Subject: Re: Exponential Data Fitting (2-nd Posting; Modified problem)
From: Gunnar Isaksson
Date: Mon, 20 Jan 1997 10:45:17 +0100
Diane wrote:
>
> Dear all,
>
> Due to your reactions, Thanks to all of you , I just begin to understand how to find
> a function that approximates (math-people call Fitting) the given data. Due to my
> lack of mathematic discipline (I am studying Telecommunication), my problem
> described in previous posting was not clear, Therefore I post my problem
> (modified) again and hope that you can help me to find the references.
>
> My old {Xi,Yi} discrete data is modified to a new and is given below.
> The {Yi} data is now positive and in between [ 0 - 1].
> The Sampling point {Xi} is equally spaced.
>
> I want to fit the data to a fit-function YFIT1 of the form .
> YFIT1= exp { A0+ A1 X^(B1) + A2 X^(B2) +... + AN X^(BN) }; N=10 for example.
> where An, Bn are the constants to be determined.
> Take the Ln, the problem reduce to find the fit function YFIT2
>
> YFIT2 = Ln(YFIT1) = A0 + A1 X^(B1) + A2 X^(B2) + .. . + AN X^(BN). (1)
>
> My question is how can I find the How can I determine the An, Bn ?.
> By which method? . Where can I find the theory about this?.
>
> Note that : A particular case of YFIT2 : when B1=1, B2=2, B3=3,... Bn= n the
> YFIT2 becomes the general polynomial of degrees ‘n’ i.e.
> YPOLYN = Ln(YFIT1) = A0 + A1 X^(1) + A2 X^(2) + . . . + AN X^(N). (2)
> I tested the case of (2) the fitting function become very bad if ‘N’ lager than 7
> (and the error at the sampling points is too big even for N smaller than 7, specially
> at both ends of the data, where the data oscillate rather fast. .
>
> Due to stringent demand in my telecom-application, the accuracy of the model
> function (fitting function), at any sampling event at on the three regions
> {-180,-178,.....-140}, {-20,-18,...,0, ...20} and {140,172,.....-180} MUST be smaller
> than 10^ (-5), the errors at other sampling point outside the these regions are not
> critical for the application. Can we achieve this requirement. ? Is this possible ?
> and How?.
>
> I will be eternally thankful to you for your idea, hint, help etc,.
>
> Thank you in anticipation.
>
>
>
Well, I have checked the your data and by just simple eyeballing it
looks like
your function is an allmost even function f(x)=f(-x). The greatest
deviations
from that fact is around x=0. Measurement errors?
If you have other well founded reasons to believe that the function is
even,
you could fix the data by computing a new function z(x)=(f(x)+f(-x))/2
which
will fix your data to be an exactly even function of the argument x.
Now that the data is transformed to be an even function you just have to
consider
polynom or other function approximations with even functions.
Good Luck,
Gunnar Isaksson
--
=============================================
Gunnar Isaksson Hamradio: SM5IUF
Internet: Gunnar.Isaksson@saab.se
_____________________________________________
Subject: Re: cobyla: do_fio, e_wsfe, s_wsfe etc; where can I find these routines?
From: spellucci@mathematik.th-darmstadt.de (Peter Spellucci)
Date: 20 Jan 1997 11:56:04 GMT
In article , jost@psisun.u-psud.fr (Christian Jost) writes:
|> I am trying to use the c-version of cobyla (from netlib) on a Macintosh
|> with the CodeWarrior C-compiler, and after all the compilation I got a
|> couple of link errors referring to the functions e_wsfe, s_wsfe, do_fio,
|> etc. Where can I find these functions? They are not in my ANSI libraries,
|> neither are they in the f2c code. Anybody having done this already?
|>
|> Any help is welcome, Christian.
|>
|> --
|> Christian Jost, Université Paris-Sud XI, Orsay, France
|> jost@psisun.u-psud.fr
these routines should be in the f2c-libraries which accompany the f2c-
distribution
hope this helps
peter
Subject: Re: Eigenvalues
From: Dmitry Tikhonov
Date: Mon, 20 Jan 1997 09:54:33 -0800
John Hench wrote:
>
> Dmitry Tikhonov wrote:
>
> > I'm afraid it does'nt possible in simple way.
> > The only Tr() functional of matrix are linear.
> > Tr(A+B)=Tr(A)+Tr(B)
> > Tr(AB)=Tr(A)*Tr(B)
>
> Uh, no. Let A=[0,1;0,0] and B=A' (in MATLAB
> notation). Trace(A)=Trace(B)=0, but trace(AB)=
> 1. Note that trace(AB)=trace(AA')=frobenius norm
> of A in this case. Since A is not identically
> zero, trace(AB) must be greater than zero, right?
>
> On the other hand: det(A)*det(B)=det(AB).
>
> -------------------------------------------------
> Dr. J.J. Hench
> Dept. of Mathematics, Univ. of Reading, England
> Institute of Informatics and Automation, Prague
> -------------------------------------------------
Of course, I was wrong writting:
Tr(AB)=Tr(A)*Tr(B),
Really I mean:
Tr(AB)=Tr(BA)
Sorry, for my mistake.
Dmitry Tikhonov.
Subject: Re: Calculate polynomial of two variables
From: spellucci@mathematik.th-darmstadt.de (Peter Spellucci)
Date: 20 Jan 1997 12:04:41 GMT
In article <5bm05c$nus$2@salico.udc.es>, alfonso@unico.udc.es (Alfonso Castro Martinez) writes:
|> Hi! I am working with X-ray images and I want to calculate a polynomial to
|> data fitting of some points of the images. I read some books of numerical
|> methods but they show data fitting using a polynomial of one variable.
|>
|> Does anybody know any numerical method to calculate a polynomial of two
|> variables? Does anybody know any algorithm?
|>
|> Thank you in advance. Yours sincerely :
|>
|> Alfonso Castro Martinez
|> alfonso@udc.es
|>
|> P.D: Best regards!
|>
if you use data fitting by least squares, then there is _no_ difference at all.
e.g. if your "ansatz" is
z = a + b*x +c*y + d*x*y
and you have data ponits (x_i, y_i) with measurement z_i
then you assemble your basis functions 1,x,y,x*y evaluated on the given points
in a matrix, A, say
( 1 , x_0 , y_0 , x_0*y_0 )
A= ................................
( 1 , x_n , y_n , x_n*y_n )
the measurements z in a vector, b, say , b=(z_0,....,z_n)^T, ^T=transpose,
and fed A, b into a linear least squares solver, e.g. from LAPACK. that's all.
working with discrete maximum or sum-norm is similar .
hope this helps
peter
Subject: Re: Computational cost of SVD
From: Hans D Mittelmann
Date: Mon, 20 Jan 1997 07:08:17 -0700
Miroslav D Trajkovic wrote:
>
> Hi,
> Does somebody know what is tha computational cost of the
> SVD for the matrix W of dimensions n times k.
> References most welcome !
>
> Thanks,
>
> Miroslav
Hi,
yes, as someone points out, the answer is in Golub&van; Loan. However,
there is a new 3rd edition (1996). In that on page 254 is the table of
workcounts. If you use the Golub-Reinsch algorithm and want to compute
Sigma, U, V for n*k, then the work is
4n^2k + 8nk^2 + 9k^3
--
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
Subject: Re: Where are Optimized BLAS routines for Pentium Chip
From: molagnon@ifremer.fr (Michel OLAGNON)
Date: 20 Jan 1997 13:54:05 GMT
In article , mark.horridge@buseco.monash.edu (mark.horridge@buseco.monash.edu.au) writes:
>subject line says it all
Reposted from this group:
From mlkessle@cip.physik.uni-wuerzburg.de (Manuel Kessler)
Newsgroups: sci.math.num-analysis,comp.os.msdos.djgpp
Subject: Pentium Optimized BLAS
Date: Wed Nov 13 17:10:20 MET 1996
Organization: CipPool der Physikalischen Institute, Uni Wuerzburg
Hi to all number crunchers,
I'm glad to announce the first test release of my pentium optimized
BLAS level 1 library. The most important functions of the BLAS level 1
(xAXPY, xCOPY, xDOT, xNRM2, in both single and double precision)
have been completely rewritten in GNU assembler for achieving full
performance on intel pentium processors. The library is available in
both binary and source form for DOS/DJGPP and UNIX/LINUX at
http://cip.physik.uni-wuerzburg.de/~mlkessle/blas1.html
See there for further informations. Suggestions, bug reports,
comments etc. are very welcome, email adress see below.
Thanks,
Manuel
--
Manuel Kessler
Graduate Student at the University of Wuerzburg, Germany, Physics Department
SNAIL: Zeppelinstrasse 5, D-97074 Wuerzburg, Germany
EMAIL: mlkessle@cip.physik.uni-wuerzburg.de
WWW: http://cip.physik.uni-wuerzburg.de/~mlkessle
Michel
--
| Michel OLAGNON email : Michel.Olagnon@ifremer.fr|
| IFREMER: Institut Francais de Recherches pour l'Exploitation de la Mer|
| Centre de Brest - B.P. 70 phone : +33-02-9822 4144|
| F-29280 PLOUZANE - FRANCE fax : +33-02-9822 4135|
| http://www.ifremer.fr/ditigo/molagnon/molagnon.html |
Subject: Re: Re:Exponential Data Fitting (2-nd Posting, modified problem).
From: spellucci@mathematik.th-darmstadt.de (Peter Spellucci)
Date: 20 Jan 1997 13:55:44 GMT
In article <5bnvne$30l@news.tudelft.nl>, D.P.Tran@et.tudelft.nl (Diane) writes:
|> In article <5blq0d$1037@rs18.hrz.th-darmstadt.de>, Peter Spellucci, spellucci@mathematik.th-darmstadt.de wrote...
|>
|> >|> your data are _negative_ and real. this is impossible for exp on the complex
|> >|> numbers. Therefore I assume you meant -exp{..} . take the logarithm of -Y
|> >|> and you get a linear least squares problem (statisticians don't like that,
|> >|> it introduces bias into the estimated parameters, but well, it is simple)
|> >|> then you have a polynomial fit. but degree 20??? this would come up
|> >...............................................................
|> >oh oh, I have a bad day. simply add _i*pi_ to A_0 and you get your "-",
|> >if you really wanted it in that form.
|> >If fft (it interpolates the data) is not adequate for your purposes,
|> >you may have a linear least squares fit using a low order fourier sum.
|> >This would in any event come up with a well conditioned problem.
|> >
|> >peter
|>
|> Dear all,
|>
|> Due to your reactions, Thanks to all of you , I just begin to understand how
|> to find a function that approximates (math-people call Fitting) the given data.
|> Due to my lack of mathematic discipline (I am studying Telecommunication),
|> my problem described in previous posting was not clear, Therefore I post
|> my problem (modified) again and hope that you can help me to find the
|> references.
|>
|> My old {Xi,Yi} discrete data is modified to a new and is given below.
|> The {Yi} data is now positive and in between [ 0 - 1].
|> The Sampling point {Xi} is equally spaced.
|>
|> I want to fit the data to a fit-function YFIT1 of the form .
|> YFIT1=exp{ A0+ A1 X^(B1)+ A2 X^(B2)+...+ AN X^(BN) }; N=10 for example.
|> where An, Bn are the constants to be determined.
|> Take the Ln, the problem reduce to find the fit function YFIT2
|>
|> YFIT2 = Ln(YFIT1) = A0 + A1 X^(B1) + A2 X^(B2) + . . . . + AN X^(BN). (1)
|>
|> My question is how can I find the How can I determine the An, Bn ?.
|> By which method? . Where can I find the theory about this?.
|>
|> Note that : A particular case of YFIT2 : when B1=1, B2=2, B3=3,... Bn= n the
|> YFIT2 becomes the general polynomial of degrees ‘n’ i.e.
|> YPOLYN = Ln(YFIT1) = A0 + A1 X^(1) + A2 X^(2) +.. . + AN X^(N). (3)
|> I test the case of (3) the fitting function become very bad if ‘N’ lager than 7,
|> the error at the sampling points is too big even for N smaller than 7,
|> specially at both ends of the data, where the data oscillate rather fast. .
|>
|> Due to stringent demand in my telecom-application, the accuracy of the
|> model function (fitting function), at any sampling event at on the three
|> regions {-180,-178,.....-140}, {-20,-18,...,0, ...20} and {140,172,.....-180} MUST
|> be smaller than 10^ (-5), the errors at other sampling point outside the these
|> regions are not critical for the application.
|> Can we achieve this requirement. ?
|> Is this possible ? and How?.
|>
|> I will be eternally thankful to you for your idea, hint, help etc,.
|>
snip snip ..
your data look a little bit skew and twiggled. Of course you could
smooth them beforehand, but this might simply obscure the problem.
your data suggest that in principle you have a periodic process and
therefore you should use a periodic fit. well, using imaginary values for
the b[i]'s, you will get a perodic fit. whether taking the logarithm
of the original data is useful, depends e.g. on the accuracy of your data
and the kind of errors present. whatever you finally will do:
it makes little sense to approximate the data with a precision which is higher
than the precision of your data. and _taking theoretical periodicity given_
this is not as high a 10^(-5). simply compare the first and the last point.
If it is _not_ periodic, then you must think very carefully on the choice
of the correction, which should compensate for this
If you want to be able simply to reproduce the data,
well, take a fourier fit (using fft, e.g. efftf, efftb from netlib/fftpack)
or a spline-interpolation. but makes this really sense? as someone else
already stated, your data look almost as from an even function, but
near zero the deviations are rather large. a polynomial fit with a real
polynomial (after taking the log) clearly makes no sense.
on the otherside, taking an exponential fit or a trigonometric fit with
unknown frequencies (your b[i]'s) you will get a lot of trouble if
you increase N beyond 3 (yes, 3 (!)) , even if you take a very good
least squares code (public domain least squares code's are listed
at http://plato.la.asu.edu/guide.html). therefore I would propose
to try a linear least squares fit of the logarithmic data first with
a cos-fit, say a0 + a1*cos(phi) + an*cos(n*phi), increasing n
and adding some sin-components (to cope with the skewness) on a trial
and error basis.
hope this helps
peter
Subject: Re: Two Mathematicians
From: maucher@hrz.uni-kassel.de (R Maucher)
Date: 20 Jan 1997 15:18:14 GMT
In message - email@domain
com (Tom) writes:
>Can anybody help me with this problem?
>
>Two mathematicians each think of a number andwhisper it to a mutual
>friend. The friend says, "You each are thinking of a positive integer.
>The product of your numbers is either 8 or 16."
>The first mathematician says, "I don't know your number."
>The second mathematician says, "I don't know your number."
>The first one says, "Give me a hint."
>The second one says, "I know your number."
>What are the numbers.
>
>Thanks for your help.
>
>Kevin Rhine
Hi everybody,
this was really a nice riddle! It has a suprising outcome since you can't
tell the numbers of the two mathematicians, you just can tell how it happens
that the second Mathematician knows the number.
The possible number combinations are:
1 16
1 8
2 4
2 8
4 2
4 4
8 1
8 2
16 1
M1 says he doesn't know, what excludes (16,1) obviously!
Then M2 says he doesn't know either, that excludes (1,16) by the same
reason and (8,1) because if M2 had 1 M1 could only have 16 or 8 but he does
not 16 so he must have 8. That would mean M2 knows the numbers but he doesn't!
M1 asks for a hint what means, he still doesn't know!
That exculdes (1, 8) because if M1 had 1 M2 could only have 8 or 16. But M2
doesn't have 16, so he must have the 8. But then M1 would not ask for a hint.
With equivalent reasoning you can exclude (8,1).
That leaves us with the following possibilities:
2 4
2 8
4 2
4 4
Now M2 says he knows the numbers, that means he does not have a 4 since in
that case M1 could have 2 or 4. By looking at the table you can see what the
numbers are, depending on M2's number, which know of course!
Got it? Any comment?
Roland
Subject: Re: Exponential Data Fitting;
From: D.P.Tran@et.tudelft.nl (Diane)
Date: 20 Jan 1997 16:24:28 GMT
In article <32E33EAD.3002@saab.se>, Gunnar Isaksson, Gunnar.Isaksson@saab.se wrote...
>Well, I have checked the your data and by just simple eyeballing it
>looks like your function is an allmost even function f(x)=f(-x).
> The greatest deviations from that fact is around x=0.
>Measurement errors?
>
>If you have other well founded reasons to believe that the function is
>even, you could fix the data by computing a new function
>z(x)=(f(x)+f(-x))/2 which will fix your data to be an exactly even
>function of the argument x.
>
>Now that the data is transformed to be an even function you just have to
>considerpolynom or other function approximations with even functions.
Thank you very much for your reply.
Exactly, as you quoted my data must be symmetric (even function).
Due to some measurement error (you saw it immediately ! ).
Your recommendation on fixing the data {i.e z(x)=(f(x)+f(-x))/2 }.
before fitting is very (VERY) helpful. I will do that, thanks.
Simple polynomial fitting may be good enough (I’ve done that)
FITFUNC= exp{A0 + A1 X^1 + A2 X^2 + . . . + AN X^N }.
However, orthogonal polynomial fitting
FITFUNC= exp { A0 + A1 X^B1 + A2 X^B2 + . . . + AN X^BN }
may give more better result, and futhermore I would like to learn
the algorithm of orthogonal polynomial fitting (zernike or Legrend
for example; I would appreciate it very much if you can help me on
tracking the related reference.)
Thanks all of you again.
I learn from all of you more (much more) than in the college.
Kindest regards,
Tran
Subject: Re: *SPOILER* Re: Interesting question...
From: Tadeusz Liszka
Date: Mon, 20 Jan 1997 10:49:09 -0600
That seems to be ok till the last moment: only the person with 2 can
answer as there could be 4 and 4
Thomas Sebastian Szymkiewicz wrote:
>
> On Tue, 14 Jan 1997, Thomas Sebastian Szymkiewicz wrote:
>
> > On Tue, 14 Jan 1997, Wayne Hinkin wrote:
> >
> > > Two mathematicians are each given a positive whole number. Each knows
> > > his / her own number but neither one knows the other. They are told
> > > that the product of their numbers is either 8 or 16.
> > >
> > > At some point, one of the mathematicians knows the other number.
> > > What is the number and logically explain / prove it.
> > >
> > > Just for fun...
> > >
> > >
>
> I think I've finally got it now, sorry about the previous confusion:
>
> The possibilities are: 1,2,4,8,16
>
> If one person has 16 then they say that the other person has 1.
> else:
> Neither person has 16, since no one said they did. If one person has 1
> then the other person must have 8.
> else:
> Neither person has 1 or 16, since no one said they did. If one person has
> 8, then the other person can't have 4 so has 2.
> else:
> Neither person has 1, 16 or 8, since no one said they did. We are left
> with two numbers 2 and 4, thus prompting one of the mathematicians
> (whoever has better judgement of how soon to speak up, allowing the other
> mathematician just barely to figure out the answer) to say either 2 or 4.
>
> I hope this works,
> Tom.
>
> --
> Tom Szymkiewicz (mrtom@wpi.edu)
> WPI - Worcester Polytechnic Institute
> USA
>
> WWW: http://www.wpi.edu/~mrtom
> PHYSICS!
--
+-----------------------------+----------------------------+---------+
|Tadeusz J. Liszka |Computational Mechanics Co. |`.\`. |
|phone: (512) 467-0618 ext 526| http://www.comco.com | .\ `. |
|fax: (512) 467-1382 | mailto://tad@comco.com | . \ `.|
+-----------------------------+----------------------------+---------+
|Obligatory disclaimer: |Obligatory cool quote: |
| These opinions are my own, | The public opinion should be really |
| I sometimes agree with them,| alarmed by its own nonexistence |
| no one else has to. | Stanislaw Jerzy Lec, transl. TJL |
+-----------------------------+--------------------------------------+
Subject: 3D Multigrid Help
From: 9879@mne.net (C. Ara Pehlivanian)
Date: Mon, 20 Jan 1997 17:30:23 GMT
I am coding a 3D multigrid solver for the heat equation in (x,y,z,t). I need
some help with the restriction and interpolation operators (restriction
transfers problem from fine to coarse grid, while interpolation transfers
problem from coarse to fine grid). In particular, I would like formulas for
the 3D "half-weighting" restriction operator, and the bilinear interpolation
operator. In 2D, these operators are given by, in stencil form:
R: 0 .125 0 I: .25 .5 .25
.125 .5 .125 .25 1.0 .25
0 .125 0 .25 .5 .25
I think I have the right generalization of the interpolation operator, I, to
3D, but it is not clear how the R operator generalizes, and it seems that I
have it wrong.
Any suggestions? Any other restriction-interpolation pairs I should try? Any
help would be appreciated.
Thank you
Ara Pehlivanian
9879@mne.net
