HDTenEightyP/NetNews huggingface dataset:
From: ingber@umiacs.umd.edu (Lester Ingber)
Newsgroups: sci.math.num-analysis
Subject: Re: Nonlinear systems of equations
Date: 19 Dec 92 18:49:22 GMT
Organization: UMIACS, University of Maryland, College Park, MD 20742


Perhaps I can be of some help since I developed the code.  It might be
of use to the general community if you would post all your replies.

In article <1992Dec18.193249.22059@murdoch.acc.Virginia.EDU> you write:
>Hi,
>
>  I am thinking about using VFSR (A Simulated Reannealing code from
>  NETLIB) on my problem which is to minimize a nonlinear function
>  subject to nonlinear inequality constraints. In addition, the
>  constraints are nondifferentiable wrt the parameters although
>  the objective function is differentiable.
>
>  I dont know anything about the theory behind VFSR and I wanted to
>  use it as a black box. My question is:
>  Can I use VFSR for this purpose? 

I've used similar constraints.  One case that was published was in
%A L. Ingber
%T Statistical mechanics of neocortical interactions:
A scaling paradigm applied to electroencephalography
%J Phys. Rev. A
%N 6
%V 44
%P 4017-4060
%D 1991

>  It is my understanding that VFSR requires the user to supply a
>  subroutine which returns with the cost fuction and a flag which
>  determines whether the (parameter) state at that point violates some
>  constraints or not. Apparently, if the constraints are violated, a
>  new set of trial parameters are generated. Is there a guarantee that
>  the new set of trial parameters are valid? If not, then it would seem
>  to me that you could get stuck at such points.

Yes, you could get stuck if the space being sampled had no other
good points.  I.e., on return from the cost function with an invalid
flag, a new random point is selected.  If you think your space might be
flooded with invalid points, you might set up some adaptive schedule to
change the ranges during the run.  I have done this in other problems,
e.g., by passing the parameter maximum and minimum values along with
the cost function.  On lines 133-135 in vfsr.h I have placed some
comment for this possibility.

>  I would appreciate any information on the issues above.
>  Thanks.
>
>
>  -sriram srinivasan
>   (ss4i@kelvin.seas.virginia.edu)

Lester
-- 
|| Prof. Lester Ingber                         [10ATT]0-700-L-INGBER ||
|| Lester Ingber Research                        Fax: 0-700-4-INGBER ||
|| P.O. Box 857                           Voice Mail: 1-800-VMAIL-LI ||
|| McLean, VA  22101                EMail: ingber@alumni.caltech.edu ||
Newsgroups: sci.math.num-analysis,sci.engr.mech
From: a722756@roper.mc.ti.com (W. Donald Rolph)
Subject: Re: Nonlinear systems of equations
Originator: a722756@roper.mc.ti.com
Organization: Texas Instruments / Attleboro Mass / USA
Date: Thu, 17 Dec 1992 22:30:36 GMT


It depends on how you are failing, and the nature of the nonlinearity.  If the
solution is continuous in slope,  you might explore using full newton-raphson
with line search.  The line search provides a dynamically computed over/under
acceleration factor which can sometimes stablize certain problems, and can
sometimes speed up the rate of convergence.
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
Newsgroups: sci.math.num-analysis
From: dhart@bronze.ucs.indiana.edu (dave hart)
Subject: Re: Chebyshev vs. Least Squares Polynomials
Organization: Indiana University
Date: Thu, 17 Dec 1992 17:23:02 GMT

In article <1992Dec2.170529.24897@news2.cis.umn.edu> 
 pires@milli.cs.umn.edu (Luiz S. Pires) writes:
>
> Please forgive me if these are trivial questions. I am not a mathematician.
>
> When approximating a function with a polynomial, should one use the
> least squares approximating polynomial or a Chebyshev polynomial?
>
> Does L.S. minimize overall error and Chebyshev minimize maximum error?
>
> Which is more commonly used in approximating transcendental functions?
>
> Also, when are other methods (eg Rational Chebyshev approximation) used?
> Any good references?
>
	The idea of "least squares" is the same as orthogonal projection--
"the shortest distance between a point and a line..."  This is tied up
with geometry, ie a metric [inner product].  The Chebyshev polynomials
form an orthonormal basis [q.v.] for the L^2 metric;  other functions
form orthonormal bases for other metrics.

	I cannot guess what you mean by "least squares approximating
polynomial," but approximation implies a notion of distance; best 
approximation(s) follow from that.

Newsgroups: sci.math.num-analysis,sci.engr.mech
From: muthu@mecad.uta.edu (S.N. Muthukrishnan)
Subject: Mesh generation schemes (was Re: FEA help!)
Reply-To: muthu@mecad.uta.edu
Organization: University of Texas at Arlington
Date: Wed, 16 Dec 1992 23:38:49 GMT

In article <1992Dec16.164320@roper.mc.ti.com> a722756@roper.mc.ti.com (W.  
Donald Rolph) writes:
> 
> In article <1992Dec15.225304.11571@porthos.cc.bellcore.com>,  
bhatia@bigcat.bellcore.com (Raman Bhatia) writes:
> |> In article <1992Dec14.181725@roper.mc.ti.com>, a722756@roper.mc.ti.com (W.  
Donald Rolph) writes:
> |> |> 
> |> |> This is by no means a trivial question.  You must first ask the class  
of FEA to
> |> 
> |> Raman Bhatia (bhatia@bigcat.bellcore.com)
> 
> I have an interesting proof that show that the triangularization schemes all
> violate controllability theory at any given point in the model.  The effect  
is
> that in 3D, with typcial dimensions in orthoganoal directions which have
> significantly different scales (1.4 in thick by 5 inches wide by 20 inches  
long),
> the triangularizations schemes produce highly unsatisfactory meshes for many
> classes of problems, particularly nonlinear problems.  It is interesting to  
note
> the CFd codes are gradually adopting Laplacian mesh generation schemes,  
schemes
> which were popular as early as 1960 for fea.  It should be noted that CFD can  
be
> very suseptible to poor meshes, and perhaps there is a message here!  
> 
> I have simplistic proof which suggests that Laplacian mapping produces the
> "optimal mesh" fgor certain classes of problems, based on the givven mesh  
density
> provided as input.  I also have s ample program which generates hexahedral  
meshes
> off an arbitray boundary rep trimmed nurb solid model, demonstrating that the
> technogy is feasible, and the input is not overly burdensome.
> -- 
> 
> Regards.
>  
> Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263

Yes. That is true. However, it is possible to control the "aspect ratio"
in meshes, be it 2-D or 3-D. I assume that in 2-D it is triangular elements
and in 3-D it is tetrahedral elements. Laplacian mapping is one way to improve
the aspect ratio. There is also a very good paper in mesh smoothing. I shall
mail it or post it if there is any interest. I don't remember it off my head.

-Muthu
--
---------------------------------------------------------------------
S. N. Muthukrishnan                | muthu@mecad.uta.edu |
Dept.of Mech. Engg., UT, Arlington | muthu@fem.uta.edu   |(NeXT mail)
Arlington, TX  76019               | muthu@newton.uta.edu|
_____________________________________________________________________
"Perception is a reality. Reality is myth"
---------------------------------------------------------------------
Newsgroups: sci.math.num-analysis
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Subject: RE: finite element limitations??
Originator: rcpshdb@dutrun2.tudelft.nl
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Date: Wed, 16 Dec 1992 09:58:58 GMT

In article <1992Dec15.092610@roper.mc.ti.com> W. Donald Rolph:
> You are indeed confused.

I am never confused about these things.

>  Any approximation procedure must have trial functions.

No, no.

> In traditional FEA they are called shape functions, choose the name you like
> but all FEA procedures must have some approximating function over the domain
> space.

*All* procedures?

Gee, how then could I calculate the temperature distribution in a sodium pump
by an direct FE method, WITHOUT ever needing such "approximating functions"?
(An accompanying article is called "Multimode Heat Transfer in Sodium Pumps".
It is found in the Proceedings of the 0Xth Conference on Numerical Methods
in Thermal, or whatever, Problems, Swansea U.K., 197X. Look it up ;-)

I also did some work on pure advection without shape functions. But who am I?
There exists CFD code named "FLOTRAN", which has a "streamline upwind" scheme
in it, derived directly without shape functions. It's all in my SUNA articles.

And almost any Finite Element book starts with the assembly of resistor-like
elements, without approximations. The element-matrix of an electrical resistor
is derived directly by applying the laws of Ohm and Kirchhoff, giving:

         | +1/R  -1/R |
         | -1/R  +1/R |

Just define the connectivity (no coordinates) of your resistor-network,
two voltages as a boundary condition, and go ...
-- 
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood
Newsgroups: sci.math.num-analysis
From: lallu@cc.utah.edu (Upmanu Lall, Civil Eng)
Subject: shortest chain on a sphere / shortest around the world trip
News-Software: VAX/VMS VNEWS 1.4-b1  
Organization: University of Utah Computer Center
Date: 13 Dec 1992 16:44 MST  


Does anyone know if this is a solved problem ?
 It seems like it ought to be.
There are n arbitrary locations on the surface of the earth, each of which must
be visited
once and only once, starting from any one of them.
At the end of the trip one is back at the starting point.
It seems to me that the choice of the starting point does not matter,
i.e. there is a unique shortest chain, and you should be able to find it
irrespective of where you start on it.
Is this assertion correct ?
Is there an algorithm for determining the shortest path efficiently,
and the second shortest, etc ?
I recall from a network optimization class taken 2 decades ago, that
a chain network is the dual problem to a flow network
Can this be used to setup the problem such that one could
establish whether the primal or the dual is easier to solve, and to then solve
the problem ?
Distances are Euclidean on the sphere (or the Earth). LAts and Longs of the
points are known , and the number of locations is also known a priori (i.e.
this is not a stochastic process).

Any references, or suggestions to bodies of literature shall be appreciated.
Deterministic  or Monte Carlo techniques would be fine.


Thanks very much for your help
Please respond to
ulall@pub.uwrl.usu.edu  
Newsgroups: sci.math.num-analysis
From: kdeacon@nmsu.edu (DEACON)
Subject: Multipole algorithms?
Organization: NMSU Computer Science
Distribution: sci.math.num-analysis
Date: Fri, 18 Dec 1992 00:35:16 GMT

Does anyone have any references to the multipole near neighbor
algorithm?
Thanks,Keith
Newsgroups: sci.math.num-analysis
From: seewald@iw.uni-hannover.de (Georg Seewald)
Subject: GEAR algorithm
Reply-To: seewald@iw.uni-hannover.de
Organization: Institut fuer Werkstoffkunde, Uni Hannover
Date: Fri, 18 Dec 1992 11:55:31 GMT


Hi, 
I am programming in C and looking for the "GEAR algorithm" to solve 
systems of Differential Equations. I am interesting in public domain 
source code package. 
Or do know better numerical algorithms?
Can anybody help me ?

Thanks in advance,
                   Georg

Newsgroups: sci.math.num-analysis,sci.math.stat,sci.math.symbolic
From: davidm@questor.Rational.COM (David Moore)
Subject: Re: mathematica stepwise regression package
Keywords: mathematica stepwise regression
Organization: Rational
Date: Tue, 15 Dec 1992 20:30:00 GMT

andersh@maths.lth.se (Anders Holtsberg) writes:


>gary mcclelland mcclella@yertle.colorado.edu:

>You mean what is bad? Let's say we want to make predictions.
>Is the idea of using a subset of the predictors that is bad or
>do you mean the stepwise way to pick them? If the latter: do
>you know any better way (except trying all combinations)?

What about canonical co-ordinates? I always felt that this method
gave you a better feel for what might be going on, but it is
years since I did any econometrics.

What are the drawbacks of this method, assuming you use it
intelligently. (Obviously, the comments about the need for
thought and intuition apply equally well whatever method
you use).

Newsgroups: sci.math.num-analysis
From: yaonan@duteinp.et.tudelft.nl (Yaonan Zhang)
Subject: NEEDED: algorithm for quadratic programming 
Keywords: request for quadratic program
Organization: Delft University of Technology, Dept. of Electrical Engineering
Date: Fri, 18 Dec 1992 14:43:30 GMT

Hi world,

I am looking for the program (C preferred) of quadratic programing.
Any pointer or seggestion is greatly appreciated.

Cheers!


Yaonan

Newsgroups: sci.math.num-analysis
From: andy@laguna.santafe.edu (Andy Mell)
Subject: Re: finite element limitations??
Date: Sat, 19 Dec 92 02:17:09 GMT
Organization: Santa Fe Institute

In article <1992Dec16.095858.5370@dutrun2.tudelft.nl> rcpshdb@dutrun2.tudelft.nl (Han de Bruijn) writes:
>
>Gee, how then could I calculate the temperature distribution in a sodium pump
>by an direct FE method, WITHOUT ever needing such "approximating functions"?
>(An accompanying article is called "Multimode Heat Transfer in Sodium Pumps".

A much better method for this type of heat problem is to use the Transmission
Line Matrix method (TLM) - W.J.R. Hoefer "Huygens and the Computer",
Proc. IEEE, vol. 79, no. 11, October 1991.

I'd be very interested to hear from anyone else who is working with TLM,
please do email me, it might be useful to have some sort of discussion forum.

Regards

Andy

-- 
-------------------------------------------------------------------------------
Andy Mell, Applied Physics Department,  |   andy@santafe.edu
University of Hull, Hull, HU7 6RX       |   amell@nyx.cs.du.edu
England.                                |   torq@gnu.ai.mit.edu
Newsgroups: sci.math.num-analysis
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Subject: RE: finite element limitations??
Originator: rcpshdb@dutrun2.tudelft.nl
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Date: Mon, 14 Dec 1992 13:35:30 GMT

In article <1992Dec11.184635@roper.mc.ti.com> Don Rolph:
> Excellent response - so long as one chooses appropriate shape function
> (ah now there's the rub!!) FEA methods dont "fail"!

Sorry Don. The gist of an FEA method is definitely _not_ in an appropriate
choice of the shape functions. There exist many working FEM methods which
do not even refer to something like a shape function, yet are successfull.
-- 
* Han de Bruijn; Applications&Graphics | "A little bit of Physics * No
* TUD Computing Centre; P.O. Box 354   | would be NO idleness in  * Oil
* 2600 AJ  Delft; The Netherlands.     | Mathematics" (HdB).      * for
* E-mail: Han.deBruijn@RC.TUDelft.NL --| Fax: +31 15 78 37 87 ----* Blood
Newsgroups: sci.math.num-analysis
From: bhatia@bigcat.bellcore.com (Raman Bhatia)
Subject: Re: FEA help!
Organization: Bellcore, Livingston, NJ
Date: Tue, 15 Dec 92 22:53:04 GMT

In article <1992Dec14.181725@roper.mc.ti.com>, a722756@roper.mc.ti.com (W. Donald Rolph) writes:
|> 
|> This is by no means a trivial question.  You must first ask the class of FEA to
.......
|>  
|> It is also imperative that you have  a good mesh generator and graphical post
|> processor.  The difference between a good mesh generator and an ok mesh generator
|> is about 20X in productivity.
........
|> 
|> A typical system ight cost the following (rough costs):
|>   
|>      1)  FEA program $5K - $10K year
|> 
|>      2)  Mesh generator $10K yr
|> 
|>      3)  Workstation:  $25K
|>  
|>      4)  color postscript printer $5K to $10K
|> 
|> This is nearly a 1/3 cost reduction and 10,000 performance improvement over the
|> system I put together is 1983!
|> 
|> 
|> -- 
|> 
|> Regards.
|>  
|> Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263

Thanks for a very informative post. Here's another question. What are
the latest in Mesh Generation techniques? 

The last time I dabbed in FEM, the finite-octree (w Delaunay triangulation) 
appeared the most promising but there were some concerns about robustness. 
I know that companies like PDA engineering (developers of PATRAN) were 
looking at "paving". What is the current status of these two techniques, 
and are there any new ones out there (AI approaches etc.)?

Raman Bhatia (bhatia@bigcat.bellcore.com)
Newsgroups: sci.math.num-analysis,sci.math.stat,sci.math.symbolic
From: mcclella@yertle.Colorado.EDU (Gary McClelland)
Subject: Re: mathematica stepwise regression package
Keywords: mathematica stepwise regression
Organization: University of Colorado, Boulder
Date: Tue, 15 Dec 1992 06:09:18 GMT

bjk@edsr.eds.com (Brian Kodl) writes:

>Sorry if this is a frequently asked question, but has anyone
>written a package for Mathematica that does stepwise regression?

>None of the packages that came with Mathematica 2.1 for NeXT 
>include stepwise, forward, or backward regression (at least that
>I could find).

Stepwise regression is a tool of the devil.  Anyone smart enough
to be using Mathematica ought to be able to decide which
questions he or she wants to ask of the data rather than letting
a demonstrably suboptimal stepwise algorithm decide which
questions to ask.  One day there may be good AI programs for
doing statistical analysis, but stepwise regression won't be
among them.

gary mcclelland
univ of colorado
mcclella@yertle.colorado.edu
Newsgroups: sci.math.num-analysis
From: landman@splat.physics.wayne.edu
Subject: FEA help!
Organization: Wayne State University, Detroit
Date: Mon, 14 Dec 1992 21:25:05 GMT


Hello Net.People:


    I have a question about FEA, specifically software and hardware
platforms.  A friend is looking into the possibility of purchasing 
hardware and software to perform FEA calculations for his company.
I was wondering what software exists (commercial and PD) and what
platforms are recommended.

 Thanks

Joe Landman

Landman@splat.physics.wayne.edu
landman@hal.physics.wayne.edu

Newsgroups: sci.math.num-analysis
From: yaonan@duteinp.et.tudelft.nl (Yaonan Zhang)
Subject: Re: numerical integration
Keywords: c codes for num-integration
Organization: Delft University of Technology, Dept. of Electrical Engineering
Date: Fri, 18 Dec 1992 14:41:30 GMT


Newsgroups: sci.math.num-analysis
From: andrew@tasman.cc.utas.edu.au (Andrew S. L. Bainbridge-Smith)
Subject: help wanted: SVD of sparse matrices
Keywords: SVD, sparse matrices
Organization: University of Tasmania, Australia.
Date: Tue, 15 Dec 1992 23:33:15 GMT

Hi,

I'm trying to solve a problem involving the hessian matrix
(second order partical derivatives) of a least squares type
cost function.  In particular I wish to find the spectral
decomposition of this matrix. Currently I have been using 
some fortran code (linpack/eispack) that finds the singular
value decomposition (SVD).  The problem is that some times
the hessian matrix can be sparse and the fortran code never
returns -- ever-thing just hangs up.  So I'm asking for a 
little bit of your time to email me (I don't read this group
often + I'll be away for a few weeks) any suggestions,
helpful advice, solutions etc. I'm not to clue-y on this
(SVD) stuff, but I'll take any sort of help that's offered.

Thanks heaps in advance,  and I'll post a summary.

Andrew.

----
Andrew Bainbridge-Smith
Dept. Electrical & Electronic Eng.
Uni. of Tasmania (Australia)
email: andrew@tasman.cc.utas.edu.au
Newsgroups: sci.math.num-analysis
From: a722756@roper.mc.ti.com (W. Donald Rolph)
Subject: RE: finite element limitations??
Originator: a722756@roper.mc.ti.com
Organization: Texas Instruments / Attleboro Mass / USA
Date: Tue, 15 Dec 1992 14:26:10 GMT


You are indeed confused.  Any approximation p[rocedure must have trial functions.
 In traditional FEA they are called shape functions, choose the name you like but
all FEA procedures musat have some approximating function over the domain space. 
Now we csan talk about how to derive these functions  and how to apply them!
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
From: ravi@bobcat.ent.ohiou.edu (Pattalachinti)
Newsgroups: sci.math.num-analysis
Subject: Re: Nonlinear systems of equations
Date: 19 Dec 92 18:42:54 GMT
Organization: College of Engg. & Tech., Ohio University, Athens, Ohio

In article <92Dec17.174419.27750@acs.ucalgary.ca> mgh@hobbes.phys.ucalgary.ca (Mike Henderson) writes:
>I am trying to solve a set of 8 nonlinear equations in 8 unknowns.
>I have tried the Newton-Raphson technique given in NR (I know
>roughly what the solution should be) but havent had much luck. Are
>there any more robust methods to choose from?
>
>Mike Henderson
>Dept. of Physics and Astronomy
>University of Calgary
>mgh@hobbes.phys.ucalgary.ca
Is it possible in your case to calculate the analytical derivative?
If so ::::---->
I had similar kinda problem.  I had 10 NLE's and 12 variables 
Initially it was not working because of a few errors in the 
calculations of the analytical derivatives required for the 
Jacobian.  You can check the derivatives by calculating the 
numerical derivatives and they must be very close.If there
is large difference in numerical and analytical derivatives then
you are doing some mistake in calculation of the analytical
derivative or may be some other simple problem such as matrix
size.  Check if your analytical and numerical derivatives are same
I hope it will work because it is working in my case when
the numerical and analytical derivatives were very nearly equal(3 digits


ravi
From: reese@vax.oxford.ac.uk
Newsgroups: sci.math.num-analysis
Subject: Shooting to a Fitting Point
Date: 14 Dec 92 17:07:54 GMT
Organization: Oxford University VAX 6620



I wonder if anyone could tell me where I could find a Fortran programme
that does Shooting to a Fitting point (a la Numerical Recipes), for a
boundary value problem with 4 1st order odes, integrating from both 
ends of the range?

Thanks, in advance.

Jason
From: frost@sdsc.edu (Richard Frost)
Newsgroups: sci.math.num-analysis
Subject: Re: function minimization references
Date: 17 Dec 1992 18:07:36 GMT
Organization: San Diego Supercomputer Center @UCSD.
Distribution: world
Reply-To: frost@sdsc.edu
NNTP-Posting-Host: stimpy.sdsc.edu


Gradient decent is fine if you can show there exist only a small
number of extrema in the search domain.

Here are some references which discuss alternative approaches.

SIMULATED ANNEALING:

J.R. Harland and P. Salamon "Simulated annealing: A review of the
thermodynamic approach" Nuclear Physics B (Proc. Suppl.) 5A (1988) pp. 109-115

G. Ruppeiner, et al "Ensemble approach to simulated annealing"
J. Phys. I vol. 1 (1991) 455-470

K.H. Hoffman, et al "Optimal ensemble size for parallel implementations
of simulated annealing" Appl. Math. Lett. vol. 3, no. 3 (1990) 53-56

SIMULATED RE-ANNEALING:

L. Ingber and B. Rosen "Genetic algorithms and very fast simulated
annealing: a comparison" Mathematical and Computer Modelling, 16(11)
1992, 87-100

L. Ingber "Very fast simulated re-annealing" Mathematical and Computer
Modelling, 12(8) 1989, 967-973

GENETIC ALGORITHMS:

K.A. De Jong "Genetic algorithms are NOT function optimizers" in
Foundations of Genetic Algorithms: Proceedings 24-29 July 1992, D.
Whitley (ed.) Morgan Kaufman

D.H. Ackley "An empirical study of bit vector function optimization" in
Genetic Algorithms and Simulated Annealing, L. Davis (ed.) Morgan
Kaufman (1987)


Newsgroups: sci.math.num-analysis
From: wfk@lynx.spa.umn.edu (Bill Ketzeback)
Subject: PCA code wanted
Organization: University of Minnesota
Date: Fri, 18 Dec 1992 17:01:04 GMT



I am looking for some public domain or CHEAP statistical package that
incorporates Principal Component Analysis. I need it to run on a Unix 
system. I would like for it to be able to do matrix calculations up to
m=10 to 15 fast. Therefore it should have the Householder or Givens methods.
Any pointers to where I might locate such code would be helpful. Thanks in
advance.

Bill Ketzeback

wfk@aps1.spa.umn.edu


Newsgroups: sci.math.num-analysis,comp.arch