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convergence of QR-algorithm
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Paul O
science forum beginner


Joined: 01 Feb 2006
Posts: 11

PostPosted: Tue Jul 18, 2006 7:03 pm    Post subject: Re: An Interesting Subject Reply with quote

theo3 wrote:
Quote:
This is a webpage that basically gives all answers that people are
looking for. If you are interested in looking at it please check it
out. For the German vrsion please click on the German flag.
http://www.theomonistik.net/ee/374.htm



Yawn...




Paul D Oosterhout
(from SAIC)
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Titus Piezas III
science forum Guru Wannabe


Joined: 10 Mar 2005
Posts: 102

PostPosted: Tue Jul 18, 2006 5:54 am    Post subject: Re: a^2+b^2 = c^2+d^2 Reply with quote

Robert Israel wrote:
Quote:
In article <44bb3bc6$0$990$ba4acef3@news.orange.fr>,
Patrick Coilland <pcoilland@pcc.fr> wrote:

Robert,

According to the initial problem "let P be the number of primitive
solutions to the Pythagorean triple a^2+b^2 = c^2 with hypotenuse c less
than a bound N. Then P/N = 1/(2pi) as N -> infinity", I think theat the
OP question is :

"let P be the number of primitive solutions to the integer equation
a^2+b^2 = c^2+d^2 = z with z *less than* a bound N. Then does the ratio
p/Sqrt(N) approach a real constant as N -> infinity? Can it be expressed
as a rational multiple of pi?".

Hello Patrick and Robert,

My apologies for the confusion. I'll try to make it clearer. Given the
equation,

a^2+b^2 = c^2+d^2 = z

let p be the number of primitive solutions with *hypotenuse sqrt[z]
less than* a bound N.

Case 1:

For c=d=0, and z a square, Lehmer proved the function F(N): = p/N =
1/(2pi) as N -> infinity. Example, for N = 10, we only have the
solution (3,4,5) giving F(N) = 1/10 = 0.1. The sequence of F(N) for N =
10, 10^2, 10^3, 10^4,... is

F(N) = 0.1, 0.16, 0.158, 0.1593, 0.15919, ...

converging to 1/(2pi) = 0.159154...

Case 2:

However, let (a,b,c,d) all non-zero with z not necessarily a square.
(Hence we are looking for a pair of triangles with integral legs with
the same hypotenuse.) Example, for N = 10, there are only 3 primitive
solutions,

1^2+7^2 = 5^2+5^2 = 7.07^2
1^2+8^2 = 4^2+7^2 = 8.06^2
2^2+9^2 = 6^2+7^2 = 9.21^2

so F(N) = p/N = 3/10 = 0.3.

Questions: 1) What is the sequence of F(N) for this? 2) Does it
converge to some constant? 3) If yes, can it be expressed in terms of
pi, analogous to Lehmer's result?

I hope it is clearer now.

P.S. The sequence of F(N) for case 2 for N = 10, 10^2, 10^3,... can
easily be ascertained by anyone with programming skills which I hope
someone in this channel would be curious enough to do so.

--Titus
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Phil Carmody
science forum Guru Wannabe


Joined: 05 Jun 2005
Posts: 267

PostPosted: Sat Jul 08, 2006 12:53 am    Post subject: Re: Newton Raphson variants... Reply with quote

Peter Luschny <spamgrube@luschny.de> writes:
Quote:
Phil Carmody schrieb:
"Hugo Pfoertner" <nothing@abouthugo.de> wrote

And where did Ernst Schröder lose the German Umlaut "ö"? The only
acceptable transcription of the German Umlaut derived from "o" is "oe".
^^^^^^^^^^^^^

i.e. Schröder --> Schroeder

Schroeder's original article

Ignore Hugo, he's full of old-Europe huff and nonsense. In a
_German_ medium the only acceptable transliteration is as he
^^^^^^^^^^^^^^^
indicates. _Usenet is not a German medium_. It is the target that
decides the rules for transliteration therein, not the source
language.

Ignore Phil, he does not know the difference between
'transcription' ans 'transliteration'.

Given that the process being performed was precisely transliteration,
that's the word I used. And I also know how wide space characters are,
unlike you.

Phil
--
The man who is always worrying about whether or not his soul would be
damned generally has a soul that isn't worth a damn.
-- Oliver Wendell Holmes, Sr. (1809-1894), American physician and writer
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sonnenrain17@gmx.de
science forum beginner


Joined: 02 Jul 2006
Posts: 2

PostPosted: Sun Jul 02, 2006 8:25 pm    Post subject: Re: Newton Raphson variants... Reply with quote

For equations having one unknown there are numerous variants which are
more efficient than Newton's method, the most well known ones mentioned
before. For systems of equations the situation changes, there are not
as many. In any case one has to compare carefully the number of
function evaluations and derivative(s) used for each iteration in order
to derive reasonable conclusions. E.g. there is a method having order
1+sqrt(2) (for "single" roots) which uses one evaluation of f and one
for its first derivative per step which works for systems too. The
computational costs are comparable to Newton's method if implemented
reasonably. See Numerische Mathematik, 32 (1979) and 38 (1982). As far
as I know there is no method using one value of f and one evaluation of
the jacobian per step which has higher order.
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Richard Mathar
science forum beginner


Joined: 23 May 2005
Posts: 45

PostPosted: Sun Jul 02, 2006 6:33 pm    Post subject: Re: Elliptic integral of the seconnd kind with complex amplitude Reply with quote

In article <1hhrlv3.1xijemp15dioq3N%cbarron3@ix.netcom.com>,
cbarron3@ix.netcom.com (Carl Barron) writes:
Quote:
Gerald I. Evenden <gerald.evenden@verizon.net> wrote:

Is anyone aware of a technique for the solution of the incomplete
elliptic integral of the second kind with a complex amplitude:

E(x+iy,m)

m is small, in the range of 0.006.

If there is a library reference, is should be usable under either
a GNU or MIT license.

Thanks in advance.

There is an article on elliptic integrals and elliptic functions.

Bulirsch Numerishe Mathematik vol 7 p78-90,vol 7 p353-354,vol 13,
p 305-315.

Either the first most likely since it contain algol 60 code provides an
algorithm to compute first and second elliptic integrals with complex
argument and real modulus. If its the first the second looks like
errata, and the third I don;t recall seeing.

its at guttenburg(sp??) online library.


See also

@article{CarlsonNumMath33,
author = {B. C. Carlson},
title = {Computing Elliptic Integrals by Duplication},
journal = {Num.\ Math.},
volume = {33},
pages = {1--16},
year = 1979
}

@article{CarlsonMathComp53,
author = {B. C. Carlson},
title = {A Table of Elliptic Integrals: Cubic Cases},
journal = {Math.\ Comput.},
volume = {53},
number=187,
pages = {327--333},
year = 1989
}

@article{Carlsonarxiv94,
author = {B. C. Carlson},
title = {Numerical Computation of real or complex Elliptic Integrals},
journal = {arxiv:math.CA/9409227},
year = 1994
}
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Gottfried Helms
science forum Guru


Joined: 24 Mar 2005
Posts: 301

PostPosted: Fri Jun 30, 2006 12:54 pm    Post subject: Re: Best Fit Circle made Easy Reply with quote

Am 29.06.2006 09:28 schrieb Han de Bruijn:
Quote:
Peter Spellucci wrote:

In article <4ac1e$44a2739b$82a1e228$1666@news1.tudelft.nl>,
Han de Bruijn <Han.deBruijn@DTO.TUDelft.NL> writes:
http://hdebruijn.soo.dto.tudelft.nl/jaar2006/kromming.pdf
you seem to assume that R^2 approx a^2+b^2 ??

No approximations involved: R^2 = a^2 + b^2
if R = radius and (a,b) = midpoint .

where is a problem with circle fit?
[ ... ]
this is clearly a nonlinear least squares problem.
[ ... ]

While my formulation is _linear_ all over the place.

this should give at least a reasonable first guess for the nonlinear problem
and will in many cases be already a very satisfacting solution.

Yes. But the point is that I _hate_ nonlinear problems. :-(

...

Han de Bruijn

Hmm, another try which i did recently:


Let the coordinates of all N points yi,xi
and let yi',xi' be the coordinates of the set of points
translated to their mean.

denote
ßi = arctan(yi'/xi')

then the minimization means in terms of a projected circle with
center (0,0) and optimal radius r to estimate:

Sum (xi' - r*cos(ßi))²+(yi'-r*sin(ßi)² = min

Derivative:

Sum 2(r*cos²(ßi) -xi*cos(ßi) + 2(r*sin²(ßi) -yi*sin(ßi)) = 0

2r Sum(cos²(ßi)+sin²(ßi)) = 2 Sum (xi*cos(ßi)) + yi*sin(ßi))
denote

laei = sqrt(xi'² + yi'²)

then

2r Sum 1 = 2 Sum (xi'² / laei) + yi'²/laei)

Sum (xi'² + yi'²)/laei
r = ------------------------
N

Sum laei
r = --------
N

r = Mean (laei)

... which generalizes easily to more dimensions.

Gottfried Helms
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Himanshu Chauhan
science forum beginner


Joined: 15 Jun 2006
Posts: 2

PostPosted: Thu Jun 15, 2006 3:47 pm    Post subject: Re: ANNOUNCE: comp.soft-sys.octave has been created Reply with quote

Martin Eisenberg wrote:
Quote:
Himanshu Chauhan wrote:

This group hasn't been added by google yet. I am, atleast,
waiting for them to do so.

What do you mean?
http://groups.google.com/group/sci.math.num-analysis


Martin


Oh My fault in posting. Actually I meant "comp.soft-sys.octave" hasn't
been added by google yet.

Himanshu

--
----------------------------------------
Himanshu Chauhan
MCA (Final Year)
I.G. National Open University
Jaipur (India)

Mobile:(+91)-(98292)-(92757)
Web: http://members.lycos.co.uk/hschauhan
Email: hs.chauhan@gmail.com

"Education is what remains after one
has forgotten everything he learned
in school." -- A. Einstein.
----------------------------------------
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Jean-Claude Arbaut
science forum Guru


Joined: 13 Jun 2005
Posts: 573

PostPosted: Thu Jun 15, 2006 2:03 pm    Post subject: Re: ANNOUNCE: comp.soft-sys.octave has been created Reply with quote

Martin Eisenberg wrote:
Quote:
Himanshu Chauhan wrote:


This group hasn't been added by google yet. I am, atleast,
waiting for them to do so.


What do you mean?
http://groups.google.com/group/sci.math.num-analysis

smna, yes, but not the new Octave group Smile
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Martin Eisenberg
science forum beginner


Joined: 11 May 2005
Posts: 28

PostPosted: Thu Jun 15, 2006 2:00 pm    Post subject: Re: ANNOUNCE: comp.soft-sys.octave has been created Reply with quote

Himanshu Chauhan wrote:

Quote:
This group hasn't been added by google yet. I am, atleast,
waiting for them to do so.

What do you mean?
http://groups.google.com/group/sci.math.num-analysis


Martin

--
Quidquid latine scriptum sit, altum viditur.
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Himanshu Chauhan
science forum beginner


Joined: 15 Jun 2006
Posts: 2

PostPosted: Thu Jun 15, 2006 9:51 am    Post subject: Re: ANNOUNCE: comp.soft-sys.octave has been created Reply with quote

J.V.Ashby wrote:
Quote:
This may be of interest to some here.

john



This group hasn't been added by google yet. I am, atleast, waiting for
them to do so.

I wasn't aware of this group earlier so I didn't cross post the RFDs
here. I am sorry for that.

--
----------------------------------------
Himanshu Chauhan
MCA (Final Year)
I.G. National Open University
Jaipur (India)

Mobile:(+91)-(98292)-(92757)
Web: http://members.lycos.co.uk/hschauhan
Email: hs.chauhan@gmail.com

"Education is what remains after one
has forgotten everything he learned
in school." -- A. Einstein.
----------------------------------------
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J.V.Ashby
science forum beginner


Joined: 10 Jun 2005
Posts: 12

PostPosted: Thu Jun 15, 2006 7:23 am    Post subject: Re: ANNOUNCE: comp.soft-sys.octave has been created Reply with quote

This may be of interest to some here.

john


The Big-8 Management Board wrote:

Quote:
The Big-8 Management Board has voted to create comp.soft-syst.octave.
The group was created on 12 Jun 2006.


NEWSGROUPS LINE:

comp.soft-sys.octave GNU Octave, language for numerical
computations.


CHARTER: comp.soft-sys.octave

General GNU Octave questions and its application to specific fields
such as digital signal processing, digital communications, medical
imaging, etc., as well as development issues such as bugs, fixes,
workarounds, etc., in the main Octave kernel as well as developer- and
user-supplied scripts (.m files) which extend Octave's functionality.

The general issues include:

* Addressing new user queries.
* Exchange of source code in public domain, especially .m scripts.
* Addressing the applications of GNU Octave in various fields.

Usage specific in fields like Signal Processing:

* Exchange of public domain source code to strengthen the signal
processing part of GNU Octave.
* Addressing the beginner to advanced queries regarding various
signal
processing related function available with GNU Octave, as built-in
or as separate packages.

Commercial Postings

Commercial advertising related to Octave from software vendors,
publishers, etc. will be permitted in a very limited fashion.

Unwelcome posts

Flames and other non-constructive criticism are harmful when directed
to new readers. The octave group is meant to be a friendlier place on
the net for beginners, intermediate, advanced users. Flames and other
non-constructive criticism aimed at those seeking help are unwelcome.

END CHARTER.


PROPONENT:

Himanshu Singh Chauhan <hs.chauhan@gmail.com


HISTORY:

2006-06-12 Group created.
2006-06-09 Vote ends; group will be created on 12 Jun 2006.
2006-06-02 Third RFD/last-call-for-comments posted.
2006-05-20 Second RFD submitted; reformatted by Tim Skirvin.
2006-05-18 Added several more newsgroups.
2006-03-15 Reformatted by Tim Skirvin for posting as an RFD.
2006-03-14 Submitted by Himanshu Singh Chauhan.
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astanoff@yahoo.fr
science forum beginner


Joined: 21 Oct 2005
Posts: 17

PostPosted: Fri Jun 09, 2006 6:27 am    Post subject: Re: Normal distribution function with skew and kurtosis Reply with quote

About the skew-normal distribution,
you can have a look at :

http://www.bus.lsu.edu/economics/faculty/chill/personal/aie/DeLuca.pdf


v.a.
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Tony Lance
science forum beginner


Joined: 18 May 2005
Posts: 30

PostPosted: Mon May 22, 2006 12:09 pm    Post subject: Re: Big Bertha Thing blogs Reply with quote

Big Bertha Thing letter
Cosmic Ray Series
Possible Real World System Constructs
http://web.onetel.com/~tonylance/letter.html
4K Web Page
Astrophysics net ring Access site
Newsgroup Reviews including sci.physics.particle

Letter to a friend explaining Pastures and quest.

From Pastures Software Package Documentation.
(Particle Structure Results Program in Fortran 77.)
Sub-atomic Mesons, Baryons and Leptons Classification System.
(C) Copyright Tony Lance 1997
Distribute complete and free of charge to comply.

Big Bertha Thing folder

Keep busy in your mailbox and put all postings
in a PIP Research folder for safe-keeping in perpetuity,
every night.

Part of your chores would be to read each missive,
subject by subject. Answer it, in a weekly PIP Research
Newsletter or not, as the case may be, which would again,
go rapidly from the mailbox to the folder.

(C) Copyright Tony Lance 1997.
To comply with my copyright,
please distribute complete and free of charge.

Tony Lance
tonylance@webandmail.co.uk


Big Bertha Thing debit

UK big banks have now made debit cards mandatory.
1. Every cheque book has a debit card.
2. You do not sign agreement for debit card.
3. Lost or stolen cards must be cancelled.
4. Insure debit cards for 1000 pounds sterling.
5. For every 8 pounds loss charged to the customer, the bank can
lend 92 pounds. (Multiplier)
6. Cheque guarantee card included.
7. Cash Card included.
8. Phantom withdrawals included.
9. 50 pounds cashback at supermarkets.
10. 250 pounds 'Hole-in-the-wall' withdrawals.
11. 100 pounds Post Office and bank withdrawals.
12. How do you prove that you have cut the card up and disposed of it?
13. How do you prove that you did not use it before destruction?
14. Cut the card up and give to your solicitor or third party for
safe-keeping. (Escrow)
15. The survival of the banks depends on putting customers to the
sword. (Barbaric) Obviously not all customers, just enough.
16. Unauthorized access to debit card accounts, includes both
criminals and bank insiders, which seem to be synonymous.
17. Authorized access to debit card accounts, includes bank
insiders, grazing on the customers like milk cows, in their
official capacity, as per job description.
18. You no longer need to prove this in a court of law, since it is
endemic and ubiquitous. (Common knowledge) Case law sufficient
for class action.
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Jean-Marc Gulliet
science forum beginner


Joined: 28 May 2005
Posts: 38

PostPosted: Sun May 21, 2006 3:00 pm    Post subject: Re: Trig identity Reply with quote

Andrew Poelstra wrote:
Quote:
On 2006-05-20, Kaimbridge <Kaimbridge@gmail.com> wrote:
Borked Pseudo Mailed wrote:
1-cos(x)
-------- = tan(x/2)^2
1+cos(x)

1-cos(x)^2
---------- = ?????
1+cos(x)^2

Does anyone know what function this reduces to?
I presume you mean a single function, in which case I don't think there
is one.
However, besides the ones that others have pointed out, some more
equivalencies are:

sin(x)^2 .5*sin(x)^2 2*sin(.5*x)^2*cos(.5*x)^2
------------ = --------------- = -----------------------------,
2 - sin(x)^2 1 - .5*sin(x)^2 1 - 2*sin(.5*x)^2*cos(.5*x)^2

2
= -------------------------;
tan(.5*x)^2 + cot(.5*x)^2


~Kaimbridge~

Here's an equivilancy with the benifit of being easy to type:
(cos(x) - 1)(cos(x) + 1)
--------------------------
-(cos(x) - i)(cos(x) + i)

Here is what I get with Mathematica 5.2


In[1]:=
TrigReduce[(1 - Cos[x]^2)/(1 + Cos[x]^2)]

Out[1]=
1 - Cos[2 x]
------------
3 + Cos[2 x]

HTH,
Jean-Marc
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Gottfried Helms
science forum Guru


Joined: 24 Mar 2005
Posts: 301

PostPosted: Fri May 05, 2006 8:14 pm    Post subject: Re: not positive-definite covariance matrix Reply with quote

ANother approach, besides the already given ones,
may be a SVD-decomposition. Since it is a covariance,
it is symmetric and

M = S D T = S D S'

where D is diagonal.
Then you could inspect D and see, whether you may
apply rounding to zero for very small negative D.
After that the cholesky-decomposition is

C = S * D^(1/2)

and

M = C * C'

THis way you may have more control over the numerical
problems, and I think, Matlab should provide the tools
to deal with this problem sensibly.

Gottfried Helms
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