
ACCOMMODATIONS: To make reservations call the Austin Marriott at the Capital
[(512) 478-1111 or (800) 228-9290] or mail the enclosed card for rooms at the
special conference rate of $55 for single or double rooms. All reservations
are handled on a first-come-first-served basis. Reservations must be received
by October 4, 1988. Reservations made after this date are subject to room
availability.
REGISTRATION: Conference advanced registration is $95 (SIAG/LA or SIAG/SC
member $85), if received by September 1, 1988. If paid after this date or
at the conference, the registration fee is $125. This fee includes morning
and afternoon coffee breaks, a reception, a banquet honoring Professor Young,
and other activities. The student registration fee is $15 which allows
admission to the technical sessions and coffee breaks only. To register,
detach and mail the registration form below. For additional information,
contact the CNA at the address below or at Tel: (512) 471-1242;
Arpanet: sheri@sally.utexas.edu; Bitnet: sheri@uta3081.
Mrs. Katy Burrell
Conference Secretary
Center for Numerical Analysis
RLM Bldg 13.150
University of Texas at Austin
Austin, Texas 78713-8510
.........................Detach and mail.............................
Name:
Organization:
Address:
Telephone:
Email:
Number of guest banquet tickets ($27.50 per person):
Amount Enclosed:
(Make checks payable to The University of Texas at Austin)
___ Yes, I plan to attend. Enclosed please find my registration fee of $_____
___ I am not ready to register yet, but send me additional information as it
becomes available.
------------------------------
From: Bob Ward <ward@msr.epm.ornl.gov>
Date: Wed, 8 Jun 88 10:13:39 EDT
Subject: First SIAG/LA Linear Algebra Prize
PRESENTATION OF THE FIRST
SIAG/LA LINEAR ALGEBRA PRIZE
The first SIAG/LA Linear Algebra Prize was presented to James
Demmel for his paper "On Condition Numers and the Distance to the
Nearest Ill-Posed Problem," Numer. Math 51 (1987) 251-289. The
Prize Plaque and Citation were presented to Demmel by SIAG/LA
Chair Bob Ward and Vice-Chair David Carlson at the Third SIAM
Conference on Applied Linear Algebra held in Madison, Wisconsin
May 23-26, 1988. After the ceremony, Demmel gave a stirring
half-hour account of his work in the area of the prize paper.
The winning paper was selected by a Prize Committee consisting of
Dave Carlson (chair), Ake Bjorck, Gene Golub, Roger Horn, and Tom
Kailath. Altogether there were 23 1986 and 1987 papers in
applicable linear algebra nominated. Preliminary judging by 3
subcommittees narrowed the field to 6 finalist papers. Demmel's
paper, together with
T. Ando, Roger Horn, and Charles Johnson, The singular values of
a Hadamard product: a basic inequality, Lin. Multilin. Alg.
21(1987) 345-366.
Thomas Coleman and Alex Pothen, The null space problem I.
Complexity, SIAM J. Alg. Disc. Meth. 7 (1986) 527-537.
Pamela Coxson, Loren Larson, and Hans Schneider, Monomial
patterns in the sequence A sup k b, Lin. Alg. Appl. 94 (1987)
89-101.
Chris Paige, Computing the generalized singular value
decomposition. SIAM J. Sci. Stat. Comput. 7 (1986) 1126-1146.
Ion Zaballa, Matrices with prescribed rows and invariant factors,
Lin. Alg. Appl. 87 (1987) 113-146
(Horn did not take part in the final judging.) Future Linear
Algebra Prizes will be awarded every two years by the SIAG/LA.
------------------------------
From: Edmond Ng <esmond@msr.EPM.ORNL.GOV>
Date: Fri Jun 3 12:48:34 1988
Subject: Source for DEROOT and ODERT
Several people at the Oak Ridge National Laboratory are looking for
DEROOT and ODERT which are ode solvers written by Shampine and
Gordon. If anyone knows where and how I can get a copy of the
routines, please let me know.
Esmond Ng (esmond@msr.epm.ornl.gov)
Mathematical Sciences Section
Oak Ridge National Laboratory
P.O. Box 2009, Bldg. 9207-A
Oak Ridge, TN 37831-8083
------------------------------
From: Peter Aitchison <AITCHIS%Uofmcc.Bitnet@Forsythe.Stanford.EDU>
Date: Fri, 03 Jun 88 14:10 CDT
Subject: Winnipeg Numerical Analysis Meeting
CAMS/SCMA TENTH ANNUAL MEETING
AND SYMPOSIUM ON ASYMPTOTIC & COMPUTATIONAL ANALYSIS
-
THE TENTH ANNUAL MEETING OF THE CANADIAN APPLIED MATHEMATICS SOCIETY
WILL BE HELD AT THE UNIVERSITY OF MANITOBA, WINNIPEG, CANADA ON
JUNE 7, 8 AND 9, 1989 (NEXT YEAR).
THE MAJOR THEMES AND PRINCIPAL SPEAKERS OF THE MEETING INCLUDE:
1. MATRIX COMPUATION
A.J. GEORGE
G. GOLUB
W.G. STRANG
2. TWO-POINT BOUNDARY VALUE PROBLEMS
F.V. ATKINSON
H.B. KELLER
3. APPLICATIONS OF SPECIAL FUNCTIONS TO STATISTICS
C.R. RAO (PITTSBURGH)
-
PAPERS ARE NOW BEING SOLICITED FOR PRESENTATION ON THE ABOVE ASPECTS
OF APPLIED MATHEMATICS.
THE CAMS/SCMA MEETING WILL OVERLAP WITH AN INTERNATIONAL SYMPOSIUM
ON ASYMPTOTIC AND COMPUTATIONAL ANALYSIS WHICH WILL BE HELD ON JUNE
5, 6 AND 7, 1989. FOR FURTHER DETAILS AND PAPER SUBMISSION TO EITHER
THE MEETING OR THE SYMPOSIUM CONTACT:
PROFESSOR RODERICK WONG
DEPARTMENT OF APPLIED MATHEMATICS
UNIVERSITY OF MANITOBA
WINNIPEG, MANITOBA, CANADA C3T 3A6
PHONE (204) 474-8167
E-MAIL: AITCH@UOFMCC.BITNET
OR AITCH@SCORE.STANDFORD.EDU
SIGNED P.W. AITCHISON (ORGANIZER OF THE MATRIX COMPUATION SESSION)
------------------------------
From: Paul Nevai <pgn@osupyr.mast.ohio-state.edu>
Date: Sat, 4 Jun 88 07:36:50 EDT
Subject: NATO Institute on Orthogonal Polynomials
NATO Advanced Study Institute (Paul Nevai, Director) on
"Orthogonal Polynomials and Their Applications"
The Ohio State University, Columbus, Ohio 43210, U. S. A.
May 22, 1989 - June 3, 1989.
Main Subjects: Orthogonal Polynomials and Their Applications in
Approximation Theory, Coding Theory,Combinatorics, Communication
Engineering, Continued Fractions, Difference Equations, Julia Sets,
Nuclear Physics, Numerical Analysis, Pade Approximations, Potential
Theory, Prediction Theory, Rational Approximation, Recursions,
Representation Theory, Scattering Theory, Signal Processing, Solid
State Physics and Special Functions.
Organizing Committee: M. Ismail, P. Nevai (Director) and D. Stanton
Principal Speakers: D. Bessis, W. Gautschi, Y. Genin, R. Haydock,
T. Koornwinder, D. S. Lubinsky, I. Macdonald, E. A. Rahmanov,
E. B. Saff, H. Stahl, and G. Viennot
Attendance is strictly by application/invitation only, and it is limited to
approximately one hundred participants.
Partial financial support available for graduate students and scientists
from NATO nations (and possibly from other countries as well)
on a "first come first serve" basis.
Information:
OPsConf
c/o Paul Nevai
Department of Mathematics
The Ohio State University
231 West Eighteenth Avenue
Columbus, Ohio 43210-1174
United States of America
Email: ts1171@ohstvma.bitnet
pgn@osupyr.mast.ohio-state.edu
Phone: 1-(614)-292-4975
1-(614)-292-5310
------------------------------
From: David Hough <dgh@Sun.COM>
Date: Fri, 3 Jun 88 21:46:57 PDT
Subject: Contributions to Floating-Point Indoctrination
The Floating-Point Indoctrination Lectures, being presented by
W. Kahan at Sun Microsystems in Mountain View, are nearing the halfway
point with about 58 paying attendees, 21 absentees, and some full-time students.
Of the total 32 are Sun employees and the remainder are from a variety of other
organizations. The backgrounds and objectives of the participants are
sufficiently varied to defy generalization.
The purpose of this note is to solicit donations of technical
information about commercial products associated with na-net readers.
If your product falls within the realm of floating-point hardware
or mathematical software and you are willing to donate 100 copies
of literature describing it then please mail the material to
David Hough
MS 12-40
Sun Microsystems
2550 Garcia Av
Mountain View, CA 94043
It would be prudent to mail by early July to insure reaching me before the
last class session. What is most useful from the participants' point
of view is technical material like user's reference manuals.
------------------------------
From: William LeVeque <LEV@MATH.AMS.COM>
Date: Tue 7 Jun 88 10:42:09-EDT
Subject: AMS Reviews in Partial Differential Equations
ANNOUNCING
A New Publication from the
American Mathematical Society
REVIEWS IN PARTIAL DIFFERENTIAL EQUATIONS, 1980-86
Introduction by Murray H. Protter
Comprising a significant portion of present-day research in analysis,
the area of partial differential equations encompasses a broad spectrum
of topics, from classical work in linear second-order equations to more
recent work in a general nonlinear setting. In addition, the subject
has deep and fundamental ties to a wide variety of scientific areas
outside mathematics. This important reference work makes the vast
subject of partial differential equations much more accessible both to
specialists working in this area and to those interested in related areas
of mathematics and its applications.
The five volumes of REVIEWS IN PARTIAL DIFFERENTIAL EQUATIONS, 1980-86,
contain the more than 19,200 reviews that appeared in Mathematical
Reviews from 1980 through 1986 and have a primary or a secondary
classification in Partial Differential Equations (classification number 35).
Relevant cross-references are provided with each review. The fifth volume
of this set contains author and key indexes which make it very easy to locate
items written by a specific author or to get information about collections
or conference proceedings dealing with partial differential equations.
It is expected that this work will be followed by sequels, in the future.
4,040 pages in 5 volumes, Softcover, June 1988
ISBN 0-8218-0103-1, LC 88-6681
List $295, AMS Institutional Member $236, AMS Individual Member $177,
Reviewer $148
To order, please specify REVPDE/86NET
PREPAYMENT IS REQUIRED. Send orders with remittance to the American
Mathematical Society, Annex Station, P. O. Box 1571, Providence, RI
02901-9930 USA or order by phone in the continental U.S. and charge your
books to VISA or MasterCard -- call 800-556-7774.
------------------------------
From: Axel Ruhe <ruhe%cs.chalmers.se@relay.cs.net>
Date: 8 Jun 88 14:31 +0200
Subject: Two Tenured Positions at University of Umea
Announcing two tenured chairs as professor in
1. NUMERICAL ANALYSIS with special emphasis on
parallel computations
2. DISCRETE MATHEMATICS with computing applications
at the University of Umea Sweden.
Umea is far enough north to need no street lights this part
of the year. It is small enough to be traversed by bicycle
but large enough to have an Opera House.
The university started as a medical school in 1959,
and is strongest in the biomedical area, but is now also
expanding strongly into the computation field with a o
an educational program for specialists in Scientific Computation.
Skelleftea just to the north has a supercomputer center and
a parallel machine for experimentation is under delivery to Umea.
Applications will be screened by 3 experts, before
decision by the university.
Deadline for application is June 27, 1988. A written application
must have reached the University of Umea at that time.
Postal address: Universitetet, S-90187 Umea, Sweden
Telex: 54005 UNIVUME S
Telefax: int-46-90165488
The application should be supported by a c.v., list of publications,
copies of relevant publications and a short description of the
applicants research achievments and future plans. This supporting
material may arrive 3 weeks after the application date, and should
be sent in 4 copies.
Information can be obtained from
Bo Kagstrom, Telephone int-46-90165419, email bokg@seumdc51.bitnet
Per Ake Wedin, Tel int-46-90165439
and of course from myself who worked there the best part of
my carreer 1970-83,
Axel Ruhe
email: ruhe@cs.chalmers.se
Telephone int-46-31721096 (office)
int-46-31483168 (home)
------------------------------
From: Willard Miller <miller@csfsa.cs.umn.edu>
Date: Wed, 8 Jun 88 11:31:20 cdt
Subject: Minnesota IMA Summer Program on Signal Processing
INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
University of Minnesota
514 Vincent Hall
206 Church Street S.E.
Minneapolis, Minnesota 55455
(612)624-6066 ima_staff%csfsa@umn-cs.cs.umn.edu
Summer Program on
SIGNAL PROCESSING
June 27-August 5, 1988
Organizers: T. Kailath (Chairman), L. Auslander
A. Grunbaum, W. Helton, P. Khargonekar, S. Mitter
The basic objective of the IMA program is to interest a broad set of
mathematicians in research into the challenging problems posed by
Signal Processing and by bringing mathematicians and other experts
in the field together for an extended period, to make significant
research progress toward the solution of these problems.
The program will be an integrated approach to one-dimensional and
multidimensional problems in Signal Processing. Thus the first two
weeks will be devoted to a broad range of problems and methods.
An effort will be made to promote new interaction also within
one-dimensional research groups (such as the linear control and
the nonlinear control people) and within the multi-dimensional
groups (such as radar, sonar and medical imaging).
The first two weeks will be concerned with broad general issues.
The next four weeks will be somewhat less structured and will include
periods of concentration in both 1-dimensional and multidimensional
topics.
OUTLINE
June 27-July 10 (Weeks 1 and 2)
Expository lectures on Problem Areas & Methods
July 11-24 (Weeks 3 and 4}
Period of concentration: Digital filter & VLSI implementation.
Integrable circuit modelling
July 25-31 (Week 5)
Period of concentration: Robust & nonlinear control with aerospace
applications
August 1-5 (Week 6)
Period of concentration: Problems in radar, sonar & medical imaging
During July 11-15 (week 3) there will be a special session on ``Signal
Processing''at the 1988 annual meeting of the Society for Industrial
and Applied Mathematics (SIAM). The annual meeting is being
held in downtown Minneapolis. The special session will consist of:
(1) One hour expository lecture by Gerard Bricogne (Mathematical
Crystallography), Paris/Sud and CNRS
(2) Two half day mini-symposia organized by:
(i) Donald E. McClure, Brown University
(ii) Mostafa Kaveh, University of Minnesota
PROGRAM FOR WEEK 1
Expository lectures on problem areas and methods
Monday, June 27
9:00 am Sanjoy Mitter MIT
Variational and Probabilistic Problems in Image Analysis, Part I
10:30 am William Helton UC, San Diego
Worst Case Design in the Frequency Domain
2:30 pm Tom Kailath Stanford University
A Displacement Structure of Matrices and its Applications
Tuesday, June 28
9:00 am Marvin Bernfeld Raytheon Co.
Signal Processing Concepts in Radar
10:30 am Alberto Grunbaum UC, Berkeley
Solitons in Signal Processing
2:30 pm Thomas S. Huang U. of Illinois-Urbana
Linear Algorithms for 3-D Motion Estimation from Image Sequences
Wednesday, June 29
9:00 am Sanjoy Mitter MIT
Variational and Probabilistic Problems in Image Analysis, Part II
10:30 am Israel Gohberg Tel Aviv University
Interpolation Problems for Matrix Functions
2:30 pm Richard Roy Stanford University
Large Geometric Methods and Invariance Techniques in Multidimensional
Signal Processing
Thursday, June 30
9:00 am Jonathan Allen MIT
Computer Architecture and VLSI Implementation for Digital Signal
Processing, I
10:30 am Louis Auslander CUNY
Nonstationary Processes
2:30 pm Jonathan Allen MIT
Computer Architecture and VLSI Implementation for Digital Signal Processing, II
Friday, July 1
9:00 am Pramod Khargonekar University of Minnesota
Robust Stability and Stabilization
10:30 am Allen Tannenbaum University of Minnesota
Generalized Interpolation in the Robust Control of Uncertain Systems
------------------------------
From: Horst Simon <simon@orville.nas.nasa.gov>
Date: Wed, 8 Jun 88 14:03:17 PDT
Subject: Conference on Applications of the Connection Machine
Scientific Applications of the Connection Machine
Conference Announcement and Call for Papers
NASA Ames Research Center, Moffett Field, CA 94035
September 12-14, 1988
Organized by
Numerical Aerodynamics Systems Division,
NASA Ames Research Center
With Support from
Thinking Machines Corporation
PROGRAM COMMITTEE
Bruce Blaylock -- NASA Ames Research Center
Creon Levit -- NASA Ames Research Center
Jill Mesirov -- Thinking Machines Corporation
Oliver McBryan -- University of Colorado, Boulder
Horst Simon -- NASA Ames Research Center
ORGANIZING COMMITTEE
Carol Bee-Latty -- Thinking Machines Corporation
Lyz Dunham -- NASA Ames Research Center
Judy McWilliams -- NASA Ames Research Center
Objective:
NASA's Numerical Aerodynamic Simulation (NAS) Program, located at NASA
Ames Research Center near Mountain View, California, will be sponsoring
a conference on Scientific Applications of the Connection Machine from
September 12 to September 14, 1988. Focus of this conference will be the
application of the Connection Machine to the solution of large scale
computational problems in physics, in particular computational fluid
dynamics, chemistry, and engineering.
Program:
On the first day of the conference (Monday, 9/12/88), Thinking Machines
will present an all-day tutorial on using the Connection machine.
The purpose of the tutorial is to familiarize the audience with the
architecture and programming environment of the Connection Machine, and to
introduce data parallel programming as a methodology for utilizing a
massively parallel SIMD machine. One of the presenters of the tutorial
will be Guy Steele, Jr. from Thinking Machines Corporation.
For experienced users of the Connection Machine there will be a workshop
on CFD application organized by Creon Levit. For details on the workshop
contact Creon Levit directly.
The actual conference will be held on Tuesday and Wednesday, September 13-14.
The following researchers have been invited to give presentations
about their research using the Connection Machine:
Alan Egolf -- United Technologies
Oliver McBryan -- University of Colorado, Boulder
James Sethian -- University of California, Berkeley
Creon Levit -- NASA Ames Research Center
Gary Demos -- Whitney and Demos
David Rogers -- RIACS
CONTRIBUTED PAPERS:
Contributed presentations will be selected from the submitted abstracts.
If you are interested in presenting a 15 minute paper at the conference please
submit a 200 word abstract by July 15, 1988 to Lyz Dunham, NASA Ames
Research Center, Moffett Field, CA 94035; or by e-mail to
dunham@prandtl.nas.nasa.gov. Selected speakers will be notified by July 29,1988.
PROCEEDINGS:
Proceedings of the conference will be published. Each speaker will be asked
to provide a camera-ready copy of his paper by October 1, 1988.
GENERAL INFORMATION
LOCATION
All technical sessions will be held at the NASA Ames Research Center,
Moffett Field, CA 94035.
TRANSPORTATION
40 minutes from San Francisco International Airport
20 minutes from San Jose International Airport
Direct or connecting flights to every major city in the
United States.
CLIMATE
Bay Area temperatures in the fall range from warm days (75 degrees F)
to cool nights (47 degrees F), with the average daytime temperature
about 62 degrees F.
HOTELS
The following Hotels are conveniently located within NASA Ames Research
Center area. Arrangements should be made directly with the Hotel of
your choice.
The County Inn (415) 961-1131
850 Leong Drive, Mountain View, CA
Rates*: Government - $57.00 Corporate - $65.00/$67.00
Sundowner Inn (408) 734-9900
504 Ross Drive, Sunnyvale, CA 94089
Rates*: Government - $57.00 Corporate - $72.00
Comfort Inn 800-228-5150
1561 El Camino Real West, Mountain View, CA 94040
Rates*: Government/Corporate - $55.00
Best Western Sunnyvale Inn 800-528-1234
940 Weddell Drive, Sunnyvale, CA 94089
Rates*: Government - $54.00 Corporate - $51.00 Regular - $56.00
* All rates based on room availability
REGISTRATION FEES
The regular registration fee is $75. This fee includes the tutorial and
conference, lunch on Tuesday and Wednesday, a dinner on Tuesday
evening, and refreshments during the breaks.
ADVANCE REGISTRATION
Please use this form or a facsimile to pre-register. Advance
registration closes August 15, 1988. Early registration is encouraged,
since available space is limited. Participants will be registered on
a first come basis. Late registration is based on a space available
basis and subject to a $25 late fee.
Please mail your completed form with check or international money order
(US funds) payable to "Connection Machine Conference" to:
Connection Machine Conference
NASA Ames Research Center
c/o Ms. Lyz Dunham
Mail Stop 258-6
Moffett Field, CA 94035
Requests for refunds will be honored until September 1, 1988.
For further information, please call (415) 694-4370, or send
electronic mail to: dunham@prandtl.nas.nasa.gov
Registration Form
Scientific Applications of the Connection Machine
Name (last name first) ________________________________
Affiliation ___________________________________________
Address _______________________________________________
_______________________________________________________
City _______________________ State ____________________
Zip/Postal Code ____________ Country __________________
Phone number __________________________________________
Electronic mail address (if applicable) _______________
I will attend _____ conference and tutorial
_____ tutorial only
_____ conference only
Total enclosed:
$ ______ U.S.
------------------------------
From: Dimpy Pathria <clyde!watmath!watdragon!violet!dpathria@bellcore.com>
Date: 7 Jun 88 16:54:16 GMT
Subject: Rational Approximation of Floating Point Numbers
Does anyone know of numerical algorithms which, when given a number
x in [0..1), return the rational number (numerator/denominator)
that approximates x to within machine tolerance?
I know of a few methods for solving this problem (continued fractions, etc),
am interested in trying some different algorithms. Thanks.
Dimpy Pathria.
dpathria@violet.waterloo.edu
[Editors Note: Of course, every floating point number is already rational,
and so is its own best rational approximation. We must add the requirement
that the numberator and denominator be "small" integers.]
------------------------------
End of NA Digest
**************************
-------
Reposted by
--
Kenneth R. Jackson, krj@csri.toronto.edu (csnet)
Department of Computer Science, uunet!csri.toronto.edu!krj (uucp)
University of Toronto, krj@csri.toronto.cdn (ean x.400)
Toronto, Canada M5S 1A4 krj%csri.toronto.edu@relay.cs.net (arpa)
(416) 978-7075 krj@csri.utoronto (bitnet)