Subject: NA Digest, V. 93, # 11 NA Digest Sunday, March 14, 1993 Volume 93 : Issue 11 Today's Editor: Cleve Moler The MathWorks, Inc. moler@mathworks.com Today's Topics: Daniel Pierce Honored Networks via Mean Value Analysis Numerical Algorithms for Generalized Eigenvalue Problem F90 Compiler Performance Benchmark Project CUTE -- Constrained and Unconstrained Testing Environment Guranteed Accuracy in Numerical Linear Algebra New Book on Fractal Image Compression South African Symposium on Numerical Mathematics Mathematics of Computation 1943-1993 Inverse Problems Conference in Austria Workshop on Reliability Computations European Multigrid Conference Position at University of Virginia Position at Heriot-Watt University Conttents: IMA Numerical Analysis Contents: Linear Algebra and its Applications Submissions for NA Digest: Mail to na.digest@na-net.ornl.gov. Information about NA-NET: Mail to na.help@na-net.ornl.gov. ------------------------------------------------------- From: Biswa Datta Date: Tue, 9 Mar 93 13:17:29 CST Subject: Daniel Pierce Honored DANIEL PIERCE of Boeing Computer Services is the recipient of the 1993 Outstanding Young Alumni Award from Northern Illinois University. Dan received his B.S and M.S degrees from NIU, and wrote his master thesis under the direction of Biswa Datta. He obtained his Ph.D from North Carolina University and wrote his Ph.D dissertation under the direction of Robert Plemmons. Dan is currently a Senior Principal Scientist at Boeing Computer Services. CONGRATUALTIONS, DAN ! ------------------------------ From: Chris Skiscim Date: Tue, 9 Mar 93 18:17:36 EST Subject: Networks via Mean Value Analysis If anyone has an efficient implementation for the exact or approximate solution of queueing networks (open and closed chains) by Mean Value Analysis, please contact me. I am attempting to implement MVA for client-server applications involving a large numbers of chains and customers. Thanks in advance. Chris Skiscim skiscimc@wmavm7.vnet.ibm.com ------------------------------ From: Alex Stankovic Date: 07 Mar 1993 19:54:10 -0500 (EST) Subject: Numerical Algorithms for Generalized Eigenvalue Problem I would be very grateful for suggestions and pointers regarding the numerical methods for the generalized eigenvalue problem (possibly large scale). Specifically, I am looking for review and research papers on the topic, and I am aware of the standard references (Goluv&Van Loan, Horn&Johnson). Thanks in advance for any help, Alex Stankovic astankov@hilbert.coe.northeastern.edu ------------------------------ From: John Prentice Date: Fri, 12 Mar 93 18:41:12 MST Subject: F90 Compiler Performance Benchmark Project For some months I have been working on a set of Fortran 90 codes for the purpose of performing a benchmark of the ParaSoft, NAG, and Pacific Sierra Research Fortran 90 compilers (or translators, if you prefer). The thrust of this project is to determine the performance of these compilers on a variety of numerical problems, many of which are actual production codes. I am performing my benchmark runs on a single processor Sparc 10. I am nearing completion of my F90 performance benchmark suite and will soon be running the final benchmark tests and writing up the results. With that in mind, I am assembling an e-mail list to keep people up to date on this project and for dissimination of the results. If you are interested in being on this list, please send me some e-mail letting me know. If you know of any other people who might be interested in this project, please send me their e-mail address and I will contact them. ------------------------------ From: Philippe Toint Date: Thu, 11 Mar 93 14:34:24 +0100 Subject: CUTE -- Constrained and Unconstrained Testing Environment CUTE is now available CUTE (Constrained and Unconstrained Testing Environment) is a set of Fortran subroutines, system tools and test problems in the area of nonlinear optimization and nonlinear equations. Its purpose is to - provide a way to explore an extensive collection of problems (over 500 different test problems today) - provide a way to compare existing packages (MINOS, LANCELOT, OSL, ...) - provide a way to use a large test problem collection with new packages, - provide motivation for building a meaningful set of new interesting test problems, - provide ways to manage and update the system efficiently, and - do all the above on a variety of popular platforms (SUNs, DEC, IBM workstations, CRAY, ...) CUTE has been written by I. Bongartz, A.R. Conn (both at IBM, Watson Research Center), Nick Gould (CERFACS, France) and Ph. Toint (FUNDP, Belgium). A LaTeX manuscript detailing the package may be obtained by email from any of the authors. It is also included in the distribution. CUTE is written is standard ANSI Fortran 77. Single and double precision versions are available. Machine dependencies are carefully isolated and easily adaptable. Automatic installation procedures are available for DEC ULTRIX, DEC/OSF/1, SUN SunOS, CRAY UNICOS and IBM AIX. The package may be obtained in one of two ways. Firstly, the reader can obtain CUTE electronically (and free of charge) via an anonymous ftp call to the account at the Rutherford Appleton Laboratory camelot.cc.rl.ac.uk (Internet i.d. 130.246.8.61, in the directory pub/cute), or at Facultes Universitaires Notre-Dame de la Paix (Namur) thales.math.fundp.ac.be (Internet i.d. 138.48.4.14, in the directory cute). We request that the userid is given as the password. This will serve to identify those who have obtained a copy via ftp. Secondly, the package can be obtained on a floppy disk or magnetic tape at a minimal price, intended to recoup the costs of media, packaging, preparation and courier delivery. Potentially interested parties should contact Ph. Toint for obtaining a suitable order form. Ingrid Bongartz bongart@watson.ibm.com Andy Conn arconn@watson.ibm.com Nick Gould gould@cerfacs.fr Philippe Toint pht@math.fundp.ac.be ------------------------------ From: S. M. Rump Date: Tue, 9 Mar 1993 14:55:02 +0100 Subject: Guranteed Accuracy in Numerical Linear Algebra Viktor Kostin reviewed the book "Guaranteed Accuracy in Numerical Linear Algebra" by S.K.Godunov et al. He mentions that the methods described over there are almost unknown to English speaking readers. There are a number of books on the subject covering numerical linear algebra, nonlinear equations, global optimization, differential equations and others. For example Adams.,E.,Ansorge,R.,Grossmann,Ch.,Roos,H.-G.: Discretization in Differential Equations and Enclosures, Akademie-Verlag, Vol. 36, Berlin 1987. Alefeld,G.,Herzberger,J.: An Introduction to Interval Computations, Academic Press, New York 1983. Ames,W.F.(Ed.): Numerical and Applied Mathematics, J.C.Baltzer Scientific Publications, Basel 1989. Brezinski,C.,Kulisch,U.(Eds): Computational and Applied Mathematics I - Algorithms and Theory, Proceedings of the 13th IMACS World Congress, Dublin, Ireland, Elsevier Science Publishers 1992. Herzberger,J.,Atanassova,L.(Eds.): Computer Arithmetic and Enclosure Methods, Elsevier Science Publishers, Amsterdam 1993. Klatte,R.,Kulisch,U.,Neaga,M.,Ratz,D.,Ullrich,Ch.: Pascal-XSC - Language Reference with Examples, Springer Verlag, Berlin, Heidelberg, New York 1991. Moore,R.E.(Ed.): Reliability in Computing: The Role of Interval Methods in Scientific Computing,Academic Press, San Diego 1988. Neumaier,A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge 1990. Ullrich,Ch.(Ed.): Computer Arithmetic and Self-Validating Numerical Methods,Academic Press, San Diego 1990. This is an uncomplete (authors: please, don't blame me for not being mentioned) list of the newer books, others are cited there. ------------------------------ From: Lyman Hurd Date: Wed, 10 Mar 93 16:48:03 EST Subject: New Book on Fractal Image Compression I am pleased to announce the arrival of the book: Fractal Image Compression by Michael Barnsley and Lyman Hurd. hardbound 244 pages, 16 color plates ISBN 1-56881-000-8 A sequel to Fractals Everywhere, Fractal Image Compression describes the mathematics underlying the Fractal Transform an image compression method discovered by Michael Barnsley and developed at Iterated Systems. Topics include: * Iterated Function Systems * Local IFS and the Black and White Fractal Transform * The Grayscale Fractal Transform * Information Theory * Huffman codes * Arithmetic Codes * JPEG Image Compression Example programs in C are interspersed throughout the text. Copies can be ordered directly from the publisher by the following methods: 1) Email: send a message to kpeters@geom.umn.edu for price and credit card information. 2) Mail: AK Peters, Ltd. 289 Linden St. Wellesley, MA 02181 3) Phone: (617) 235-2210. ------------------------------ From: WSKDPL Date: Mon, 08 Mar 93 12:45 Subject: South African Symposium on Numerical Mathematics 19th South African Symposium on Numerical Mathematics The Nineteenth South African Symposium on Numerical Mathematics, organized by the South African Society for Numerical Mathematics (SANUM) and the Department of Computer Science of the University of Natal, Durban, will be held at the Protea Hotel, San Lameer, near Margate on the South Coast, on 12, 13 and 14 July 1993. The Symposium offers numerical and applied mathematicians an opportunity to discuss their research, establish contacts and exchange ideas with other delegates. Presentations of 20 minutes duration on any aspect of numerical mathematics and its applications are hereby solicited. All presentations submitted will be published in the unrefereed Proceedings after the Symposium, subject to approval by the SANUM Committee. The deadline for submission of titles and summaries of presentations (one A4 page) as well as the necessary registration form and payment is 17 May 1993. In order to stimulate research among our younger colleagues, SANUM has decided that an award will be made for the best paper presented at the Symposium. This award is limited to candidates under the age of 35 years. Please direct all queries to: Dr Shirley Abelman Secretary: SANUM Dept. of Computational & Applied Mathematics University of the Witwatersrand, Johannesburg P O Wits 2050 South Africa Tel: 27-11-7163298 Fax: 27-11-3397965 e-mail: 076ABNS@WITSVMA.WITS.AC.ZA ------------------------------ From: Walter Gautschi Date: Wed, 10 Mar 1993 09:12:13 -0500 Subject: Mathematics of Computation 1943-1993 *THIRD ANNOUNCEMENT*: *MEETING AND CALL FOR PAPERS* NOTE: This announcement differs from the preceding second announcement only in the e-mail address to which inquiries with regard to registration and housing should be directed. Under the auspices of the American Mathematical Society (AMS), and in celebration of the 50th anniversary of the journal Mathematics of Computation, an international conference MATHEMATICS OF COMPUTATION 1943-1993 A half-century of computational mathematics devoted to all aspects of computational mathematics will be held on the campus of the University of British Columbia, Vancouver, Canada, August 9-13, 1993, immediately prior to the joint AMS/MAA/CMS Summer Meeting. As part of the meeting there will be a two-session minisymposium on Computational Number Theory dedicated to the memory of D.H. Lehmer. Invited speakers will be presenting survey and state- of-the-art lectures in plenary sessions. In addition there will be poster sessions and 15-minute talk sessions for contributed papers. The invited (and confirmed) speakers are: James H. Bramble, Cornell University Johannes Buchmann, Universitaet des Saarlandes Margaret Cheney, Rensselaer Polytechnic Institute Bjorn Engquist, University of California at Los Angeles Donald Goldfarb, Columbia University James N. Lyness, Argonne National Laboratory J.C. Nedelec, Ecole Polytechnique Palaiseau Andrew M. Odlyzko, AT&T Bell Laboratories Frank W.J. Olver, University of Maryland Linda R. Petzold, University of Minnesota Carl Pomerance, University of Georgia Larry L. Schumaker, Vanderbilt University Hans J. Stetter, Technical University of Vienna G.W. Stewart, University of Maryland Roger Temam, Indiana University The deadline for submission of contributed papers is April 1, 1993. Abstracts should be prepared on AMS abstract forms (available at most universities or obtainable from the AMS upon request) and should indicate whether they are submitted for a poster session or for a short talk session. They should be sent in duplicate to the Chairman of the Organizing Committee. Decisions on acceptances will be made by May 1, 1993. All invited lectures and contributed papers are to appear (subject to the usual reviewing procedures) in a Proceedings volume published by the American Mathematical Society. The deadline for registration/housing is June 6, 1993. Registration and housing forms can be obtained from the Mathematics Meetings Service Bureau, P.O. Box 6887, Providence, RI 02940-6887. There will be a registration fee of $50.00. Inquiries with regard to registration and housing should be directed to that address, or by e-mail to wsd@math.ams.org, or tel: 401-455-4143. Other inquiries should be sent to the Chairman of the Organizing Committee. Sponsorship by the National Science Foundation is pending; additional funds for travel and support are sought from the National Security Agency and the Department of Energy. The Organizing Committee: James H. Bramble Walter Gautschi (Chairman) Department of Computer Sciences Purdue University West Lafayette, IN 47907, USA e-mail: wxg@cs.purdue.edu Eugene Isaacson Vidar Thomee Hugh C. Williams ------------------------------ From: Heinz W. Engl Date: Thu, 11 Mar 1993 10:42:15 MEZ Subject: Inverse Problems Conference in Austria Between June 26 and July 2, 1994, a conference on Inverse Problems will be held on Lake St.Wolfgang, Austria. This conference will be the first one in a series of conferences on inverse problems to be organized for SIAM and GAMM by D.Colton (Delaware), H.W.Engl (Linz), A.Louis (Saarbruecken), and W.Rundell (Texas A&M). The local organizer for the St.Wolfgang conference will be H.W.Engl. More detailed information will follow later. Prof.Dr.Heinz W. Engl E-Mail: engl@indmath.uni-linz.ac.at Industriemathematik or na.engl@na-net.ornl.gov Institut fuer Mathematik secretary: nikolaus@indmath.uni-linz.ac.at Johannes-Kepler-Universitaet Phone: +43-(0)732-2468; ext.9219 or 693, Altenbergerstrasse 69 secretary: 9220; home: +43-(0)732-245518 A-4040 Linz Fax: +43-(0)732-246810, attn.:Prof.Engl Oesterreich / Austria Telex: 2-2323 uni li a ------------------------------ From: Valerie Fraysse Date: Thu, 11 Mar 1993 19:01:30 GMT Subject: Workshop on Reliability Computations CERFACS INRIA Centre Europ\'een Institut National de Recherche et de de Recherche Formation Avanc\'ee et de en Informatique en Calcul Scientifique et Automatisme WORKSHOP ON RELIABILITY OF COMPUTATIONS New trends in the assessment of quality for numerical linear algebra software and algebraic processes. Second Announcement March 30--31, April 1 1993, TOULOUSE (FRANCE) For an engineer or a researcher, having confidence in the results obtained from large-scale computational codes and the robustness of the underlying models is of the utmost importance. Indeed, the widespread availability of supercomputers has replaced physical experimentation by intensive numerical simulations, even for problems at the edge of instability. Unfortunately, tools for obtaining information on the quality and the reliability of computer results are rarely used in industrial environments. However, there exist some new tools, developed in the framework of linear algebra, which can offer the best possible solution to a given problem. One has to discover and learn how to use them. The aim of the workshop ``RELIABILITY OF COMPUTATIONS'' is then twofold : * to present an overview of the new concepts influencing the design of LAPACK, the Linear Algebra Library released in 1992, * to show how the user can get meaningful information even from results where all the digits can be wrong. Such computational problems are considered so difficult that they are called ``impossible''. List of speakers: M. Arioli, IAN--CNR Pavia and CERFACS J. C. Berg\`es, CNES J. P. Boujot, CISI Ing\'enierie T. Braconnier, CERFACS F. Chatelin, THOMSON--CSF and CERFACS J. Cullum, IBM Yorktown V. Frayss\'e, CERFACS S. Hammarling, NAG N. Higham, University of Manchester I. Ipsen, Yale University B. K{\aa}gstr\"{o}m, University of Ume\aa Organizing Committee: M. Arioli, IAN--CNR Pavia and CERFACS F. Chatelin, THOMSON--CSF and CERFACS, I. S. Duff, Rutherford Appleton Laboratory and CERFACS, V. Frayss\'e, CERFACS. This workshop is organized by CERFACS with the support of INRIA and IBM. Registration fees: Regular participants : 2500 FF Students : 1000 FF The registration fees include : participation at the workshop, breakfasts, lunches, coffee-breaks, a cocktail and relevant documentation. Possibility for accomodation on the campus (47 FF per night). Scientific coordination and registration : CERFACS Val\'erie Frayss\'e Tel : (33) 61 19 30 18 42 ave. G. Coriolis Fax : (33) 61 19 30 00 31057 Toulouse CEDEX Email : fraysse@cerfacs.fr FRANCE ------------------------------ From: Piet Wesseling Date: Fri, 12 Mar 93 10:13:30 MET Subject: European Multigrid Conference EMG'93 EUROPEAN MULTIGRID CONFERENCE July 6-9, 1993 Amsterdam The Netherlands Call for Registration Following earlier conferences in 1981, 1985 and 1990, EMG'93, the fourth European Multigrid Conference will be held in Amsterdam, July 6-9, 1993. The conference aims at providing a forum for the presentation and discussion of recent research in the general area of multilevel methods. Participation will be worldwide. Themes of the conference will range from basic research to industrial applications, including but not limited to the following subjects: * computational fluid dynamics, * reservoir engineering, * statistical physics, * parallel computing, * adaptive computing, * numerical analysis of multigrid methods. PROGRAM The program of the fourth European Multigrid Conference will consist of invited lectures in plenary sessions and some 50 selected presentations in 2 parallel sessions. Program chairmen are P.W. Hemker (CWI/University of Amsterdam) and P. Wesseling (Delft University of Technology). Invited lectures will be given by: D. Braess (Bochum, Germany), Multigrid for nonconforming finite elements A. Brandt (Rehovot, Israel), Multiscale computations in physics and chemistry C. Douglas (New Haven, USA), A generalized multigrid theory L. Fuchs (Stockholm, Sweden), Multigrid methods for calculation of flows in multi-components geometries W. Hackbush (Kiel, Germany), The frequency-decomposition multigrid method P. Lauwers (Bonn, Germany), Multigrid algorithms in lattice gauge theories K. Stueben (Sankt Augustin, Germany), Adaptive multigrid on distributed memory computers S. Vandewalle (Leuven, Belgium), Multicomputer solution of parabolic problems C.H. Venner (Enschede, The Netherlands), Multilevel solution of integral and integro-differential problems G. Wittum (Heidelberg, Germany), On robust multigrid methods CONTRIBUTED PAPERS The program of contributed papers will be made available through anonymous ftp from "ftp.cwi.nl" (Conferences/EMG93) and mgnet (caspar.cs.yale.edu). If you don't have ftp-facilities you may contact the conference administrator (Beau) at fax +31-20-5924199 to get the program. ORGANIZING COMMITTEE P.W. Hemker F. Snijders P. Wesseling S. van der Wolff CONFERENCE ADMINISTRATOR EMG'93 - European Multigrid Conference 1993 c/o CWI Ms. Simone van der Wolff P.O. Box 4079 1009 AB Amsterdam The Netherlands Tel. +31-20-5924189 Fax. +31-20-5924199 email: simone@cwi.nl COMPLETE ELECTRONIC ANNOUNCEMENT Available via e-mail from: simone@cwi.nl ------------------------------ From: Lois Mansfield Date: Wed, 10 Mar 93 14:02:46 EST Subject: Position at University of Virginia The Department of Applied Mathematics in the School of Engineering and Applied Science at the University of Virginia has a tenure-track opening for an assistant professor in the area Numerical Analysis/Scientific Computing/Mathematical Modelling. We seek candidates who are willing and able to work with colleagues throughout the School of Engineering & Applied Science in applying mathematics to the understanding of real-world phenomena, especially to those problems that arise in modern technology. That is, we seek candidates who demonstrate an interest in and potential for mathematical modelling and the development and application of efficient numerical methods for predicting the behavior of such models. Finally, we seek candidates with a strong commitment to the education of both undergraduates and graduates within an environment dedicated to engineering and applied science. Please send resumes and supporting materials to: Chair of the Search Committee Department of Applied Mathematics Thornton Hall, University of Virginia Charlottesville, VA 22903 The University of Virginia does not discriminate on the basis of race, creed, gender, or physical disability. Applications from underrepresented minorities are especially encouraged. ------------------------------ From: Dugald Duncan Date: Thu, 11 Mar 93 08:28:47 GMT Subject: Position at Heriot-Watt University Research Associateship in Nonlinear Systems Heriot-Watt University, Edinburgh, Scotland Applications are invited for a postdoctoral research associateship sponsored by the UK SERC. The research associate will work on computational problems arising in the study of phase transformations in solids under the direction of Professors J M Ball & O Penrose and Dr. D B Duncan. A background in scientific computation or numerical analysis is essential, and experience in partial differential equations, solid mechanics or statistical physics would be an advantage. The appointment will commence on 1 October 1993 and will be for a period of 2-3 years at a starting salary of between 13,400-14,962 pounds on the RA 1A scale. The position will involve a small amount of teaching duties. Application forms, and further particulars, are available from the Director of Personnel, Heriot-Watt University, Riccarton, Edinburgh, Scotland, EH14 4AS to whom completed forms should be returned by 10 April 1993. Overseas applicants can instead send a curriculum vitae and names of 3 referees, if desired by FAX (44-31-451 3249) or e-mail (J.M.Ball@cara.ma.hw.ac.uk). Please quote reference number 08/93. ------------------------------ From: Iain Duff Date: Mon, 8 Mar 93 18:20:09 GMT Subject: Conttents: IMA Numerical Analysis Contents of IMA Journal of Numerical Analysis. Volume 13 Number 2. Govaerts W and Pryce J D Mixed block elimination for linear systems with wider borders Papamichael N, Soares M J and Stylianopoulos A numerical method for the computation of Faber polynomials and starlike domains. Atkinson K and Flores J The discrete collocation method for nonlinear integral equations. Pickering W M and Harley P J FFT solution of the Robbins problem Hoppe R H W A globally convergent multi-grid algorithm for moving boundary problems of two-phase Stefan type Zhao N and Dai J Uniformly third-order accurate TVNE interpolation Humphries A R Spurious solutions of numerical methods for initial value problems. Torelli L and Vermiglio R On the stability of continuous quadrature rules for differential equations with several constant delays Weiner R, Arnold M, Rentrop P, and Strehmel K Patritioning strategies in Runge-Kutta type methods ------------------------------ From: Richard Brualdi Date: Thu, 11 Mar 93 07:29:29 CST Subject: Contents: Linear Algebra and its Applications Contents Volume 184 Chjan C. Lim (Troy, New York) Nonsingular Sign Patterns and the Orthogonal Group 1 Vlastimil Ptak and Zdenek Vavrin (Prague, Czech Republic) Bezout, Hankel, and Loewner Matrices 13 Bryan L. Shader (Laramie, Wyoming) Convertible, Nearly Decomposable, and Nearly Reducible Matrices 37 Daniel Hershkowitz (Haifa, Israel) The Relation Between the Jordan Structure of a Matrix and Its Graph 55 Roy Mathias (Minneapolis, Minnesota) An Arithmetic-Geometric-Harmonic Mean Inequality Involving Hadamard Products 71 Jiang Zeng (Strasbourg, France) A Bijective Proof of Muir's Identity and the Cauchy-Binet Formula 79 Bernard F. Lamond (Quebec, Canada) An Efficient Basis Update of Asymptotic Linear Programming 83 Robert E. Hartwig (Raleigh, North Carolina) and Michael Neumann (Storrs, Connecticut) Bounds on the Exponent of Primitivity Which Depend on the Spectrum and the Minimal Polynomial 103 Zhe-xian Wan (Lund, Sweden) On the Orthogonal Invariants of a Subspace of a Vector Space Over a Finite Field of Odd Characteristic 123 Zhe-xian Wan (Lund, Sweden) On the Orthogonal Invariants of a Subspace of a Vector Space Over a Finite Field of Even Characteristics 135 W. Hasselbarth (Berlin, Germany) and E. Ruch (Gottingen, Germany) On the Extension of Linear Contractions: Some Results of the Hahn-Banach Type 145 Roland W. Freund (Murray Hill, New Jersey) and homas Huckle (Wurzburg, Germany) A Restricted Signature Normal Form for Hermitian Matrices, Quasi-Spectral Decompositions, and Applications 165 Scott McCullough (Gainesville, Florida) Minimal Separators of 2-Chordal Graphs 187 Michael I. Gil' (Beer Sheva, Israel) On Inequalities for Eigenvalues of Matrices 201 Wang Xinmin (Beijing, People's Republic of China) Generalized Stein-Rosenberg Theorems for the Regular Splittings and Convergence of Some Generalized Iterative Methods 207 LeRoy B. Beasley (Logan, Utah) and Larry J. Cummings (Waterloo, Ontario, Canada) The Structure of Generalized Permanent Semigroups 235 Bo-Ying Wang and Ming-Peng Gong (Beijing, People's Republic of China) Some Eigenvalue Inequalities for Positive Semidefinite Matrix Power Products 249 Cheng Wang, Dao-Sheng Zheng, Guo-Liang Chen, and Shu-Qin Zhao (Shanghai, People's Republic of China) Structures of p-Isometric Matrices and Rectangular Matrices With Minimum p-Norm Condition Number 261 ------------------------------ End of NA Digest ************************** -------