Newsgroup sci.math.num-analysis 29605

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longam@hotmail.com wrote:
>I need to find the inverse of a Hermitian covariance matrix. The 
>covariance matrix is formed by averaging the sample covariances of the 
>form: R = XX'  Each sample covariance is formed by multiplying a sample 
>column vector by its transpose.
I am also interested in finding an efficient way to solve for an
inverse covariance matrix.  Most of the straight forward  methods are
N^3 in complexity, and I would like to hear from anyone you has
encountered and tackled this problem.  Can Woodbury's identity be used
here?
Also, does anybody know why using an adaptive filter (such as a
transversal FIR using LMS) to solve for the inverse is not a good
solution?  I thought that since the solution to the matrix equations
is just the well-known Wiener-Hopf equations, one can find the optimal
filter weights iteratively instead of taking a huge inverse of a
matrix.
Any help is greatly appreciated.
Regards,
Jim
Jim Shima
Advanced Signal Processing
Texas Instruments, Inc.
email:  shima@ti.com
*********************************************************
* Opinions stated are mine and not representative of TI *
*********************************************************

In article <01bc0230$91ac3880$ef5857c2@cyril.orc.ru>, "Cyril Nickanorov"  writes:
|> Hello!
|> 
|> Does anybody knows where on the Internet I can find algorithms library
|> especially in numerical differentiation & integration, math phisics etc.
|> 
|> Best regards.
|> -- 
|> 
|> Cyril Y. Nickanorov 
|> E-mail: cyril@orc.ru
http://netlib.no should be a good starting point for you, 
hope this helps
peter

In article <5bish0$m2p@news.cc.ucf.edu>, drollins@pegasus.cc.ucf.edu (David Rollins) writes:
> Anyone got any references to the details of doing numerical simulations
> of the initial value problem for Korteweg-deVries (and related equations)
> equation.
> Texts?, papers? web resourses?
 Here are a few. Hope they help.
 Mark.
@article{BDK,
  author  = {Bona, J.L. and Dougalis, V.A. and 
             Karakashian, O.A. and McKinney, W.R.},
  title   = {Conservative, High-Order Numerical Schemes for the 
             Generalised {K}orteweg-de {V}ries Equation},
  pages   = {107-164},
  journal = {Philosophical Transactions of the Royal Society of London. 
             Series A},
  volume  = {351},
  year    = {1995}  }
@article{KM,
  author  = {Karakashian, O.A. and McKinney, W.R.},
  title   = {On Optimal High-Order in Time Approximations for 
             the {K}orteweg-de {V}ries Equation},
  pages   = {473-496},
  journal = {Mathematics of Computation},
  volume  = {55},
  number  = {192},
  year    = {1990}  }
@article{VL,
  author  = {Vliegenthart, A.C.},
  title   = {On Finite-Difference Methods for the 
             {K}orteweg-de {V}ries Equation},
  pages   = {137-155},
  journal = {Journal of Engineering Mathematics},
  volume  = {5},
  number  = {2},
  year    = {1971}  }
@article{GM,
  author  = {Greig, I.S. and Morris, J.L.},
  title   = {A Hopscotch Method for the {K}orteweg-de-{V}ries Equation},
  pages   = {64-80},
  journal = {Journal of Computational Physics},
  volume  = {20},
  year    = {1976}  }
@article{SCH3,
  author  = {Schoombie, S.W.},
  title   = {Spline {P}etrov-{G}alerkin Methods for the Numerical Solution 
             of the {K}orteweg-de {V}ries Equation},
  pages   = {95-109},
  journal = {{IMA} Journal of Numerical Analysis},
  volume  = {2},
  year    = {1982}  }
@article{GGA,
  author  = {Gardner, L.R.T. and Gardner, G.A. and Ali, A.H.A.},
  title   = {Simulations of Solitons Using Quadratic Spline Finite Elements},
  pages   = {231-243},
  journal = {Computer Methods in Applied Mechanics and Engineering},
  volume  = {92},
  year    = {1991}  }

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,.
Thank you in anticipation.
Tran
  {Xi}                        {Yi}
-1.8000000e+002  1.4082994e-001
 -1.7800000e+002  1.3807657e-001
 -1.7600000e+002  1.2646053e-001
 -1.7400000e+002  1.0663137e-001
 -1.7200000e+002  8.1553011e-002
 -1.7000000e+002  5.8885043e-002
 -1.6800000e+002  5.3761948e-002
 -1.6600000e+002  7.1532763e-002
 -1.6400000e+002  9.6270893e-002
 -1.6200000e+002  1.1611010e-001
 -1.6000000e+002  1.2617840e-001
 -1.5800000e+002  1.2463213e-001
 -1.5600000e+002  1.1178409e-001
 -1.5400000e+002  9.0862567e-002
 -1.5200000e+002  6.9171947e-002
 -1.5000000e+002  5.8640140e-002
 -1.4800000e+002  6.4677764e-002
 -1.4600000e+002  7.6760006e-002
 -1.4400000e+002  8.3990468e-002
 -1.4200000e+002  8.3095977e-002
 -1.4000000e+002  7.7102773e-002
 -1.3800000e+002  7.4057662e-002
 -1.3600000e+002  7.9620518e-002
 -1.3400000e+002  8.9855569e-002
 -1.3200000e+002  9.8298072e-002
 -1.3000000e+002  1.0198587e-001
 -1.2800000e+002  1.0482382e-001
 -1.2600000e+002  1.1275739e-001
 -1.2400000e+002  1.2704571e-001
 -1.2200000e+002  1.4246724e-001
 -1.2000000e+002  1.5268452e-001
 -1.1800000e+002  1.5510829e-001
 -1.1600000e+002  1.5375173e-001
 -1.1400000e+002  1.5715927e-001
 -1.1200000e+002  1.6860871e-001
 -1.1000000e+002  1.8195752e-001
 -1.0800000e+002  1.9100072e-001
 -1.0600000e+002  1.9537320e-001
 -1.0400000e+002  1.9864379e-001
 -1.0200000e+002  2.0414794e-001
 -1.0000000e+002  2.1255435e-001
 -9.8000000e+001  2.2230539e-001
 -9.6000000e+001  2.3125174e-001
 -9.4000000e+001  2.3694361e-001
 -9.2000000e+001  2.4035048e-001
 -9.0000000e+001  2.4648756e-001
 -8.8000000e+001  2.6012375e-001
 -8.6000000e+001  2.7789133e-001
 -8.4000000e+001  2.9281276e-001
 -8.2000000e+001  3.0313773e-001
 -8.0000000e+001  3.1342959e-001
 -7.8000000e+001  3.2825020e-001
 -7.6000000e+001  3.4673685e-001
 -7.4000000e+001  3.6661903e-001
 -7.2000000e+001  3.8719523e-001
 -7.0000000e+001  4.0715990e-001
 -6.8000000e+001  4.2356006e-001
 -6.6000000e+001  4.3800090e-001
 -6.4000000e+001  4.5340886e-001
 -6.2000000e+001  4.7098275e-001
 -6.0000000e+001  4.9011726e-001
 -5.8000000e+001  5.0982955e-001
 -5.6000000e+001  5.3017592e-001
 -5.4000000e+001  5.5058579e-001
 -5.2000000e+001  5.7106429e-001
 -5.0000000e+001  5.9202494e-001
 -4.8000000e+001  6.1304168e-001
 -4.6000000e+001  6.3289257e-001
 -4.4000000e+001  6.5039920e-001
 -4.2000000e+001  6.6653046e-001
 -4.0000000e+001  6.8282592e-001
 -3.8000000e+001  7.0044655e-001
 -3.6000000e+001  7.2056805e-001
 -3.4000000e+001  7.4366099e-001
 -3.2000000e+001  7.6952016e-001
 -3.0000000e+001  7.9572866e-001
 -2.8000000e+001  8.2081446e-001
 -2.6000000e+001  8.4330563e-001
 -2.4000000e+001  8.6322697e-001
 -2.2000000e+001  8.8141411e-001
 -2.0000000e+001  8.9861776e-001
 -1.8000000e+001  9.1399748e-001
 -1.6000000e+001  9.2777999e-001
 -1.4000000e+001  9.3963676e-001
 -1.2000000e+001  9.5150265e-001
 -1.0000000e+001  9.6355165e-001
 -8.0000000e+000  9.7505699e-001
 -6.0000000e+000  9.8497453e-001
 -4.0000000e+000  9.9230459e-001
 -2.0000000e+000  9.9718331e-001
  0.0000000e+000  9.9962015e-001
  2.0000000e+000  1.0000115e+000
  4.0000000e+000  9.9924044e-001
  6.0000000e+000  9.9745888e-001
  8.0000000e+000  9.9513041e-001
  1.0000000e+001  9.9132258e-001
  1.2000000e+001  9.8602971e-001
  1.4000000e+001  9.7850940e-001
  1.6000000e+001  9.6835589e-001
  1.8000000e+001  9.5535548e-001
  2.0000000e+001  9.3956104e-001
  2.2000000e+001  9.2078874e-001
  2.4000000e+001  8.9958043e-001
  2.6000000e+001  8.7570938e-001
  2.8000000e+001  8.5031531e-001
  3.0000000e+001  8.2359746e-001
  3.2000000e+001  7.9637937e-001
  3.4000000e+001  7.6950244e-001
  3.6000000e+001  7.4396927e-001
  3.8000000e+001  7.2030263e-001
  4.0000000e+001  6.9889189e-001
  4.2000000e+001  6.7905508e-001
  4.4000000e+001  6.5978889e-001
  4.6000000e+001  6.3931516e-001
  4.8000000e+001  6.1603445e-001
  5.0000000e+001  5.9092178e-001
  5.2000000e+001  5.6650663e-001
  5.4000000e+001  5.4418304e-001
  5.6000000e+001  5.2474704e-001
  5.8000000e+001  5.0689733e-001
  6.0000000e+001  4.8874801e-001
  6.2000000e+001  4.6929400e-001
  6.4000000e+001  4.4859559e-001
  6.6000000e+001  4.2841039e-001
  6.8000000e+001  4.1109292e-001
  7.0000000e+001  3.9694926e-001
  7.2000000e+001  3.8415810e-001
  7.4000000e+001  3.7016470e-001
  7.6000000e+001  3.5319130e-001
  7.8000000e+001  3.3369523e-001
  8.0000000e+001  3.1533342e-001
  8.2000000e+001  3.0091581e-001
  8.4000000e+001  2.8971101e-001
  8.6000000e+001  2.7985624e-001
  8.8000000e+001  2.6964042e-001
  9.0000000e+001  2.5948365e-001
  9.2000000e+001  2.5143747e-001
  9.4000000e+001  2.4498246e-001
  9.6000000e+001  2.3674456e-001
  9.8000000e+001  2.2540316e-001
  1.0000000e+002  2.1413615e-001
  1.0200000e+002  2.0619114e-001
  1.0400000e+002  2.0250232e-001
  1.0600000e+002  2.0098100e-001
  1.0800000e+002  1.9766738e-001
  1.1000000e+002  1.8897093e-001
  1.1200000e+002  1.7561435e-001
  1.1400000e+002  1.6406275e-001
  1.1600000e+002  1.6046303e-001
  1.1800000e+002  1.6172792e-001
  1.2000000e+002  1.5928688e-001
  1.2200000e+002  1.4848925e-001
  1.2400000e+002  1.3100564e-001
  1.2600000e+002  1.1436944e-001
  1.2800000e+002  1.0698059e-001
  1.3000000e+002  1.0764156e-001
  1.3200000e+002  1.0781893e-001
  1.3400000e+002  1.0191544e-001
  1.3600000e+002  9.1119221e-002
  1.3800000e+002  8.2397683e-002
  1.4000000e+002  8.2040821e-002
  1.4200000e+002  8.7507444e-002
  1.4400000e+002  9.0505506e-002
  1.4600000e+002  8.6501771e-002
  1.4800000e+002  7.6713182e-002
  1.5000000e+002  6.8523573e-002
  1.5200000e+002  7.2806474e-002
  1.5400000e+002  8.9517924e-002
  1.5600000e+002  1.0899711e-001
  1.5800000e+002  1.2293768e-001
  1.6000000e+002  1.2721696e-001
  1.6200000e+002  1.2042593e-001
  1.6400000e+002  1.0464055e-001
  1.6600000e+002  8.4970857e-002
  1.6800000e+002  7.0164107e-002
  1.7000000e+002  6.9935068e-002
  1.7200000e+002  8.4199595e-002
  1.7400000e+002  1.0372060e-001
  1.7600000e+002  1.2214902e-001
  1.7800000e+002  1.3529757e-001
  1.8000000e+002  1.4115296e-001

Michel OLAGNON writes:
> optimized BLAS that I know of:
> Sun Solaris : libsunperf  (unfortunately not free, from Sun, or licensed from
...[details on other vendors]
HP-UX:    /opt/fortran/lib/libblas.a & /opt/fortran/lib/pa1.1/libblas.1
It is included with the fortran compiler without additional charge.
(not an official response)
-- 
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
Tim Butler             Massachusetts Language Lab - Hewlett Packard  
e-mail: butler@apollo.hp.com

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

OK it sounds like you have decided to take logs and now just want to do
polynomial fit.  The best way is to do orthogonal polynomial fit.  I
personally have done this to order 60 on around 500 data points and the
routine worked fine.
See the following book for explanation and Fortran code (p.264).
S.D. Conte & C. de Boor (1980) Elementary numerical analysis: An
algorithmic approach.  NY: McGraw-Hill.
Do you realise there are other ways to approximate your set of points?
For example splines?
Bill Simpson

Hartmut Schmider wrote:
> 
See:
http://www.ncgia.ucsb.edu/pubs/spherekit/main.html  .
> Hello,
> 
>         I hope I am not too off-topic here, but I tried to pick the two
> newsgroups that seem close enough. I am looking for fast algorithms for
> interpolations on the surface of a sphere. The grid I have consists of
> spherical triangles, but has no discernible regularity. I guess the best
> way to proceed is to find the three points closest to the interpolation
> point (that is the first part of the problem; how to do that without
> calculating the distance from all mesh points?), and then to do the actual
> interpolation (seems to me the graphics folks must be doing that all the
> time).
> 
> Does anyone have suggestions or literature I can look at? If you send me
> answers by email, I collect them and repost them after a while.
> 
> Regards, Hartmut "for the ones just as clueless as myself" Schmider
> 
> --
> Hartmut Schmider                   ~  Remember there's a big difference
> Dept. Chem. Queen's University     ~  between kneeling down and bending
> Kingston, Ontario, K7L 3N6 CANADA  ~  over. FZ
> hasch@ct3a.chem.queensu.ca         ~
-- 
_________________________________________________________________
Robert Raskin                         ECS Science Liaison at JPL
Mail Stop 525-389                     raskin@searider.jpl.nasa.gov
Jet Propulsion Laboratory             (phone)(818) 306-6061
California Institute of Technology    (fax)  (818) 306-6929
Pasadena, CA 91109

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