Subject: NA Digest, V. 93, # 36 NA Digest Sunday, September 26, 1993 Volume 93 : Issue 36 Today's Editor: Cleve Moler The MathWorks, Inc. moler@mathworks.com Today's Topics: IEEE Trap Handling on SUN Systems Help for IEEE Floating Point Woes Reality check for an IEEE arithmetic critic Code for Cubic Schroedinger Equation Fortran 90 Introduction and Conversion One day MATLAB Course at RMCS (Cranfield) Mathematics-in-Industry Study Group Gelfand Celebration Postions at King Saud University Two Post-Doctoral Positions at Napier University, SCOTLAND Research and Development Programmer Positions Faculty Positions at SUNY-Buffalo Contents, Linear Algebra and its Applciations Contents, SIAM Mathematical Analysis Contents, IMA Numerical Analysis 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: Richard Lehoucq Date: Tue, 21 Sep 93 17:03:55 CDT Subject: IEEE Trap Handling on SUN Systems The following is the code needed to be able to trap Floating Point exceptions within your code on SUN systems in FORTRAN. Note: for you C programming types, this also works within C software. Compile the following integer function along with the rest of your code. You need to compile this and the routines you want to look at with the -g option if using the debugger. This has proved instrumental in debugging. integer function handler(sig,code,sigcontext) integer sig integer code integer sigcontext(5) print *, ' ieee exception code ',loc(code), & ' occurred at pc ', sigcontext(4) call abort end Notes: the name 'handler' is user defined. The first line of your main program needs the followinf stmt: ieee= ieee_handler('set','common',handler) or(or both): ieee= ieee_handler('set','underflow',handler) The last line of main program needs the following stmt. ieee= ieee_handler('clear','all',handler) notes: 'common' flags overflow, NaN, division by zero. See man page on ieee_handler for more info. User defined 'handler' needs to be declared as an external integer function within the main program unit. the integer variable 'ieee' is user defined. Now complie with -g switch, fire up the debugger and make sure the first command issued within the debugger is: catch FPE and type run. Upon encountering an exception, the code pauses execution and the debugger points to the trapped line and informs user of what exception occured and at what line number. To continue type cont and you will encounter the next exception(if any). An excellent reference is the SUN manual: Numerical Computation Guide. DISCLAIMER: I don't profess that this is the best or only way to trap for floating point exceptions under SUN systems. -- Richard Lehoucq ------------------------------ From: Jim Van Zandt Date: Wed, 22 Sep 93 08:02:52 -0400 Subject: Help for IEEE Floating Point Woes Richard Brankin writes: >The following may be of help, for SGI Indigo users anyway, > >setenv TRAP_FPE "UNDERFL=ZERO;OVERFL=TRACE(1),EXIT;DIVZERO=TRACE(1),EXIT;INVALID=TRACE(1),EXIT" > >and don't forget to link with "-lfpe" when building an executable. This >has the effect "underflow to zero; bomb out with a traceback when either >overflow, or divide by zero, or an invalid operand is detected", with >which I expect most people would be happy. Can anyone suggest how to get similar effects with Turbo/Borland C? -- Jim Van Zandt ------------------------------ From: Dean Schulze Date: Sat, 25 Sep 1993 16:07:42 -0700 (MST) Subject: Reality check for an IEEE arithmetic critic Joe Grcar's complaints about IEEE arithmetic need to be addressed because some of the things that he says are simply wrong. He made this complaint in August of 1991: >There is no machine epsilon. Yes there is. Numbers can underflow, however, to de-normalized values that result in a loss of precision. If you think this is a problem how are you dealing with non-IEEE arithmetic that simply underflows to zero without telling you about it? Two weeks ago Grcar published part 2 of his complaint about IEEE arithmetic which included the following: >A colleague with a PhD in mechanical engineering just walked >into may office and said: >"I like my new SGI Indigo, but I really don't like those NaN >things it has. I'm rewriting a piece of software and when I >looked at the output it was all full of NaN's. I found out >this means I'm dividing by zero somewhere, but where? ..." Wrong. Under IEEE arithmetic dividing a finite, non-zero number by zero produces an Inf. Division by zero produces a NaN only if the numerator is zero, Inf, or NaN. NaNs can also be the result of common programming errors like reading beyond the bounds of an array. >By chance I once asked SGI technical support where to find >documentation on IEEE arithmetic. Here is an excellent reference that describes IEEE arithmetic: "What every computer scientists should know about floating-point arithmetic", David Goldberg, ACM Computing Surveys, v23, #1, pp. 5-48, 1991. In response to his posting, I sent Joe Grcar a message that included this citation. He responded with: >Pardon me for saying so, but I think your opinion >of ieee arithmetic would be different if you had >a real job instead of an academic one. Scientists >and engineers don't hae time to read arcane >documentation about trivia in some other field. Well, I am a scientist and I have to read many papers every month, and I found that reading Goldberg's paper was beneficial. But if Grcar doesn't have time to read one paper where does he get the time to write complaints about IEEE arithmetic for NA Digest? It may be easier to complain than it is to educate yourself, but then why bother asking for documentation on IEEE arithmetic in the first place. And why waste the bandwidth on the network if you are going to be churlish when someone responds with a reference that you were trying to find? Also, I think that the people at Motorola, Intel, Sun, the Free Software Foundation, Symantec, etc., etc., who implement IEEE arithmetic in hardware and software, will be quite surprised to hear that they don't have real jobs. I see two problems with IEEE arithmetic, neither one inherent to the IEEE standard. First, there is a great deal of misinformation floating around about it (i.e. getting a NaN means that you're dividing by zero). Probably the worst example I've seen of misinformation about IEEE arithmetic is P.J. Plauger's book __The Standard C Library__. He makes some ambiguous comments about IEEE arithmetic in his chapter on the ANSI C math library that lead you to believe that ANSI C requires IEEE arithmetic. It doesn't, and he should have explicitly said so instead of making misleading comments. Goldberg's paper is the cure for this problem. The second problem is that examples of how to use the features of IEEE arithmetic in C and Fortran source code are not readily available. Unfortunately, Goldberg's paper is of no help here. Does anyone have any examples of profitable use of IEEE arithmetic in their source code that they can share? Dean Schulze ------------------------------ From: William E. Schiesser Date: Fri, 24 Sep 1993 13:46:31 EDT Subject: Code for Cubic Schroedinger Equation A complete Fortran code for the method of lines (MOL) integration of the cubic Schroedinger equation (CSE) is available (on a DOS-formatted diskette). This code demonstrates the solitons of the CSE, and gives a comparion of the MOL solution with the analytical solution for the one-soliton case. A paper pertaining to the CSE, the analytical solution and the MOL approximation is included with the code. Requests should be sent to: W. E. Schiesser Iacocca Hall, D307 Lehigh University 111 Research Drive Bethlehem, PA 18015 USA (215) 758-4264 (office) (215) 758-5057 (fax) wes1@lehigh.edu (Internet) ------------------------------ From: Bert Buckley Date: Sat, 25 Sep 93 15:46:38 -0700 Subject: Fortran 90 Introduction and Conversion A preprint of a pair of papers which have been accepted for publication in TOMS is available via anonymous ftp. The papers are 1. Conversion to Fortran 90: A Case Study This describes the conversion of the published TOMS Algorithm 630 from Fortran 77 to Fortran 90. The paper can serve as an introduction to many of the features of Fortran 90. As well, it discusses some of the issues involved in doing a conversion. 2. A Fortran 90 code for Unconstrained Nonlinear Minimization This is the converted version of Algorithm 630. It makes extensive use of many of the new features of Fortran 90. The code for the Fortran 90 algorithm is also available. Further details are available in the readme file on the ftp server. The papers are stored as postscript files. Readers who do not have access to anonymous ftp or who cannot conveniently handle postscript files may directly contact the author: Bert Buckley bbuckley@agb.royalroads.ca The files are available in the directory pub/software/bbuckley/alg999 on the ftp server ftp.royalroads.ca Comments are welcome. ------------------------------ From: E. Smith Date: Thu, 23 Sep 93 13:11 +0100 (BST) Subject: One day MATLAB Course at RMCS (Cranfield) One day Course MATLAB: a tutorial introduction 1 WHEN AND WHERE 16 November 1993 Applied Mathematics and Operational Research Group, Royal Military College of Science. 2 PARTICIPANTS The course is specifically designed for scientists, engineers and lecturers who either use routine numerical/matrix calculations in their specific application area or teach the subject at various levels. No specialist knowledge of programming or computer science is required, but participants are expected to have basic knowledge of computing and to be educated to degree level. 3 COURSE MATERIAL Every participant will receive a comprehensive set of lecture notes illustrating various features of MATLAB. 4 COURSE OVERVIEW The course provides hands-on experience of the MATLAB package, and will be followed by an in-depth presentation of case studies which deal with specialist topics from Signal Processing, Control System Design, Numerical Computations, Splines and Optimization. Most of the afternoon is devoted to tutorials and the participants will have ample opportunity to work on a specialist topic of their choice. 5 COURSE STAFF The course lectures will be given by the teaching and research staff of the Applied Mathematics and Operational Research Group under the direction of Dr. Venkat V.S.S. Sastry. 6 COURSE FEES The fee of #150 (pounds) includes morning coffee, afternoon tea, a copy of lecture notes and all computer facilities used. (Academic staff: #112.50 (pounds), research students: #75 (pounds)). Cheques are payable to RMCS(Cranfield). 7 BOOKING To reserve a place on the course please write to Mrs. E. Smith, Applied Mathematics and Operational Research Group, RMCS, Shrivenham, Swindon, Wilts., SN6 8LA or Tel: (0793) 785317 or Fax: (0793) 782179 or e-mail: smithe@uk.ac.cran.rmcs ------------------------------ From: Kerry Anne Landman Date: Fri, 24 Sep 93 09:42:23 EST Subject: Mathematics-in-Industry Study Group MATHEMATICS-IN-INDUSTRY STUDY GROUP 31 January -- 4 February 1994 at the University of Newcastle The tenth annual Mathematics-in-Industry Study Group (MISG) will be held at the University of Newcastle from Monday January 31 to Friday February 4, 1994. This meeting will be the first under the aegis of the Department of Mathematics at the University of Melbourne. This department has recently become the new sponsor of the MISG for a period of three to five years, with Dr Kerry A. Landman as director. The MISG brings mathematicians together to work exclusively on real-life problems facing business and industry. Since 1984 the Study Group has worked on 63 problems from 39 organisations, with participating companies ranging from small electronics firms to mining giants. Past problems include differential equation modelling, operations research, software design, computational problems, communication theory, statistical problems, and more. The applications have spanned Australian industry. The major prerequisites are that the problems should be of practical importance and amenable to mathematical modelling. The Study Group is an opportunity for mathematicians to apply their skills to real problems that present exciting challenges and new research topics, and for industry to explore the problem solving power of mathematics. The format of the Study Group will remain as in recent years. On the first day, a set of scientific problems of industrial origin will be described by industrial representatives and the assigned problem moderators. The mathematicians will then work collaboratively with the industry participants on the problems for three days before presenting their results on the final day. Please note the following items - Financial support will be available on a needs basis for students, unemployed mathematicians, and academic mathematicians. This will be in the form of grants for travel costs and a contribution towards accommodation costs. - The MISG is keen to ensure greater participation by young mathematicians, including those at Honours level. Seminars to describe the Study Group and to encourage participation will be given at a number of universities in spring this year. - The MISG is hopeful that the Study Group will stimulate Honours projects and eventually higher degrees; sponsorship for these projects will be sought from major companies. - A student technical essay competition with prize money from a sponsor is again planned to be held in association with the Study Group. - If you have any suggestion for suitable Study Group problems, or industrial contacts who may have a problem for the Study group, please get in touch with Kerry Landman To have your name entered on the mailing list for the next circular, contact Dr Kerry Anne Landman Department of Mathematics, University of Melbourne, Parkville, Victoria 3052 Australia, ph (03) 344 6762, fax (03) 344 4599 or email misg@mundoe.maths.mu.oz.au ------------------------------ From: Harriet Cohen Date: Fri, 24 Sep 93 10:57:25 EDT Subject: Gelfand Celebration This is a preliminary announcement of the next Statistical Mechanics Meeting which will take place at Rutgers on December 15th, 16th, and 17th, 1993. A complete program will be sent out in the middle of October. The tentative list of speakers includes: M. Avellaneda, N. Berker, W. Bialek, O. Biham, E. Bodenschatz, H. Brezis, L. Bunimovich, P. Cvitanovic, R. Goldstein, D. Haldane, P. Hohenberg, A. Karma, H. Koch, P. Kolodner, A. Majda, S. Orszag, Y. Pomeau, D. Tsui, D. Vanderbilt MEANWHILE, I WOULD APPRECIATE IF YOU WOULD LET ME KNOW IF YOU RECEIVE MORE COPIES OF THIS ANNOUNCEMENT THAN YOU WANT. [Even one might be too many.] Also, if you know of someone who should be on the email-mailing list but is not, please let me know. If I do not hear from you, I will assume that everything is okay and will use the same address for the announcement. With best wishes, Joel Professor Joel L. Lebowitz: ARPANET: lebowitz@math.rutgers.edu BITNET: LEBOWITZ@ZODIAC P.S. There will be a conference at Rutgers, October 24 - 27, 1993, in celebration of Israel M. Gelfand's 80th Birthday. Speakers listed are: K.Aomoto, A.Connes, I.Frenkel, M.Gromov, L.Jeffrey, M.Jimbo, M.Kapranov, D.Kazhdan, B.Kontsevich, B.Kostant, R.MacPherson, O.Mathieu, C.Moeglin, A.Polyakov, I.Singer, E.Witten, S.-T. Yau and A.Zamolodchikov For more information, please contact: Mary Anne Jablonski, Administrative Assistant Gelfand Conference RUTGERS - Department of Mathematics, Hill Center, Busch Campus, New Brunswick, NJ 08903, USA email:gelfandc@math.rutgers.edu Phone:(908)932-3921 FAX:(908)932-5530 ------------------------------ From: Omar Hamed Date: Wed, 22 Sep 93 12:39:19 +0000 (SLT) Subject: Postions at King Saud University We would like to announce the need of our department to hire two professors in mathematics with high calibre of research and academic performence specially with the experience of supervision of PH.D. students. The jobs are going to be available in January, 1994, with very attractive, tax exempt salaries and allowances for children's education and provision of free accomodation and health care beside other benefits. Interseted persons should send their c.v. with recent photo to: Mathematics Department College of Science King Saud University P.O.Box 2455 Riyadh 11451 Saudi Arabia For further information you may send e-mail to f40m033@saksu00 (bitnet) or fax us at ++ 966 1 4676512 ------------------------------ From: Gavin Pringle Date: Wed, 22 Sep 93 17:01:38 +0100 (BST) Subject: Two Post-Doctoral Positions at Napier University, SCOTLAND (I am posting this as a favour, please do not reply to this email address. Thanks, Gavin) NAPIER UNIVERSITY, EDINBURGH, SCOTLAND DEPARTMENT OF MATHEMATICS POST DOCTORAL RESEARCH ASSISTANTS (2 POSTS) 12,638 - 14,962 pounds (under review) The Mathematics Department, as a result of recent funding of SHEFC, is able to announce the appointment of two-post-doctoral Research Assistants, for a period of two years initially, to work with the Computational Applied Mathematics Group, whose interests include Engineering Geometry, Solar Magnetohydrodynamics, Rational Approximation Theory, Computational Fluid Dynamics and the Computational N-body problem. In particular, one Assistant will concentrate on the Computational N-body problem (including parallel implementation) in the context of Fluid Dynamics. It is expected that part of the duties of the second Assistant will involve numerical problems associated with Rational Approximants, although no prior expertise in this topic is required. Substantial expertise in Computational Mathematics would be an advantage in each case. The successful candidates will be expected to have, or to have submitted for, Ph.D degrees upon taking up their appointments. Appointment will normally be made within the lowest quartile of the salary scale. Application forms and further particulars are available from the Head of Personnel services, Napier University, 219 Colinton Road, EDINBURGH, EH14 1DJ, SCOTLAND. Tel: +44 31 455 4288 or +44 31 455 4564. Closing Date: Friday 15 October 1993. ------------------------------ From: Stanley Osher Date: Wed, 22 Sep 93 16:50:58 PDT Subject: Research and Development Programmer Positions At rapidly growing numerical image processing company. Bachelor or Masters degree, excellent C-programming, Unix/systems level, background in algorithms. Signal/image processing, graphics, numerics, windows and windows 3.1 development. Excellent computing/scientific environment. Hardware understanding is desirable. Please mail resume to: 2800-28th Street, Suite 101, Santa Monica, CA 90405 Stan Osher ------------------------------ From: Luis Mandel Date: Fri, 24 Sep 1993 10:41:45 +0200 (MET DST) Subject: Faculty Positions at SUNY-Buffalo STATE UNIVERSITY OF NEW YORK AT BUFFALO Faculty Positions The Department of Computer Science seeks candidates for faculty positions at the Assistant or Associate Professor level. We will consider only those candidates who demonstrate exceedingly high research promise. We are seeking candidates in applied and ex- perimental areas of computer science as well as candidates who will collaborate with researchers in other disciplines. We are especially keen on attracting faculty in parallel computing and systems in order to continue to build our current base in those areas. The department currently has 15 tenure-track faculty, 3 full-time lecturers, and 9 research and adjunct faculty members. Primary research areas include: artificial intelligence, complexity theory, computer vision, numerical linear algebra, parallel algo- rithms, pattern recognition, programming languages, systems and VLSI. Department members are actively engaged in interdisci- plinary research with the Graduate Group in High Performance Com- puting, Cognitive Science Center, Medical Foundation of Buffalo, NSF National Center for Geographic Information and Analysis, and the USPS Center of Excellence for Document Analysis and Recogni- tion. Address applications, including a cover letter, curriculum vitae, a one-page research statement, and names and addresses of three references to: Professor Anthony Ralston; Chair, Recruiting Com- mittee; 226 Bell Hall; Department of Computer Science; State University of New York; Buffalo, NY 14260-2000. E-mail: ralston@cs.buffalo.edu. Phone: (716) 645-3180. FAX: (716) 645- 3464. SUNY is an Equal Opportunity/Affirmative Action employer. ------------------------------ From: Richard Brualdi Date: Thu, 23 Sep 1993 08:28:07 -0500 (CDT) Subject: Contents, Linear Algebra and its Applciations LINEAR ALGEBRA AND ITS APPLICATIONS Volume 193 Eduardo Marques de Sa (Coimbra, Portugal) and Maria-Jose Sodupe (Bilbao, Espana) Characterizations of *Orthant-Monotonic Norms 1 Gerhard Schmeisser (Erlangen, Germany) A Real Symmetric Tridiagonal Matrix With a Given Characteristic Polynomial 11 Dietrich von Rosen (Uppsala, Sweden) Homogeneous Matrix Equations and Multivariate Linear Models 19 John A. Kaliski and Yinyu Ye (Iowa City, Iowa) An Extension of the Potential Reduction Algorithm for Linear Complementarity Problems With Some Priority Goals 35 Zinovy Reichstein (Berkeley, California) On Automorphisms of Matrix Invariants Induced From the Trace Ring 51 Karla Rost (Chemnitz, Germany) Generalized Lyapunov Equations, Matrices With Displacement Structure, and Generalized Bezoutians 75 M. C. Gouveia (Coimbra, Portugal) Generalized Invertibility of Hankel and Toeplitz Matrices 95 L. J. Landau and R. F. Streater (London, England) On Birkhoff's Theorem for Doubly Stochastic Completely Positive Maps of Matrix Algebras 107 Karlheinz Grochenig (Storrs, Connecticut) A Discrete Theory of Irregular Sampling 129 Karla Rost (Chemnitz, Germany) Generalized Companion Matrices and Matrix Representations for Generalized Bezoutians 151 Vlad Ionescu and Martin Weiss (Bucharest, Romania) Continuous and Discrete-Time Riccati Theory: A Popov-Function Approach 173 R. Kit Kittappa (Millersville, Pennsylvania) A Representation of the Solution of the nth Order Linear Difference Equation With Variable Coefficients 211 Author Index 223 Special Issues in Progress 1. Proceedings of the Second Conference of the International Linear Algebra Society at Lisbon; special editors are J. A. Dias da Silva, Chi-Kwong Li, and Graciano de Oliveira. To appear as Volume 197, January 15, 1994. 2. Special Issue Honoring Ingram Olkin: special editors are Friedrich Pukelsheim, George P. H. Styan, Henry Wolkowicz, and Ion Zaballa. To appear as Volume 199, March 1, 1994. 3. Special Issue Honoring Marvin Marcus; special editors are Bryan E. Cain, Moshe Goldberg, Robert Grone, and Nicholas J. Higham. Submission deadline: December 31, 1992. Details in Volume 161, January 15, 1992. 4. Linear Systems and Control, Third Special Issue; special editors are A. C. Antoulas, P. A. Fuhrmann, M. L. J. Hautus, and Y. Yamamoto. Submission deadline: November 30, 1992. Details in Volumes 162-164, February 1992. 5. Special Issue Honoring Chandler Davis; special editors are Rajendra Bhatia, Shmuel Friedland, and Peter Rosenthal. Submission deadline: March 31, 1993. Details in Volume 169, May 1992. 6. Proceedings of the Third Conference of the International Linear Algebra Society at Pensacola; special editors are Dianne P. O'Leary, Leiba Rodman, and Helene Shapiro. Submission deadline: June 30, 1993. Details provided with the conference announcement. 7. Proceedings of the conference ``Matrices and Graphs'' in honor of John Maybee's 65th birthday, held at Boulder, Colorado, May 7, 8, 1993. Special editors: C. R. Johnson and J. R. Lundgren. Submission deadline: August 31, 1993. Details provided with the conference announcement. 8. Fourth Special Issue on Linear Algebra and Statistics; special editors are Jeffrey J. Hunter, Simo Puntanen, and George P. H. Styan. Submission deadline: June 30, 1993. Details in Volume 177, December 1992. 9. Proceedings of the workshop ``Nonnegative Matrices, Applications and Generalizations'' and the Eighth Haifa Matrix Theory conference held at Haifa, Israel, May 31-June 4 and June 7-June 10, 1993, respectively. Special editors: S. Friedland, D. Hershkowitz, and R. Loewy. Submission deadline: September 15, 1993. Details provided with the conference announcement. 10. Special Issue Honoring Miroslav Fiedler and Vlastimil Ptak; special editors are Wayne Barrett, Angelika Bunse-Gerstner, and Nicholas Young. Submission deadline: August 31, 1993. Details in Volume 179. 11. Proceedings of the Fourth Conference of the International Linear Algebra Society at Rotterdam; special editors are Harm Bart, Ludwig Elsner, and Andre Ran. Submission deadline November 30, 1994. Details provided with the conference announcement. 12. Special issue honoring J. J. Seidel; special editors are Aart Blokhuis, Willem H. Haemers and Alan J. Hoffman. Submission deadline: August 30, 1994. Details in Volume 193, November 1, 1993. Special issues are available to individuals at a reduced rate. For further information, please contact Yusuf Guvenc, Journals Customer Service, Elsevier Science Publishing Co., 655 Avenue of the Americas, New York, NY 10010; Tel. 212-633-3955; Fax 212-633-3990. ------------------------------ From: SIAM Date: Fri, 24 Sep 93 12:03:06 EST Subject: Contents, SIAM Mathematical Analysis Table of Contents SIAM J. Math. Anal. January 1994, Vol. 25, No. 1 On Nonstationary Motion of a Compressible Barotropic Viscous Capillary Fluid Bounded by a Free Surface W.M. Zajaczkowski Solutions for Two Dimensional System for Materials of Korteweg Type Harumi Hattori and Dening Li Instability of Planar Interfaces in Reaction Diffusion Systems Masaharu Taniguchi and Yasumasa Nishiura Tikhonov Regularization for Finitely and Infinitely Smoothing Operators B.A. Mair Regularizing Microscopes and Rivers Marc Diener A New Standard Isometry of Developable Surfaces in CAD/CAM Erwin Kreyszig Existence of a Homoclinic Orbit of the Lorenz System by Precise Shooting Brian Hassard and Jianhe Zhang The Addition Formula for Continuous q-Legendre Polynomials and Associated Spherical Elements on the SU(2) Quantum Group Related to Askey-Wilson Polynomial s H.T. Koelink Basic Hypergeometric Functions and the Borel-Weil Construction for UQ(3) M.A. Lohe and L.C. Biedenharn ------------------------------ From: Iain Duff Date: Sat, 25 Sep 93 08:19:52 +0100 (BST) Subject: Contents, IMA Numerical Analysis IMA Journal Numerical Analysis. Volume 13, Number 4. October 1993. Pruess S 2 Shape preserving C cubic spline interpolation. Chandler-Wilde S N Some uniform and stability and convergence results for integral equations on the real line and projection methods for their solution. Achdou Y and Pironneau O The chi-method for the Navier-Stokes equation. Binding P A, Browne P J, and Ji X A numerical method using the Prufer transformation for the calculation of eigenpairs of two parameter Sturm-Liouville problems. Fuchs P M A global negative result on algebraic stability and a special positive result on linear stability of generalized IRK methods. Sloan I H, Dat Tran, and Fairweather G A fourth-order cubic spline method for linear second-order two-point boundary value problems. ------------------------------ End of NA Digest ************************** -------