Newsgroup sci.math.num-analysis 29052

Directory

Hello, 
 it may depend to what extent you are ready to go with this
(resources available, people, time,...)
as a starting point you may see what is the status with pi,
computation, books, digits, results, articles, what's on the
net , records...
all this at http://www.cecm.sfu.ca/pi
see also http://www.cecm.sfu.ca/projects/ISC/resources.html
this will point you to several good links related to numbers
on the net.
Simon Plouffe
Research Associate
CECM, Centre for Experimental & Constructive Mathematics
Simon fraser University
Vancouver, Canada.

Le Compte de Beaudrap (jd@cpsc.ucalgary.ca) wrote:
: On Mon, 2 Dec 1996, Willieum J. Melton wrote:
: 
: 
: > dealing with the number pi.  At some time in the U.S. there was a push by 
: > some state(s) to legislate a rational value for pi. If anyone has any 
: 
: 
: 	Oh, this is priceless! This is exactly like government! Image: You see
: a person up on a platform at a political debate/speech, say a particularly
: well established Republican (or, for my Canadian compatriots, Reform 
: Party member): "Let's cut down on wastly extra decimal places that nobody 
: uses by making new 'rational Pi value' legislation"!!
: 
: Ha! I can just see the next US election: "Vote Al Gore for 3.14" !!
: 
: 
: (Yes, I know Gore is a Democrat, but it rhymed, and that makes up for it!)
The fact that it rhymes would also be the only reason for the politicians
to choose 3.14 and not e.g. 3.1415 (although they would dream up several
other "reasons" for the public, of course). 
-- 
Ulrich Lange                       Dept. of Chemical Engineering
                                   University of Alberta
lange@gpu.srv.ualberta.ca          Edmonton, Alberta, T6G 2G6, Canada

In article <32A5AE19.1637@u.washington.edu>, rons@u.washington.edu wrote:
> Most people doing serious work use one of the commercially available
> matrix class libraries.  
Well, no, many of us are using our own libraries.  In fact I know of no
one using a commercial library with C++.
Lou Pecora
code 6343
Naval Research Lab
Washington  DC  20375
USA
 == My views are not those of the U.S. Navy. ==
------------------------------------------------------------
  Check out the 4th Experimental Chaos Conference Home Page:
  http://natasha.umsl.edu/Exp_Chaos4/
------------------------------------------------------------

In article <583hi7$mkj@b.stat.purdue.edu>,
Herman Rubin  wrote:
>I believe that matrices of the form I+K, K compact, also occur in
>quantum mechanics, and that the determinant is sometimes needed.
Yes.  But physicists, having no sense of restraint, also work with 
det(A) where A is not of the above form.  For example, in quantum 
field theory they often talk about the determinant of the 
Laplacian!  To make sense of this, people have developed a large 
battery of tools for regularizing determinants of operators, using 
zeta function tricks to make sense of divergent sums.  These 
are especially handy in applications of quantum field theory to
topology and vice versa --- the sort of thing Atiyah, Singer, Witten
and company are famous for.

That's an interesting opinion.  Personally, I'd like to see more people
agree with it.  I've had the occasion to replace IMSL functions when the
licensing terms became too onerous in the multiple site world.
Tim

I need to fit a complex function of a real variable to complex measured
data. All algorithms I found so far assume real-valued data. Of course I
could define an SSQ or chi^2 as usual and applay a general minimization
algorithm, as e.g. Nelder-Mead Simplex. Buit I wonder if there are
specialized, more efficient algorithms or even implementations of e.g.
Levenberg-Marquardt?
I would be grateful for any hints, 
=====================================================================
  Gerhard Heinzel                          E-mail:   ghh@mpq.mpg.de
  Max-Planck-Institut fuer Quantenoptik
  Hans-Kopfermann-Str. 1                    Phone: +49(89)32905-268
  D-85748 Garching                                             -252
  Germany                                     Fax: +49(89)32905-200
=====================================================================

>|> Integrate the function exp(-a/sin(x)) for x between 0 and pi/2.
>|> The parameter a is a positive constant.
The Romberg procedure succeeds, as well as the Gauss intergral with some 
interval splitting, as tested with EULER (ftp://am.ku-eichstaett.de/pub):
>gauss("exp(-10/sin(x))",0,pi/2,20)
  1.70152e-05 
>romberg("exp(-10/sin(x))",0,pi/2)
  1.70152e-05 
No problem.
Rene.

Hello,
I would like to know where can I do a PhD , a MSc or a MPhil in
Operational Research , starting in September of 1998.
Thank you.
Please send info to : noses@picasso.lc.ehu.es

Marcus wrote:
> 
> Thank you for your help.  What you have described is exactly what I'm
> currently doing, however its not very robust when noisy data is
> introduced into the algorithm.  Do you know how to do a least squares
> fit of data to the equation of a circle? (I suppose: f(x,y)=x^2+y^2)
> I have a suspicion that this is a multivariate regression problem.  I'm
> rather a newbie to this sort of thing.
This is actually a difficult problems and there
are a number of alternative solutions to this
problem. Below is one "simple" solution. I cannot
quantify the robustness of this compared with other
methods - but comments most welcome.
A generalised circle can be written as
	x^2 + y^2 + d x + e y + f = 0
We wish to minimize (standard linear least squares) for n points
	|| a z - b ||
Where
	a = | x1 y1 1 |
	    | x2 y2 1 |
	    | .  .  1 |
	    | xn yn 1 |
	z = | d |
	    | e |
	    | f |
	b = | -(x1^2 + y1^2) |
	    | -(x2^2 + y2^2) |
	    | -(.    + .   ) |
	    | -(xn^2 + yn^2) |
There are most probably a number of elegant ways to solve
the above - I simply use the routine of Lawson & Hanson*.
If you need more details/references please email me.
*c.l.lawson & r.j.hanson, jet propulsion laboratory, 1973 jun 12
to appear in 'solving least squares problems', prentice-hall 1974
solve least squares problem using algorithm hfti.
____________________________________________
Suhail A Islam
Biomolecular Modelling Laboratory
Imperial Cancer Research Fund, P.O. Box 123
44 Lincoln's Inn Fields, London WC2A 3PX
Tel: (0171) 269 3380, Fax: (0171) 269 3479
email: islam@icrf.icnet.uk
http://www.icnet.uk/bmm/
____________________________________________

Willieum J. Melton (cojomelt@juno.com) wrote:
>For my high school precalc class, a partner and I have to do a report 
>dealing with the number pi.  At some time in the U.S. there was a push by 
>some state(s) to legislate a rational value for pi.
Check the alt.folklore.urban FAQ for more details on this.
--
Richard Herring      |  richard.herring@gecm.com | Speaking for myself
GEC-Marconi Research Centre                      | Not the one on TV.

Marcus wrote:
>Thank you for your help.  What you have described is exactly what I'm 
>currently doing, however its not very robust when noisy data is 
>introduced into the algorithm.  Do you know how to do a least squares 
>fit of data to the equation of a circle? (I suppose: f(x,y)=x^2+y^2)
>I have a suspicion that this is a multivariate regression problem.  I'm 
>rather a newbie to this sort of thing.
>
>Thanks,
>Marc
Hello, it's me again! You told me that my solution to your problem was the
same as the one you were already using. Then I believe that your original
description was a bit wrong, but that won't help you any further. I spent
some time on a least squares method and I found one. I will not repeat the
derivation here since that would give me a headache typing it, but I will
give you the final result:
Suppose that you have datapoints (x1,y1), (x2,y2), ..., (xN,yN). Now form
the following sums:
M00 = N
M10 = x1+x2+...+xN
M20 = x1^2+x2^2+...+xN^2
M30 = x1^3+x2^3+...+xN^3
M01 = y1+y2+...+yN
M11 = x1*y1+x2*y2+...+xN*yN
M21 = x1^2*y1+x2^2*y2+...+xN^2*yN
M02 = y1^2+y2^2+...+yN^2
M12 = x1*y1^2+x2*y2^2+...xN*yN^2
M03 = y1^3+y2^3+...+yN^3
As you can see the first index of M gives you the power of the x's within
the summation, and the second one of the y's.
Now the quantities you want are the center (X,Y) of the circle and it's
radius R. They can be given by:
     ( + M00^2*M02*M30 - M00*M01^2*M30 + M00^2*M12*M02 - M00*M12*M01^2 +     )
     ( - M20*M10*M00*M02 - M00*M02^2*M10 - M00^2*M21*M11 + M00*M21*M01*M10 + )
     ( - M00^2*M03*M11 + M00*M03*M01*M10 + M20*M01*M00*M11 + M02*M01*M00*M11 )
X = ---------------------------------------------------------------------------
              2 * ( + M00^2*M20*M02 - M00*M01^2*M20 - M00*M10^2*M02 + )
                  ( - M00^2*M11^2 + 2*M00*M01*M10*M11                 )
     ( + M00^2*M21*M20 + M00^2*M03*M20 - M20^2*M00*M01 - M00*M20*M01*M02 +     )
     ( - M00*M10^2*M21 - M00*M10^2*M03 - M00^2*M30*M11 - M00^2*M12*M11 +       )
     ( + M00*M10*M11*M20 + M00*M11*M02*M10 + M10*M01*M00*M30 + M01*M10*M00*M12 )
Y = -----------------------------------------------------------------------------
               2 * ( + M00^2*M20*M02 - M00*M01^2*M20 - M00*M10^2*M02 + )
                   ( - M00^2*M11^2 + 2*M00*M01*M10*M11                 )
R^2 = X^2 + Y^2 + M20/M00 + M02/M00 - 2*M10/M00*X - 2*M01/M00*Y
Now I hope I didn't make any mistakes typing it. I've checked this formula for
the data points (2,1), (-3,-4), (1,-2) and (-6,5). It does indeed give X=-3, Y=1
and R=5, as it should. I believe that my derivation is correct. Use it, try it
out and tell me when anything is wrong with it.
More greetings from Dave...

Willieum J. Melton (cojomelt@juno.com) wrote:
>For my high school precalc class, a partner and I have to do a report 
>dealing with the number pi.  
>information about this issue or any other info on the number pi, such as 
>attempts to find the extent of pi's decimal expansion, I would greatly 
>appreciate it. 
If you want a few pages of digits if pi, here's a bit of obfuscated C 
which does the rounds... I can't credit the author, unfortunately. 
int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;for(;d=0,g=c*2;c
-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,f[b]=d%--g,d/=g--,--b;d*=b);}
--
Richard Herring      |  richard.herring@gecm.com | Speaking for myself
GEC-Marconi Research Centre                      | Not the one on TV.

Lee Hon Chor (kwyu) wrote:
> 
> Dear net users,
> 
> I would like to know if there are good reference books for
> sturm sequence (sturm sqquence is for diagonalizing tri-diagonal
> matrices).  Thanks in advance.
> 
> Regards
> HC Lee
Hi,
you also find it in Golub&van; Loan (3rd, NEW edition), section 8.5.2.
If you do not have this book, yet, buy it. It's a great reference.
-- 
Hans D. Mittelmann			http://plato.la.asu.edu/
Arizona State University		Phone: (602) 965-6595
Department of Mathematics		Fax:   (602) 965-0461
Tempe, AZ 85287-1804			email: mittelmann@asu.edu

Any ideas where I could lay my hands on a reasonably generic 
expression for the number of bit operations in the multiplication 
of both floating point and integer values to compare with bit shifts
and other more fundamental operations? Perhaps clock cycles is a better
angle...
I just need something for a theoretical comparison.
Pointers? references?
Thanks, Thomas.

In article <57gjs0$dad@eng-ser1.erg.cuhk.hk>, mcau@ee.cuhk.hk (Au_Yeung_Man_Ching) writes:
|> Hi,
|>  
|> I would like to ask some questions:
|>  
|> (1) Given the following problem,
|>  
|>        min J(q,S)
|>        q,S
|>  
|>     in doing the above problem, we do the
|>     above problem like this:
|>  
|>      min min J(q,S)
|>       q   S
|>  
|>     i.e. doing the problem sequentially as two
|>     parallel independent minimization problem.
|>     With a fixed q first, minimize J(q,S) w.r.t.
|>     S to obtain S*. Then, minimize J(q,S*) w.r.t.
|>     q. Finally, get the optimal point(local/global??)
|>     (q*,S*).
|>  
|>     CAN WE DO THE PROBLEM AS DESCRIBED ABOVE??
|>     IF CANNOT, how can we do?
|>  
if J is convex in the joined variable (q,S) this works and is 
known for a long time (blockwise coordinate descent).
the classical book of blum&oettli; 'mathematical optimization (or similar title)'
which appeared with Springer publisher is a good source.
Otherwise you migth try to solve grad_{q,S}J=0 by solving (analytically
or numerically) a subsystem for S=S(q) plug this into the remaining equations
and solve these for q (But I am afraid that this will not work well globally
even if your problem is quite well behaved but nonconvex)
hope this helps
peter