HDTenEightyP/NetNews huggingface dataset:
From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Newsgroups: sci.math.num-analysis
Subject: Re: Condition Number
Keywords: eigenvalues, singular values, condition number
Date: 12 Jan 93 20:34:53 GMT
Article-I.D.: daphne.thompson.726870893
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota

mckenney@cims.nyu.edu (Alan McKenney) writes:


>In article <1993Jan11.214251.12368@news.eng.convex.com>
>dodson@convex.COM (Dave Dodson) writes:
>>In article <1993Jan11.083253.9521@dutrun2.tudelft.nl>
>>rcpshdb@dutrun2.tudelft.nl (Han de Bruijn) writes:
>>>The condition number of a matrix  A  is defined as the absolute value of
>>>its largest eigenvalue divided by its smallest eigenvalue. Right?
>>>
>>>I'm at loss how this is related to the loss of decimals when solving the
>>>equations system:
>>>                      A.x = b
>>>
>>>Can someone out there provide a simple argument?

>  [Dave Dodson's good explaination deleted]

>One thing that no one has remarked on: The original poster's reference
>to *eigenvalues*.

>    As Dave Dodson points out, the condition number of A is
>norm(A) * norm(A-inverse).  No mention of eigenvalues.

>    In fact, the condition number of a matrix has (almost) nothing to
>do with the eigenvalues, as the example

>        ( 1  t )                        -1   ( 1  -t )
>    A = (      )   for t >> 1 shows:   A   = (       )
>        ( 0  1 )                             ( 0   1 )

>so the eigenvalues are both 1, but norm(A) = norm(A-inverse)
>which are approximately  t, so the condition number is approx.
>t**2.

>    What the condition number *is* related to are the *singular values*.

The singular values of A are the square roots of the eigenvalues of
the symmetric matrix AA'.  These are of course the eigenvalues of A
itself whenever A is symmetric and positive semidefinite.  So saying
that the condition number has (almost) nothing to do with the
eigenvalues is not accurate either.  The original poster may have had
the symmetric positive semidefinite situation in mind.

The eigenvalues of AA' also have an exact relationship to the norm of
A if we define the latter appropriately.  For example, if we define
the norm of A as:
                   |Ax|
	|A| = max  ---- 
               x    |x|

Then we get 

|A| = square root of the largest eigenvalue of AA' and

|Inverse(A)| = 1 / (square root of the smallest eigenvalue of AA')

and the condition number is the product of these two norms.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209
Newsgroups: sci.math.num-analysis
From: ercolessi@uimrl3.mrl.uiuc.edu (furio ercolessi)
Subject: Re: Numerical Recipes
Reply-To: ercolessi@uimrl3.mrl.uiuc.edu (furio ercolessi)
Organization: MRL - UIUC
Date: Wed, 13 Jan 1993 02:07:17 GMT
Keywords: Press et al.

In article <1993Jan12.204430.17861@mailhost.ocs.mq.edu.au>, wskelly@laurel.ocs.mq.edu.au (William
Skelly) writes:
|>Does anyone know whether the authors of numerical recipes
|>(Press, Flannery, Teukolsky and Vetterling) allow their
|>routines to be used in public domain software?

On my edition, I read:

"To purchase diskettes in IBM PC or Apple Macintosh formats, use the
 order form [...]  Unlicensed transfer of Numerical Recipes programs
 from the abovementioned IBM PC or Apple Macintosh diskettes to
 any other format, or to any other computer except a single IBM PC
 or Apple Macintosh or compatible for each diskette purchased, is
 strictly prohibited."

So, just compiling one of their routines on a IBM RS/6000 or HP or Cray ...
automatically implies that a crime has been perpetrated.

But ...wait!  Look carefully!  The note is concerned about a copy *from
the diskette* to another machine.  
But since the source is published, you can type it by hand into a file,
and then that file can be transferred here and there without restrictions :-)

But this still leaves your question open ... 

--
Furio Ercolessi
Materials Research Laboratory           |   Intl School for Advanced Studies
Univ. of Illinois at Urbana-Champaign   |   Trieste, Italy
furio@uiuc.edu                          |   furio@sissa.it
From: thompson@atlas.socsci.umn.edu (T. Scott Thompson)
Newsgroups: sci.math.num-analysis
Subject: Re: Condition Number
Keywords: eigenvalues, singular values, condition number
Date: 12 Jan 93 21:36:13 GMT
Article-I.D.: kiyotaki.thompson.726874573
Reply-To: thompson@atlas.socsci.umn.edu
Organization: Economics Department, University of Minnesota

thompson@atlas.socsci.umn.edu (T. Scott Thompson) writes:

>The singular values of A are the square roots of the eigenvalues of
>the symmetric matrix AA'.  These are of course the eigenvalues of A
                      ^^^
This should have been A'A.  Many apologies for any confusion.

>|A| = square root of the largest eigenvalue of AA' and
                                                ^^^ 
>|Inverse(A)| = 1 / (square root of the smallest eigenvalue of AA')
                                                               ^^^
Same problem and correction in these two statements.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209
Newsgroups: sci.math.num-analysis
From: ledwards@leland.Stanford.EDU (Laurence James Edwards)
Subject: Re: Advantages/Disadvantages of Galerkin Method
Organization: DSG, Stanford University, CA 94305, USA
Date: Sun, 10 Jan 93 10:49:39 GMT

In article <yg!31aq@rpi.edu>, sassoj@aix.rpi.edu (John J. Sasso Jr.) writes:
|>     The Galerkin method, which is used to approximate solutions to differential
|>    equations, appears to be a pretty interesting technique.  However, I am 
|>    not too knowledgable about it, so I'd like to know what is its advantages
|>    and disadvantages.  Also, how does it fair against nonlinear ODEs and PDEs,
|>    which tend to have chaotic solutions.  If anyone has any good introductory
|>    references to the Galerkin method, I would like to know about them.

Take a look at "Variational Methods in Mathematics, Science and Engineering"
by Karel Rektorys, for further explanation and comparison with other methods.

Larry Edwards
Newsgroups: sci.math.num-analysis
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Newsgroups: sci.math.num-analysis
Subject: MATLAB on a supercomputer?
Date: 11 Jan 93 09:02:25 GMT
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Originator: rcpshdb@dutrun2.tudelft.nl

MATLAB is available for Convex 3840 (super)computers. Personally, I am no user
of MATLAB. I know that it is a product which is good for PC's and workstations,
but I wonder if it has sense to implement it on a Convex. It depends on whether
MATLAB is used for compute- and memory- intensive work (real number crunching),
or whether it is used (in practice) only for small and/or moderate problems.

I would appriciate if some experienced people react on this question.
-- 
* 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: simardr@IRO.UMontreal.CA (Richard Simard)
Subject: Re: Aproximating a messy product ...
Reply-To: simardr@IRO.UMontreal.CA (Richard Simard)
Organization: Universite de Montreal
Date: Mon, 11 Jan 1993 19:50:57 GMT

In article <1iah0fINNjhj@solar.usc.edu> laydin@solar.usc.edu (Levent Aydin) writes:
>
>	I need to approximate a function given in terms of a product with a
>simple function that I can deal with.The expression is as follows:
>
>               n              /            \  
>              __             |    x-m_i     | 
>        g(x)= || 0.5*{ 1+ erf| -------------| }
>              i=1            |  sqrt(2)*s_i | 
>                              \            /  
>
************************************************************

I don't think it would be a good idea to approximate by a
single term of the same form , whatever m_0 and s_0.

It would be best to separate the 2 factors with m_i much
larger than the others. Consider the function erf(y). It is
a monotonously increasing func; it is practically 0 from 
- infinity to 0 - 3 standard deviations, and  practically 1 
from 0 +
3 stand. dev. to + infinity; inbetween, it increases 
smoothtly.
Thus, you could approximate one factor  in the above
product by the following
approx.:

f(x) = 0.5 , for x = - infinity to m_i - 3 stand. dev.
     = 1 , for x = + infinity to m_i + 3 stand. dev.
     = a cubic polynomial with inflexion at m_i in-between.
 
Also you could approximate the n-2 factors with the m_i close
together by the function h(x) where

h(x) 	= 0.5^(n-2) , for x= - infinity to min{m_i - a}
     	= 1 , for x = max{m_i + b} to +infinity
	= Ae^(px), in-between 

the parameter a is around 1 to 3 standard dev of  in min m_i
the parameter b is between 1 to 3 standard dev. from max m_i
and A and p are chosen so the straight line px + c1 represent
well enough the log of the product of the (n-2) factors.

--
From:  simardr@iro.umontreal.ca
From: mld900@anusf.anu.edu.au (Murray Dow)
Newsgroups: sci.math.num-analysis
Subject: Re: MINOS
Date: 5 Jan 1993 00:35:07 GMT
Organization: ANU Supercomputer Facility
Distribution: world
Reply-To: mld900@anusf.anu.edu.au
NNTP-Posting-Host: 150.203.5.3
Originator: mld900@asterix


In article <C0CJ78.IA7@ireq.hydro.qc.ca>, robin@fractal.ireq-num (Robin Duquette ) writes:
|> ---
|> 
|> I would appreciate very much if somebody could give me the E-mail
|> address of Michael Saunders or Bruce Murtagh (the authors of MINOS).
|> 
|> Thanks in advance.
|> 
|> --------------------------------------------------------------------------------
|> Robin Duquette                              E-mail: robin@ireq-num.hydro.qc.ca
|> Service Methodes numeriques                 phone : (514) 652-8239
|> Institut de recherche d'Hydro-Quebec        fax   : (514) 652-8005
|> 1800 montee Ste-Julie,
|> Varennes (Quebec) Canada J3X 1S1
|> --------------------------------------------------------------------------------
|> 

MINOS - 

   Stanford Office of Technology Licensing
    350 Cambridge Ave.
    Suite 250
    Palo Alto, CA  94306
    USA
    phone: 415-723-0651

-- 
Murray Dow                         GPO Box 4 Canberra ACT 2601 Australia
Supercomputer Facility             Phone: +61 6 2495028
Australian National University     Fax:   +61 6 2473425
mld900@anusf.anu.edu.au
Newsgroups: sci.math.num-analysis
From: mhosea@seas.smu.edu (Mike Hosea)
Subject: Re: NAG routine D02BAF uses runge-kutta-MERSON ??
Keywords: Merson, nonlinear problems
Organization: SMU - School of Engineering & Applied Science - Dallas
Date: Tue, 5 Jan 1993 17:31:51 GMT


In article <1ia4m9INN2h2@rave.larc.nasa.gov> makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes:
>Most error estimates for integrators are based on what is called
>the "linear test equation". It is extremely difficult if not
>impossible to obtain local truncation error estimates for 
>nonlinear problems. Normally error estimates for nonlinear
>problems are obtained by linearizing the equation(s).

If I understand you correctly, this isn't true.  Error estimators 
for integrators are usually based on subtracting a higher order 
result from a lower order result to get an asymptotically correct
estimate of the error in the lower order result.  The technique is
valid for linear and nonlinear problems.   

You must be thinking about linear vs. general (nonlinear) stability
analysis, since the "linear test equation" is used in that context,
and general (nonlinear) stability analysis is a lot harder than 
the linear stability analysis.
-- 
Regards,
Mike Hosea (mhosea@seas.smu.edu)
Newsgroups: sci.math.num-analysis
From: chuck@stratus.drexel.edu (Wombozni Klik)
Subject: Re: Num Recipes errata?
Organization: Drexel University Department of Physics
Date: Mon, 11 Jan 1993 20:22:18 GMT
Distribution: sci.math.num-analysis
X-Posted-From: stratus.physics.drexel.edu
NNTP-Posting-Host: sol.ctr.columbia.edu


        You have to purchase it from the publisher. It is not freely
distributable. There should be an order form in the back of the book.

Chuck 
--
Chuck Browne                            | Frisbeetarianism: The belief that  
Drexel University Department of Physics | when you die, your soul goes up
chuck@nimbus.physics.drexel.edu         | on the roof and gets stuck.
From: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Subject: Condition Number
Originator: rcpshdb@dutrun2.tudelft.nl
Reply-To: rcpshdb@dutrun2.tudelft.nl (Han de Bruijn)
Organization: Delft University of Technology
Date: Mon, 11 Jan 1993 08:32:53 GMT

The condition number of a matrix  A  is defined as the absolute value of
its largest eigenvalue divided by its smallest eigenvalue. Right?

I'm at loss how this is related to the loss of decimals when solving the
equations system:
                      A.x = b

Can someone out there provide a simple argument?
-- 
* 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
From: at913@cleveland.Freenet.Edu (Mirko Vukovic)
Newsgroups: sci.math.num-analysis
Subject: Re: Advantages/Disadvantages of Galerkin Method
Date: 8 Jan 1993 15:51:30 GMT
Organization: Case Western Reserve University, Cleveland, OH (USA)
Distribution: inet
Reply-To: at913@cleveland.Freenet.Edu (Mirko Vukovic)
NNTP-Posting-Host: hela.ins.cwru.edu


In a previous article, sassoj@aix.rpi.edu (John J. Sasso Jr.) says:

>    The Galerkin method, which is used to approximate solutions to differential
>   equations, appears to be a pretty interesting technique.  However, I am 
>   not too knowledgable about it, so I'd like to know what is its advantages
>   and disadvantages.  Also, how does it fair against nonlinear ODEs and PDEs,
>   which tend to have chaotic solutions.  If anyone has any good introductory
>   references to the Galerkin method, I would like to know about them.
>
>                          -- John
>
   My introduction to the Galerkin method as appled to a solution of
electromagnetic wave propagation in a plasma was the article
TEXTBOOK FINITE ELEMENT METHODS APPLIED TO LINEAR WAVE PROPAGATION
PROBLEMS INVOLVING CONVERSION AND ABSORPTION by Appert et al in
Computer Physics Comm. vol.40(1986)pp 73--93.  

Section 2 describes the physics and diff eq. sect.3 deals with the application
of the Galerkin method to the problem.  It took several readings, but was
worth it.
-- 
Mirko Vukovic 
University of Wisconsin -- Madison
VUKOVIC@UWMFE.NEEP.WISC.EDU
From: jon@oas.stanford.edu (Jon Claerbout)
Newsgroups: sci.math.num-analysis
Subject: Re: "Minimize any function, as long as it is named FUNC"
Date: 10 Jan 1993 02:52:17 GMT
Organization: Stanford Exploration Project
NNTP-Posting-Host: oas.stanford.edu

>
>I attempt to explain below in more detail what the problem is.
>It is quite entangled and technical,

That is why some people use C and C++ for numerical work.
From: makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni)
Newsgroups: sci.math.num-analysis
Subject: Re: NAG routine D02BAF uses runge-kutta-MERSON ??
Date: 5 Jan 1993 19:08:23 GMT
Organization: NASA Langley Research Center, Hampton, VA  USA
Distribution: world
NNTP-Posting-Host: gazelle.larc.nasa.gov
Keywords: Merson, nonlinear problems

In article <1993Jan5.173151.21651@seas.smu.edu>, mhosea@seas.smu.edu (Mike Hosea) writes:
|> 
|> In article <1ia4m9INN2h2@rave.larc.nasa.gov> makarand@gazelle.larc.nasa.gov (Makarand A. Kulkarni) writes:
|> >Most error estimates for integrators are based on what is called
|> >the "linear test equation". It is extremely difficult if not
|> >impossible to obtain local truncation error estimates for 
|> >nonlinear problems. Normally error estimates for nonlinear
|> >problems are obtained by linearizing the equation(s).
|> 
|> If I understand you correctly, this isn't true.  Error estimators 
|> for integrators are usually based on subtracting a higher order 
|> result from a lower order result to get an asymptotically correct
|> estimate of the error in the lower order result.  The technique is
|> valid for linear and nonlinear problems.   
|> 
|> You must be thinking about linear vs. general (nonlinear) stability
|> analysis, since the "linear test equation" is used in that context,
|> and general (nonlinear) stability analysis is a lot harder than 
|> the linear stability analysis.
|> -- 
|> Regards,
|> Mike Hosea (mhosea@seas.smu.edu)


Yep, you are right. I was talking about stability analysis.
Your point is well taken.

Regards
Makarand
email: makarand@puma.larc.nasa.gov.
Newsgroups: sci.math.num-analysis
From: wangly@cad17.ncku.edu.tw ()
Subject: How to obtain MINOS?
Organization: National Cheng Kung University, Tainan, Taiwan
X-Newsreader: Tin 1.1 PL3
Date: Mon, 4 Jan 1993 23:41:20 GMT

Hi,
   Does anyone know how to obtain the MINOS package?

Thanks.


-----------------------------------------------------
Wang L. Y.
National Cheng Kung Univ.
Tainan, Taiwan, ROC
From: neilfp@astro.ocis.temple.edu (Neil F. Palumbo)
Newsgroups: sci.math.num-analysis
Subject: HELP Quadratic Eqns
Date: 9 Jan 93 17:29:52 GMT
Organization: Temple University
X-Newsreader: TIN [version 1.1 PL8]

[ Article crossposted from sci.math ]
[ Author was Neil F. Palumbo (neilfp@astro.ocis.temple.edu) ]
[ Posted on Sat, 9 Jan 1993 17:25:20 GMT ]

I have a set of equations of the form:

			s'*Ai*s + Bi*s + ci = 0, i=1,2,...,100

The unknown is the vector s (the same unknown vector in all 100 eqns).
The Ai's are square (100x100) matrices, the Bi's are (100x1) vectors,
and the ci's are constants. s is of course a (100x1) vector. Can anyone
suggest an approach to simultaneously solving this set of equations? Please
e-mail me with your responses. Thanks in advance...

				Neil
From: sshankar@td2cad.intel.com (Sadasivan Shankar)
Newsgroups: sci.math.num-analysis
Subject: Re: Calculating norm of a function
Date: 4 Jan 93 20:09:41 GMT
Reply-To: sshankar@td2cad.intel.com (Sadasivan Shankar)
Distribution: usa
Organization: Intel Corporation, Santa Clara, CA USA


Your problem looks like that of finding a Cauchy sequence in the L2 norm.  Typically, this
seems to be a Hilbert-space problem, where a converging series is used to find the
norm of a function. Does anybody else have a comment on this ?

S.Shankar
Tech CAD
INTEL Corporation
From: imustafa@slate.mines.colorado.edu (MUSTAFA IBRAHIM )
Newsgroups: sci.math.num-analysis
Subject: Multiple Solutions  and   COLSYS  ?
Date: 9 Jan 93 08:04:51 GMT
Organization: Colorado School of Mines


  

Would any kind soul tell me how can we handle multiplicity in 
numerical approximations. Actually I am working on a two point boundary
value problem which involves a second order ODE. It has been solved
numerically by converting it to initial value problem but my technique
is to use the recently developed Collocation Software designed by 
Dr. Ascher and Dr. R.D. Russel (Simon Fraser University , Canada).
This is a robust software for solving bvodes. 
My computer program is not working for multiple solutions.
I will be very thankful if somebody gives me a useful information. 
Even if somebody has e-mail no. of either of the above mentioned 
professors please pass it to me. 

Thanks once again in anticipation.


  which involves a second order differential equation. It has
been solved numerically 
Newsgroups: sci.math.num-analysis
From: rbk5287@usl.edu (Kearfott Ralph B)
Subject: Conference:  Math. Modelling and Sci. Computing
Organization: University of Southwestern Louisiana
Date: Tue, 12 Jan 1993 15:59:03 GMT

Date: September 14--17, 1993
Place: Sozopol, Bulgaria
Conference Title: Mathematical Modeling and Scientific Computation
                  (MMSC-93)

Official Language: English

The purpose of the conference is to make new methodologies for
scientific computation and new software tools widely available, to
promote contacts between software companies and diverse groups of
specialists, and to train and encourage beginning researchers.  Topics
include advanced numerical methods (involving interval and parallel
techniques) and applications, mathematical models with interval-valued
input data, enclosure methods with automatic result verification,
interval methods in automatic theorem proving, program packages, and
applied modeling.

Papers, nominally four pages long, should be submitted via e-mail and in
printed form to one of the contact persons by April 1, 1993.
Notification of acceptance will be by June 15, 1993.  Preregistration
($100 U.S.) is before July 15, 1993, and on-site registration is $130.

Contact persons: S. Markov (chairman), N. Dimitrova (secretary)
                 Institute of Biophysic,s BAS
                 Acad. G. Bonchev str. bl. 21
                 BG-1113 Sofia, Bulgaria
                 email: neli@bgearn.bitnet

                 D. V. Shiriaev, Karlsruhe
                 email: dima@iamk4058.mathematik.uni-karlsruhe.de

                 I. Dimov, Sofia
                 email: ivdimov@bgearn.bitnet

                 R. B. Kearfott
                 Department of Mathematics
                 University of Southwestern Louisiana
                 U.S.L. Box 4-1010
                 Lafayette, LA 70504-1010
                 USA
                 Work phone: (318) 231-5270
                 Home phone: )318) 981-9744
                 email: rbk@usl.edu (Internet)

                 Ch. Ullrich, Basel
                 email: ullrich@urz.unibas.ch

                 A. P. Voshinin, Moscow
                 email: apv@mei.msk.su

                 K. Madsen, Lyngby
                 email: numikm@vm.uni-c.dk

                 J. Wolff von Gudenberg, Wuerzburg
                 email: wolff@informatik.uni-wuerzburg.de

Newsgroups: sci.math.num-analysis
From: a722756@roper.mc.ti.com (W. Donald Rolph)
Subject: Re: Advantages/Disadvantages of Galerkin Method
Originator: a722756@roper.mc.ti.com
Organization: Texas Instruments / Attleboro Mass / USA
Date: Fri, 8 Jan 1993 13:38:28 GMT


I will approach this from the perspective of using Galerkin methods for
approximating PDEs using fnite element techniques.

The Galerkin is an alternative formulation to the variational formulation
typically used when a variational does not exist.  If it is possible to integrate
the Galerkin form by parts one typically ends up with a variational formulation.

This statement has the following consequences:
 
     - a pure Galerkin formulatiuon typically results in nonsymmetric equations

     - a pure Galerkin forumlation typically requires trial functions of twiceUnread news in news.config                                 1 article  +  412 oldUnread news in news.config                                 1 article  +  412 old
            the order of an equivalent Variational formulation

There is also the issue that a Galerkin forumation, since it need not be
positive definite, can result in saddle solutions rather than the minima typical
of variational forms.  This has the effect in certain nonlinear PDEs for
introducing instability, although I am not aware of solid research to support
this statement for the general case,  It would  probably be an interesting
problem.  (Careful though, when evaluating convegence properties one should look
at the convergence of primary variables only, not secondary variables introduced
to enforce boundary conditions - e.g. one should not look at the convergence of
problems with Lagrange multiplier constraints by by including the behavior of the
Lagrange multipliers in the convergence study).

Hope this is useful!
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263
Newsgroups: sci.math.num-analysis