From surfer.EPM.ORNL.GOV!nacomb Sun Apr 21 14:36:07 0400 1991 Received: by pyxis; Sun Apr 21 14:36 EDT 1991 To: pyxis!ehg Received: by inet.att.com; Sun Apr 21 14:36 EDT 1991 Received: by surfer.EPM.ORNL.GOV (5.61/1.34) id AA12504; Sun, 21 Apr 91 14:36:07 -0400 Date: Sun, 21 Apr 91 14:36:07 -0400 From: nacomb@surfer.EPM.ORNL.GOV (NA-NET) Message-Id: <9104211836.AA12504@surfer.EPM.ORNL.GOV> Subject: NA Digest, V. 91, # 16 Comment: Submissions for NA News Digest, mail to na.digest@na-net.ornl.gov. Comment: Information about NA-NET, mail to na.help@na-net.ornl.gov. Comment: Comments about the NA-NET, mail to nanet@na-net.ornl.gov. Apparently-To: ehg@research.att.com NA Digest Sunday, April 21, 1991 Volume 91 : Issue 16 Today's Editor: Cleve Moler Today's Topics: Last Call for SIAG/LA Prize Sparse Solvers for Overdetemrined Systems Block Tridiagonal Matrices Shampine/Gordon Integrator in C C Source for a Stiff ODE Integrator Three Measures of Precision in Floating Point Arithmetic Meeting on Mathematics, Computations, and Reactor Physics Second NIU Conference Position at the University of Auckland Postgraduate Opportunity at University of Durham Position at Centenary College of Louisiana Position at Lawrence Livermore Laboratory ------------------------------------------------------- From: Mike Heath Date: Sun, 14 Apr 91 21:23:36 CDT Subject: Last Call for SIAG/LA Prize This is a reminder that the deadline is April 30 for nominations of papers for the SIAG/LA Prize for the best paper in applicable linear algebra for the years 1988-1990. Please see previous announcements for details and eligibility rules. Nominations should consist of a complete bibliographic citation and a brief statement justifying the nomination, and should be directed to the Prize Committee Chairman at the following address: Michael T. Heath Center for Supercomputing Research and Development 305 Talbot Laboratory University of Illinois 104 South Wright Street Urbana, IL 61801-2932 Phone: 217-244-6915 Fax: 217-244-1351 Email: heath@csrd.uiuc.edu ------------------------------ From: Rob MacLeod Date: Sun, 14 Apr 91 21:28:58 MDT Subject: Sparse Solvers for Overdetemrined Systems Hello na-netters, I am looking for a solver that will work on a slightly overdetermined linear system which is very sparse and reasonably large (ca. 3024 by 2832). It arises from a minimization problem with which I can perform inteprolation of electric potential values over a three-dimensional surface (the human thorax). I matrix is not symmetric, or shouldn't be in theory, and I have not yet checked diagonal dominance or positivity. In the past, I have used direct QR-solvers on the full matrix in problems of this type which were a bit smaller (1112 by 1048) in very tolerable amounts of time (6 minutes) on an IBM RS/6000 520. Unfortunately, the scaled-up version employing this approach for a 3024 by 2832 case ran for 12 hours before I gave up and stopped it. I need something considerably faster that will run as a two-step process so that I can either backsubstitute numerous right hand sides back into the decomposed matrix, or do an iterative computation that will run in a few minutes. Any suggestions out there? Thanks, Rob. Rob MacLeod, Ph.D. Nora Eccles Harrison Cardiovascular Research and Training Institute (CVRTI) Building 500 University of Utah Salt Lake City Utah 84112 ------------------------------ From: Tilak Ratnanather Date: Mon, 15 Apr 91 12:28:59 +0100 Subject: Block Tridiagonal Matrices The following problem arising in magnetism was brought to my attention by colleagues who'd greatly appreciate input from experienced numerical analysts: First question: is there a program that will compute the eigenvalues of a BLOCK TRIDIAGONAL (Hermitian) MATRIX. Attempts to search for one in netlib failed to yield a program. (Fortran or even C is the preferred language). Second question: at a more deeper level, the block tridiagonal Hermitian matrix M may be written in the shorthand form: (B*, A, B) where A and B are 5x5 matrices and * indicates the complex conjugate and M is 100x100. Is there a way of reducing this "large" problem to a sequence of reduced problems (or 5x5) problems? One step that has been suggested is to make use of the diagonalised form of A. The entries of A and B may be changed pending the physical configurations. Any ideas or pointers to references would be greatly appreciated. It may be possible to summarise the responses at a later date. Thanks a lot Tilak Ratnanather Dept of Mathematics City University Northampton Square London EC1V 0HB e-mail: sg391@city.ac.uk jtr%mathsun1@city.ac.uk na.ratnanather@na-net.ornl.gov ------------------------------ From: Roland England Date: Tue, 16 APR 91 14:19:12 GMT Subject: Shampine/Gordon Integrator in C In connection with Kathryn Brenan's enquiry, we were also about to start rewriting the Shampine/Gordon integration software (Adams numerical integration code for ordinary differential equations) in C. We would greatly welcome any available information, if this has already been done elsewhere, and particularly, of course, if there is a public domain version in existence which we could use. Thanks for any replies. Roland England R_ENGLAND@vax.acs.open.ac.uk NA.RENGLAND@na-net.ornl.gov Tel. +44-908-65-2329 The Open University Faculty of Mathematics Milton Keynes MK7 6AA Great Britain ------------------------------ From: Steve Nichols Date: 20 Apr 91 13:57:35 GMT Subject: C Source for a Stiff ODE Integrator I'm looking for C source code for a stiff ODE integrator. If anyone has converted LSODE (from the netlib) to C, this would be perfect. Thanks for any help, Steve Nichols Georgia Tech Physics Department gt3930b@prism.gatech.edu ------------------------------ From: Nick Higham Date: Sat Apr 13 14:31:37 PDT 1991 Subject: Three Measures of Precision in Floating Point Arithmetic Three measures of precision in floating point arithmetic by Nick Higham This note is about three quantities that relate to the precision of floating point arithmetic. For t-digit, rounded base b arithmetic the quantities are (1) machine epsilon (eps), defined as the distance from 1.0 to the smallest floating point number bigger than 1.0 (and given by eps = b**(1-t), which is the spacing of the floating point numbers between 1.0 and b), (2) mu = smallest floating point number x such that fl(1 + x) > 1, and (3) unit roundoff u = b**(1-t)/2 (which is a bound for the relative error in rounding a real number to floating point form). The terminology I have used is not an accepted standard; for example, the name machine epsilon is sometimes given to the quantity in (2). My definition of unit roundoff is as in Golub and Van Loan's book `Matrix Computations' [1] and is widely used. I chose the notation eps in (1) because it conforms with MATLAB, in which the permanent variable eps is the machine epsilon. [Ed. note: Well, not quite. See my comments below. --Cleve] The purpose of this note is to point out that it is not necessarily the case that mu = eps, or that mu = u, as is sometimes claimed in the literature, and that, moreover, the precise value of mu is difficult to predict. It is helpful to consider binary arithmetic with t = 3. Using binary notation we have 1 + u = 1.00 + .001 = 1.001, which is exactly half way between the adjacent floating point numbers 1.00 and 1.01. Thus fl(1 + u) = 1.01 if we round away from zero when there is a tie, while fl(1 + u) = 1.00 if we round to an even last digit on a tie. It follows that mu <= u with round away from zero (and it is easy to see that mu = u), whereas mu > u for round to even. I believe that round away from zero used to be the more common choice in computer arithmetic, and this may explain why some authors define or characterize u as in (2). However, the widely used IEEE standard 754 binary arithmetic uses round to even. So far, then it is clear that the way in which ties are resolved in rounding affects the value of mu. Let us now try to determine the value of mu with round to even. A little thought may lead one to suspect that mu <= u(1+eps). For in the b = 2, t = 3 case we have x = u*(1+eps) = .001*(1+.01) = .00101 => fl(1 + x) = fl( 1.00101 ) = 1.01, assuming ``perfect rounding''. I reasoned this way, and decided to check this putative value of mu in 386-MATLAB on my PC. MATLAB uses IEEE standard 754 binary arithmetic, which has t = 53 (taking into account the implicit leading bit of 1). Here is what I found: >> format compact; format hex >> x = 2^(-53)*(1+2^(-52)); y = [1+x 1 x] y = 3ff0000000000000 3ff0000000000000 3ca0000000000001 >> x = 2^(-53)*(1+2^(-11)); y = [1+x 1 x] y = 3ff0000000000000 3ff0000000000000 3ca0020000000000 >> x = 2^(-53)*(1+2^(-10)); y = [1+x 1 x] y = 3ff0000000000001 3ff0000000000000 3ca0040000000000 Thus the guess is wrong, and it appears that mu = u*(1+2^(42)*eps) in this environment! What is the explanation? The answer is that we are seeing the effect of ``double-rounding'', a phenomenon that I learned about from an article by Cleve Moler [2]. The Intel floating-point chips used on PCs implement internally the optional extended precision arithmetic described in the IEEE standard, with 64 bits in the mantissa [3]. What appears to be happening in the example above is that `1+x' is first rounded to 64 bits; if x = u*(1+2^(-i)) and i > 10 then the least significant bit is lost in this rounding. The extended precision number is now rounded to 53 bit precision; but when i > 10 there is a rounding tie (since we have lost the original least significant bit) which is resolved to 1.0, which has an even last bit. The interesting fact, then, is that the value of mu can vary even between machines that implement IEEE standard arithmetic. Finally, I'd like to stress an important point that I learned from the work of Vel Kahan: the relative error in addition and subtraction is not necessarily bounded by u. Indeed on machines such as Crays that lack a guard digit this relative error can be as large as 1. For example, if b = 2 and t = 3, then subtracting from 1.0 the next smaller floating number we have Exactly: 1.00- .111 ----- .001 Computed, without 1.00- a guard digit: .11 The least significant bit is dropped. ----- .01 The computed answer is too big by a factor 2 and so has relative error 1! According to Vel Kahan, the example I have given mimics what happens on a Cray X-MP or Y-MP, but the Cray 2 behaves differently and produces the answer zero. Although the relative error in addition/subtraction is not bounded by the unit roundoff u for machines without a guard digit, it is nevertheless true that fl(a + b) = a(1+e) + b(1+f), where e and f are bounded in magnitude by u. [1] G. H. Golub and C. F. Van Loan, Matrix Computations, Second Edition, Johns Hopkins Press, Baltimore, 1989. [2] C. B. Moler, Technical note: Double-rounding and implications for numeric computations, The MathWorks Newsletter, Vol 4, No. 1 (1990), p. 6. [3] R. Startz, 8087/80287/80387 for the IBM PC & Compatibles, Third Edition, Brady, New York, 1988. ****************** Editor's addendum: I agree with everything Nick has to say, and have a few more comments. MATLAB on a PC has IEEE floating point with extended precision implemented in an Intel chip. The C compiler generates code with double rounding. MATLAB on a Sun Sparc also has IEEE floating point with extended precision, but it is implemented in a Sparc chip. The C compiler generates code which avoids double rounding. On both the PC and the Sparc eps = 2^(-52) = 3cb0000000000000 = 2.220446049250313e-16 However, on the PC mu = 2^(-53)*(1+2^(-10)) = 3ca0040000000000 = 1.111307226797642e-16 While on the Sparc mu = 2^(-53)*(1+2^(-52)) = 3ca0000000000001 = 1.110223024625157e-16 Note that mu is not 2 raised to a negative integer power. MATLAB on a VAX usually uses "D" floating point (there is also a "G" version under VMS). Compared to IEEE floating point, the D format has 3 more bits in the fraction and 3 less bits in the exponent. So eps should be 2^(-55), but MATLAB says eps is 2^(-56). It is actually using the 1+x > 1 trick to compute what we're now calling mu. There is no extended precision or double rounding and ties between two floating point values are chopped, so we can find mu by just trying powers of 2. On the VAX with D float eps = 2^(-55) = 2.775557561562891e-17 mu = 2^(-56) = 1.387778780781446e-17 The definition of "eps" as the distance from 1.0 to the next floating point number is a purely "geometric" quantity depending only on the structure of the floating point numbers. The point Nick is making is that the more common definition of what we here call mu involves a comparison between 1.0 + x and 1.0 and subtle rounding properties of floating point addition. I now much prefer the simple geometric definition, even I've been as responsible as anybody for the popularity of the definition involving addition. -- Cleve ------------------------------ From: Milo Dorr Date: Sun, 14 Apr 91 15:34:53 PDT Subject: Meeting on Mathematics, Computations, and Reactor Physics The American Nuclear Society (Mathematics & Computation Division and Reactor Physics Division) International Topical Meeting: ADVANCES IN MATHEMATICS, COMPUTATIONS, AND REACTOR PHYSICS April 28 - May 2, 1991 Green Tree Marriott Pittsburgh, PA The banquet speaker on Wednesday, May 1 will be George E. Lindamood of the Gartner Group. The title of his lecture will be "Why Supercomputing Matters: A Perspective of the Proposed Federal High Performance Computing and Communication Program". There will also be a plenary session at 9:30am on Monday, April 29, consisting of a panel discussion on the topic "Perspectives on Advances in Supercomputing Performance" moderated by James. R. Kasdorf from Westinghouse Corporate Computer Services and the Pittsburgh Supercomputing Center. The panelists and the titles of their prepared presentations are: Gregory J. McRae (Carnegie Mellon University) "Grand Challenges in Computational Science" W. B. Barker (BBN Advanced Computers, Inc.) "Parallel Computing: Past, Present, and Future" M. L. Barton (Intel Supercomputing Systems Division) "Technology Development for Supercomputing 2000" Kenichi Miura (Fujitsu America, Inc.) "Perspectives on Advances in Supercomputer Performance - Fujitsu's View" Steve Nelson (Cray Research, Inc.) "Heterogeneous Supercomputing: Now and Then" For a copy of the Technical Program or further information, please contact the Program Chairman: I. K. Abu-Shumays RT-Mathematics, 34F Bettis Atomic Power Laboratory P. O. Box 79 West Mifflin, PA 15122-0079 (412) 476-6469, FAX (412) 476-5151 ------------------------------ From: Biswa Datta Date: Wed, 17 Apr 91 10:13:28 NZS Subject: Position at the University of Auckland THE UNIVERSITY OF AUCKLAND, NEW ZEALAND LECTURESHIP Applied and Computational Mathematics Unit The University of Auckland invites applications for a lectureship in the Applied and Computational Mathematics Unit within the Department of Mathematics and Statistics. Applicants should have a proven record in teaching and research in some branch of Applied or Computational Mathematics. Applications from candidates with expertise in a field that will strengthen and enhance the existing research interests of the Applied and Computational Mathematics Unit in differential equations and their numerical solutions and applications of scientific computing to physical and other problems are particularly welcome. The University of Auckland has a student population of about 17,000 and is the only university in Auckland, a city with about one million inhabitants. Mathematics and Statistics is the largest department. University positions are organised in a system similar to that of the UK. Thus a lectureship is intended to be a permanent appointment although, in the first instance, it is for a four year term. Most appointments are continued after that period. The salary scale for a lecturer begins at $NZ37,440 and increases in annual increments. (Details of the salary scale and prospects for promotion are available on request). It is hoped that the successful applicant will take up his or her duties by 1 August 1991 or soon after. Applications, in accordance with the "Method of Application" (available on request) should be submitted as soon as possible and no later than 30 June 1991. The University of Auckland is an Equal Opportunity Employer For further information or enquiries about the position please contact the Head of the Applied and Computational Mathematics Unit Professor J. C. Butcher using email. butcher@mat.aukuni.ac.nz or na.butcher@na-net.ornl.gov ------------------------------ From: Alan Craig Date: Thu, 18 Apr 91 16:35:46 BST Subject: Postgraduate Opportunity at University of Durham The Department of Mathematical Sciences, University of Durham and the British Gas Research Station in Northumberland have a CASE studentship in Adaptive Finite Element Analysis for Nonlinear Elastic Problems The award is for three years, tenable from October 1991 and for a programme of research leading to a PhD. Outline of project: the finite element method is the computational tool for the solution of structural analysis problems. The technique relies on replacing the underlying differential equations by an approximating set of equations. However the correct choice of this approximating set is by no means a trivial procedure. In recent years adaptive methods have been developed which attempt to choose the approximation in an automatic way linked to the problem. The analysis and implementation of these methods for linear problems is now well understood. The situation for nonlinear systems is however, less satisfactory and the interest in adaptive techniques for general nonlinear response stems from the need to maintain structural integrity in hazardous situations. In particular British Gas are interested in techniques which can be used to analyse their offshore rigs. The project will maintain a careful balance between theoretical analysis and practical implementation. Experience has shown that in this area, as in many others, the implementation can be guided and enriched by the analysis. The ultimate aim is to produce high quality algorithms for nonlinear problems. As part of the project the successful applicant will work in the British Gas Research Station for periods totaling three months under the supervision of the Industrial Supervisor Mrs. Jane Haswell. The applicant will have, or will be about to obtain, a good honours degree in Mathematics, Engineering or a related discipline. In addition to the standard SERC award British Gas will make an extra payment of 1620 pounds per annum to the student. Further information can be obtained from Dr. Alan Craig at the address below and informal enquiries are welcomed. Application should be made to the same address and should include the names and addresses of at least two referees, one of whom should be able to judge the applicants suitability for research. Dr.A.W.Craig Department of Mathematical Sciences University of Durham South Road Durham DH1 3LE England na.craig@na-net.ornl.gov ------------------------------ From: Antonio Pizarro Date: Fri, 19 Apr 91 08:35:56 IST Subject: Position at Centenary College of Louisiana Applications are invited for two tenure-track positions in MATH. A Ph.D in Mathematics is required. Linear Algebra, Topology or Applied Math are preferred fields, but all fields are considered. Beginning Fall 1991. Send resume and three letters of recommendation to: Antonio G. Pizarro, Chair Math Department Centenary College of Louisiana Shreveport LA 71134, USA. ------------------------------ From: Anthony Skjellum Date: Fri, 19 Apr 91 11:15:54 PDT Subject: Position at Lawrence Livermore Laboratory Postdoctoral Position in Parallel Computation - Lawrence Livermore A postdoctoral position in scientific parallel algorithms research is presently available in the Numerical Mathematics Group (NMG) at the Lawrence Livermore National Laboratory, Computing and Mathematics Researh Division. Candidates should be well versed in parallel computation; particularly, distributed memory machines and issues. Experience with large-scale parallel scientific algorithms and applications is strongly desired. In-depth experience with the Unix operating system and the C/C++ languages is also desired. Candidate will participate in the growing NMG research effort in parallel computation, but also will enjoy freedom to explore his or her own allied research interests. A one year position is offered, renewable for a second year if both NMG and the fellow concur. Candidate should have completed his or her Ph.D. in an appropriate discipline prior to October 1, 1991. Starting date is on or after October 1, 1991. US citizenship is required. Send three letters of recommendation, resume, and publications list to Dr. Anthony Skjellum, LLNL L-316, PO Box 808, Livermore, CA 94550. (415)422-1161. e-mail: tony@helios.llnl.gov. ------------------------------ End of NA Digest ************************** ------- From surfer.EPM.ORNL.GOV!nacomb Sat Apr 27 22:36:25 0400 1991