Newsgroup sci.math.num-analysis 29128

Directory

In article <32AAE45E.41C67EA6@imag.fr>, Rene Aid   wrote:
%Does anyone know where I can find theorem on the composition
%of asymptotic expansion of vector valued functions and norms
%of these fonctions ?
%
%Typically, I want to know wether this proposition holds :
%
%	if  f(e,x) = f_0(x) + f_1(x) e + ... + O(e^p)
%	then || f(e,x) || = || f_0(x) || + G_1(x) e + ... + O(e^p),
%	for any good norm.
%
%Thanks,
%
%Rene Aid
%-- 
%----------------------------------------------------------------
%LMC-IMAG				net : rene.aid@imag.fr
%46, Av. F. Viallet			tel : (33) 04 76 57 48 66
%38000 Grenoble 	France 			fax : (33) 04 76 57 48 03
The expansion does not exist when f(0,x) = 0, since in that case
you can choose e to make any expansion negative.
For nonzero f, the existence of the expansion depends on the
smoothness of the norm at the point f.  For example, the 1-norm
will not have an expansion if one of the components of f is zero.
Pete Stewart

In article <58g7sq$uom@hecate.umd.edu>,
Jason Stratos Papadopoulos  wrote:
%Jean Debord (jdebord@MicroNet.fr) wrote:
%: kamthan@cs.concordia.ca (KAMTHAN pankaj) wrote:
%
%: >Could anybody suggest any pointers to articles on the historical 
%: >evolution of any of the various aspects of numerical analysis: 
%: >LU decompostion, Newton's method, Simpson's rule, etc.?
%
%The more esoteric stuff like Gaussian quadrature, the Euler-Maclaurin
%formula, continued fractions and such can be found in
%
%Herman H. Goldstine, A History of Numerical Analysis from the 16th 
%Through the 18th Century. I think it mentions all the things you want,
%except maybe the LU decomp (should have something about Gaussian elim.
%though)
%
%jasonp
%
Regarding the LU decompostion, Gauss worked with positive definite
matrices and effectively computed an LDL^T decomposition.  The
general LU decomposition is due to Jacobi.  On of the following two
references contains it.
@article{jaco:1857,
   author = "C. G. J. Jacobi",
   year = "1857, posthumous",
   title = "{{\"Uber eine elementare Transformation eines in Buzug
              jedes von zwei Variablen-Systemen linearen
              und homogenen Ausdrucks}}",
   journal = " {Journal f\"ur die reine und angewandte Mathematik}",
   volume = "53",
   pages = "265-270",
   kwds = "la, lud, Sylvester-Jacobi inertia theorem, history"
}
@article{jaco:1857a,
   author = "C. G. J. Jacobi",
   year = "1857, posthumous",
   title = " {\"Uber einen algebraischen Fundamentalsatz und seine
              Anwendungen}",
   journal = " {Journal f\"ur die reine und angewandte Mathematik}",
   volume = "53",
   pages = "275--280",
   kwds = "la, lud"
Be warned that Gauss and Jacobi worked with quadratic and bilinear
forms, not matrices.
GWS

First Announcement - Call for Papers
          Advanced Concepts and Techniques in Thermal Modelling
          Eurotherm Seminar N.53
          October 8-10, 1997
          Faculte Polytechnique de Mons, Mons, Belgium.
The deadlines are :
          Abstracts due         : February 14, 1997
          Final manuscripts due : June 13, 1997
More details on the Web at :
          http://stecwww.fpms.ac.be/EURO53/
or by e-mail to :
          euro53@stecsgi.fpms.ac.be

In article <57spth$93m@ttacs7.ttu.edu>, kesinger@math.ttu.edu (Jake Kesinger) writes:
|> Could someone please comment on whether the following is a (theoretically) 
|> valid method of computing the Lobatto points of degree n+1?
|> 
|> 1.  The Lobatto points of degree 3 are {-1,0,1}.
|> 2.  If x and y are consecutive Lobatto points of degree n, then 
|>     there is exactly one Lobatto point `z' of degree n+1 in the 
|>     interval [x,y].
|> 3.  Each Lobatto point is a simple root of P'_n, so P'n changes sign
|>     at z.
|> 4.  The secant method can be used with initial endpoints [x,y] to 
|>     approximate z.
|> 5.  The other two Lobatto points are -1 and 1.
|> 
|> This method seems to work, but I have been unable to justify it.
|> 
|> I've also come across an algorithm that uses Newton's method to find
|> each Lobatto point with initial guess of cos(j*Pi/n), j=1..n-1, but
|> have not found justification for that, either.
|> 
|> Can anybody point me towards some references regarding such justification?
this comes from the theory of orthogonal polynomials
lobatto-points are -1,1 and the zeroes of pn', there pn is the n-th
legendre polynomial, the orthogonal polynomials for weight 1 on [-1,1]
orthogonal polynomials have all their roots real, simple and inside
their reference interval. by rolle's theorem, this implies the same behaviour 
for the derivatives.
for a complete proof see krylow: approximate calculation of integrals
(the rule is also given in davis&rabinowitz; numerical integration)
hope this helps, peter

In article <329CF205.4C76@softopia.pref.gifu.jp>, Joel Shellman  writes:
|> Has there been any good methods developed for numerical solution of
|> DAE's?  Is there anywhere on the net that has information about it?  I
|> read a book recently and it said there wasn't a good method for this
|> yet.  The book is a few years old, so I'm wondering what the current
|> situation is.
|> 
|> Thanks,
|> 
|> -joel
|> 
|> -- 
|> taotree Tutor and Stuff
|> Math and Physics Solver and thoughts on Creativity
|> http://www.geocities.com/CapeCanaveral/8103/
what about DASSL? there has been a lot of successful work in the field
by e.g. Gear, Leimkuhler, Petzold, Maerz,
whether the problem is "hard" depends on the so called "index" of
your system. index1 is easy and can be treated by Gear's BDF or by imlicit
Runge-Kutta-formulas. DASSl uses BDF . It is available through netlib.
For theory have a look at books e.g. by Roswitha Maerz, Hairer&Norsett;&Wanner;
etc.
hope this helps
peter

> : If all numbers are infinite, and zero is the only whole infinity, then:
> 
> : 1 is a smaller infinity than zero.
> : 2 is a smaller infinity than one.
> : 10 is a smaller infinity than nine.
> : 1 hundred,
> : 1 thousand,
> : 1 billion
> : 1 trillion, in relation to the true value of smaller numbers, or zero,
> : have
> : a smaller and smaller value.
> 
> : It follows that 1+1+1+1... is a convergent number. Each next sum,
> : in relation to the whole, has a smaller true value.
I think you are treating oo to have the same proporties as a normal
number.  In many cases, you considered oo minus oo to equal 0.this is
blatantly wrong and it can be shown in an example:
      lim  ( x^2 - x) = (oo - oo) but the limit goes to +oo
     n->oo 
By your assumtion, it would make sense that the limit goes to zero.
One should also note that infinity is not to be used as a number.  It is
merely an expression of unboundedness.  Look it up in Webster's.  Now,
when you express 0, 1, 2, 1000, etc. as being infinities, you are
violating the definition of infinity.  The number 2 is possible to count
to.  If the range of a function f(x) is entirely below 2, it is most
definately bounded.  A function g(x) increases to infinity if there DOES
NOT EXIST a number M such that the range of g(x) is entirely below M. 
Thus, saying g(x) increases to infinity is an expression of its
unboundedness.
> 
> : Html version of this paper at:
> : http://members.aol.com/spfields1/essays/math.htm
-- 
_____________________________________________________________________
thomas delbert wilkinson 038 henday lister hall university of alberta
If god  were perfect,  why  did He  create  discontinuous  functions?
http://ugweb.cs.ualberta.ca/~wilkinso/

In article <58g85b$bk3@mark.ucdavis.edu>, psalzman@landau.ucdavis.edu (Homer Simpson of the Borg) writes:
|> 
|> Hello all
|> 
|> Simple question:  Is the Crank Nicolson algorithm unitary?
|> 
|> That is, if I use C.N. to solve Schroedinger's equation and my initial
|> condition wavefunction is normalized to 1, will it stay normalized to 1
|> by the n'th iteration?
|> 
|> Thanks!
|> Peter
C.N. is nothing else than the trapezoidal rule applied to a ode.
its propagation function is (for a linear ode y'=\lambda y)
g(z)=(z+h*\lambda/2)/(z-h*\lambda/2) and hence |g(z)|=1 for z on the
imaginary axes, that is C.N. retains the amplitude (but may 
severely disturb the phase!).
hope this helps
peter

In article , j.xiao@mailbox.uq.oz.au (Jinhong Xiao) writes:
|> Does anyone knows where I can find the C source code for Non-Negative
|> Least Squares problem?
|> 
|> 
|>                                                 Many thanks in advance
|>                                                         Jim
have a look at netlib/clapack/dgelss

excuse me .... not for z, for \lambda on the imaginary axes of course
bad hacking ...
peter

Hi, 
Can anybody point to the recent references on Krylov
Integrators.
There was J. Sci. Comp. of 1989 by Tuckerman (Exponential propagation..)
was there any followup by other people?
Stas.

Henry Baker (hbaker@netcom.com) wrote:
: In article <57nf9l$12q@mathserv.mps.ohio-state.edu>,
: mcclure@math.ohio-state.edu (Mark McClure) wrote:
: > In article ,
: > M. TIBOUCHI  wrote:
: > >For more flexibility, you can define complexes as a 2x2 matrix.
: > 
: > I'm not sure I understand the advantage of defining a complex via a 2x2 
: > matrix?  Would anyone care to elaborate?
: One can 'conceive' of a complex as a 2x2 matrix, and have it inherit all
: of the usual matrix operations (whatever they may be).  You can learn a
: lot of linear algebra by specializing all that nxn stuff down to 2x2 matrices,
: and trying to understand how the general operations work in very specific
: instances.
: Enjoy!
I think you need to elaborate a bit more.  After all, representing a complex
by a 2x2 matrix uses twice as much space as it needs to; moreover, the normal
matrix multiplication it inherits would NOT correspond to complex multiplication;
I fail to see ANY advantage to doing this.  Linear operators acting on complex
numbers are nicely represented with 2x2 matrices, but not the complex numbers
themselves.
--
*------------------------------------------------------------------*
* Bill Stockwell          | "The President will keep those         *
* Computing Science       |  promises he INTENDED to keep"         *
* U. of Central Oklahoma  |       -- George Stephanopoulos         *
*------------------------------------------------------------------*

Is there an algorithm for calculating the square of a number that
is fairly quick and doesn't involve much multiplication.  I'm trying
to implement a square routine on an 8-bit processor that only does
simple addition and subtraction.
Dave

I need to solve many sets of algebraic equations. Each
one consists of five equations in five unknowns. Three
of the equations are of degree four, the other two of
degree two.
The three equations of degree four, however, are fixed;
that is, the only difference between two different
sets of equations is that the two second-order equations
are different.
Can this help to solve the systems, assuming I can do a
lot of pre-processing on the three equations which
are always the same?
Also, I am interested only in real solutions whose absolute
value is bounded by 1.
Many thanks,
-Danny Keren.

In article <58k2o6$dvs@canyon.sr.hp.com>, dbrody@sr.hp.com (Dave Brody) wrote:
>Is there an algorithm for calculating the square of a number that
>is fairly quick and doesn't involve much multiplication.  I'm trying
>to implement a square routine on an 8-bit processor that only does
>simple addition and subtraction.
If you only need to square 8-bit numbers, a look-up table won't be very big.

Hi, 
Can anybody point to refs on the
Krylov Integrators?
Thanks. 
Stas

SCI.PHYS removed from newsgroups...
On 10 Dec 1996 16:16:51 GMT, Bill Stockwell  wrote:
>Henry Baker (hbaker@netcom.com) wrote:
>: In article <57nf9l$12q@mathserv.mps.ohio-state.edu>,
>: mcclure@math.ohio-state.edu (Mark McClure) wrote:
>
>: > In article ,
>: > M. TIBOUCHI  wrote:
>: > >For more flexibility, you can define complexes as a 2x2 matrix.
....<> ......
>I think you need to elaborate a bit more.  After all, representing a complex
>by a 2x2 matrix uses twice as much space as it needs to; moreover, the normal
>matrix multiplication it inherits would NOT correspond to complex multiplication
> ;
>I fail to see ANY advantage to doing this.  Linear operators acting on complex
>numbers are nicely represented with 2x2 matrices, but not the complex numbers
>themselves.
Perhaps if M. Tibouchi had used "encode" instead of "define" as in
z=x+iy --> ( x  y )
           (-y  x )
 Then multiplication and/or addition properties are preserved, right?
Not sure what advantages...except to note the isomorphism...
Robert (FEYNMAN is just whimsical hubris)

a) What are they
b) How can I check a number (x) to see if it is a Triangular square
no.? 
Thanks - all help is greatly appreaciated!
Tim
tcs@naturally.clara.net

In article <58k2o6$dvs@canyon.sr.hp.com>, dbrody@sr.hp.com (Dave Brody) writes:
|> Is there an algorithm for calculating the square of a number that
|> is fairly quick and doesn't involve much multiplication.  I'm trying
|> to implement a square routine on an 8-bit processor that only does
|> simple addition and subtraction.
|> 
|> Dave
|> 
    Dave,
did you know that on some RISK processors floating point
multiplication takes oine cycle less than FP addition, due
to the fact that addition (or subtraction) takes a precycle 
to scale the two numbers to a compatible exponent.
                                                  W.
-- 
-----------------------------------------------------
Dr. Wolfgang M. Hartmann  SAS Institute Inc.
saswmh@unx.sas.com        SAS Campus Drive R5228
(919) 677-8000 x7612      Cary, NC 27513
-----------------------------------------------------

In article <58k2dj$ntl@frazier.backbone.ou.edu> ws@aix1.ucok.edu (Bill Stockwell) writes:
 > : One can 'conceive' of a complex as a 2x2 matrix, and have it inherit all
 > : of the usual matrix operations (whatever they may be).
...
 > I think you need to elaborate a bit more.  After all, representing a complex
 > by a 2x2 matrix uses twice as much space as it needs to; moreover, the normal
 > matrix multiplication it inherits would NOT correspond to complex multiplication;
 > I fail to see ANY advantage to doing this.
Represent a + bi as
    [ a   -b ]
    [ b    a ]
and show in what way matrix multiplication does NOT correspond to complex
multiplication.
-- 
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn  amsterdam, nederland; http://www.cwi.nl/~dik/

In article , saswmh@pascal.unx.sas.com (Wolfgang Hartmann) writes:
|> 
|> In article <58k2o6$dvs@canyon.sr.hp.com>, dbrody@sr.hp.com (Dave Brody) writes:
|> |> Is there an algorithm for calculating the square of a number that
|> |> is fairly quick and doesn't involve much multiplication.  I'm trying
|> |> to implement a square routine on an 8-bit processor that only does
|> |> simple addition and subtraction.
|> |> 
|> |> Dave
|> |> 
|> 
|>     Dave,
|> did you know that on some RISK processors floating point
|> multiplication takes oine cycle less than FP addition, due
                                   ^^^^^
                        sorry, I meant more!
|> to the fact that addition (or subtraction) takes a precycle 
|> to scale the two numbers to a compatible exponent.
|>                                                   W.
|> 
|> -- 
|> 
|> -----------------------------------------------------
|> Dr. Wolfgang M. Hartmann  SAS Institute Inc.
|> saswmh@unx.sas.com        SAS Campus Drive R5228
|> (919) 677-8000 x7612      Cary, NC 27513
|> -----------------------------------------------------
-- 
-----------------------------------------------------
Dr. Wolfgang M. Hartmann  SAS Institute Inc.
saswmh@unx.sas.com        SAS Campus Drive R5228
(919) 677-8000 x7612      Cary, NC 27513
-----------------------------------------------------

In article , saswmh@pascal.unx.sas.com (Wolfgang Hartmann) writes:
|> 
|> In article <58k2o6$dvs@canyon.sr.hp.com>, dbrody@sr.hp.com (Dave Brody) writes:
|> |> Is there an algorithm for calculating the square of a number that
|> |> is fairly quick and doesn't involve much multiplication.  I'm trying
|> |> to implement a square routine on an 8-bit processor that only does
|> |> simple addition and subtraction.
|> |> 
|> |> Dave
|> |> 
|> 
|>     Dave,
|> did you know that on some RISK processors floating point
|> multiplication takes oine cycle less than FP addition, due
Sorry, I'm not in a good shape today: mult may be faster than add
since add needs scaling
|> to the fact that addition (or subtraction) takes a precycle 
|> to scale the two numbers to a compatible exponent.
|>                                                   W.
|> 
|> -- 
|> 
|> -----------------------------------------------------
|> Dr. Wolfgang M. Hartmann  SAS Institute Inc.
|> saswmh@unx.sas.com        SAS Campus Drive R5228
|> (919) 677-8000 x7612      Cary, NC 27513
|> -----------------------------------------------------
-- 
-----------------------------------------------------
Dr. Wolfgang M. Hartmann  SAS Institute Inc.
saswmh@unx.sas.com        SAS Campus Drive R5228
(919) 677-8000 x7612      Cary, NC 27513
-----------------------------------------------------

I hope your answer was "NO!"  The angles will be equal only for
equlateral triangles.  But a counter-example to the claim that
the angles are equal for any triangle is to show that when one
side is reduced to a very small length, the angle opposite it
(whether at the vertex or at G) is reduced also to a very small
value.
On Sun, 08 Dec 1996 20:00:27 -0500 in sci.math.num-analysis,
Regis Mesnier (swann2@earthlink.net) wrote:
> Dear net users,
> Having been recently asked by one of my student if (considering any
> triangle ABC and G the center of gravity of that triangle) the angles
> AGB, BGC, and CGA were all eguals (to 120 degres, of course), I came up
> with an answer, but could not find the formal proof for it. If any of
> you could indicate me where I'd be able find it, or even show it here, I
> would sincerely appreciate it. Thanks in advance.
>       Regis Mesnier, e-mail:swann2@earthlink.net
--
Prof. Ronald J. Hartranft          http://www.Lehigh.edu
Dept. of Mech. Engr. & Mechanics         /~rjh2/rjh2.html
Lehigh University                  Phone: 610-758-4109
19 Memorial Drive West             Email: rjh2@Lehigh.edu
Bethlehem, Penn. 18015-3085

Optech Solutions is proud to announce its super efficient optimization
software.
Details can be found at:
http://www.wbm.ca/users/optimize/
Thank you. We look forward to serving you. Even Jeffrey J. Leaderer who
insists on flaming this simple announcement and taking it off the
newsgroup... thus potentially depriving many of a fine optimization
product.
JP
-- 
Dr. Jim Pulfer
President
Optech Solutions
Box 123
Delisle,  SK
S0L  0P0    Canada
E-mail: optimize@eagle.wbm.ca
http://www.wbm.ca/users/optimize/

In article <58jpcg$1krd@rs18.hrz.th-darmstadt.de>,
spellucci@mathematik.th-darmstadt.de (Peter Spellucci) wrote:
> In article ,
j.xiao@mailbox.uq.oz.au (Jinhong Xiao) writes:
> |> Does anyone knows where I can find the C source code for Non-Negative
> |> Least Squares problem?
> |> 
> |> 
> |>                                                 Many thanks in advance
> |>                                                         Jim
> have a look at netlib/clapack/dgelss
Thanks for those people who provided their helps and suggestions. Although
I could not find the C source code for NNLS directly, I did find helpful
to translate Fortran to C from these suggestions. Now the problem is
solved. 
Thanks again
Jim

http://www.geocities.com/SiliconValley/Park/1879

n8tm@aol.com writes:
>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
Haven't seen the original postings on this. I tried to get LAPACK++ working
on our DECalphas here. It failed a couple of tests. I was disappointed. 
It looks well-designed though, and I might borrow some of their ideas.
lf.

(Please display with monospaced font)
All triangular square numbers are given by,
z  = 1,   z  = 36,   z  = 34z    - z    + 2
 1         2          n      n-1    n-2
Alternatively,
                ___  n             ___  n
      ( 3 + 2 \/ 2  )  + ( 3 - 2 \/ 2  )  - 2
z  = -----------------------------------------
 n                      32
e.g.,  1, 36, 1225, 41616, 1413721, ...
Solution: Euler 1732-33
Proof of completeness: Roberts, 1879
See Martin Gardner, Scientific American, July 1974, for discussion.

Hi there !
Can anybody give me the titles of CDROMs featuring math routines for
Object/Borland Pascal (preferably those WITH their source codes)
and/or send me any interesting math web-site.
I personally work with Borland Delphi 2.0 and am looking for almost
any math unit or routine available for numerical analysis, number
theory and especially 'infinite precision (decimal) operations'.
Of course sites with similar topics are also warmly welcome.
I plan to open a math site with links to those sites in the near
future.
Please add - if available - a short description to each CDROM title or
web-site.
And those reading the various German newsgroups I posted this message
to, please forgive me for sending it in English. However I'm sure that
those German guys have all a pretty good command of the English
language.
Please send your reply via email ONLY (as I do not regularly visit all
these newsgroups and replies always tend to get lost miraculously).
My email address is:
landauf@adis.at
THANK YOU VERY MUCH ! 

On 10 Dec 1996 02:18:03 GMT, oliver@oak.math.ucla.edu (Mike Oliver)
wrote:
>In article <32A5DCE9.1DE3@grc.varian.com> mirko.vukovic@grc.varian.com writes:
>
>>There is a book on the history of Pi by Petr Bekkman (or some spelling
>>like
>>that).  Check it out.
>
>Yes, do!  But don't take everything at quite face value.
>
>It's a very enjoyable and informative book, but has some minor "crankish"
>aspects; I can't remember the details right now.
>
>By the way his surname was Beckman or possibly Beckmann.
The reference:
A HISTORY OF # (PI)
Petr Beckmann
Electrical Engineering Department,
University of Colorado
ST MARTIN'S PRESS
New York
Copyright (c) 1971 by THE GOLEM PRESS
All rights reserved. For information, write:
St. Martin's Press, Inc., 175 Fifth Ave., New York, N. Y. 10010.
Manufactured in the United States of America
Library of Congress Catalog Card Number: 74-32539
First edition preface signed: Bolder, Colorado, August 1970
Second edition preface signed: Bolder, Colorado, May 1971
Third edition preface signed: Bolder, Colorado, Christmas 1974

Hello,
Does anybody has a solution or reference to the following problem?
Let G be a symmetric and positive definite matrix, block partitioned
as
   [G_11 G_12 G_13 ... G_1n]
   [G_21 G_22 ....     G_2n]
   [...                    ]
   [...                    ]
   [G_n1 G_n2 ....     G_nn]
where all blocks are square and equally sized, and G_ii is positive
semi-definite for each i (which might be obvious?)
The problem is to minimize
   Q(a) = (\sum_{i=1}^n a_i G_ii)^{-1}
           * (\sum_{i,j} a_i a_j G_ij)
           * (\sum_{i=1}^n a_i G_ii)^{-1}
over all vectors a={a_i}, i=1,...,n, satisfying a_1+...+a_n=1 and a_i>=0
for each i. The minimization should be done in the sense of "definiteness",
i.e. if a* is the optimal vector and a is any other vector, then Q(a)-Q(a*)
is positive semi-definite.
The middle part of the expression can be viewed as a quadratic form in
the matrix blocks, while the outer parts, that are inverted, are linear
combinations of the diagonal blocks.
The questions are if a vector a* that is optimal in the sense above exists,
and, if so, if there is an algorithm to compute it?
Best wishes,
Tobias Ryden
-- 
-- Tobias Rydén                      E-mail: tobias@maths.lth.se
   Dept. of Mathematical Statistics  Tel:    int+46-46 222 4778
   Lund University                   Fax:    int+46-46 222 4623 
-- Box 118, S-221 00 Lund, Sweden    WWW:    www.maths.lth.se/matstat

Dear Netters,
I have got a problem to solve a partial differential equation using 
the Galerkin method. At the end of the calculation, I obtain a system 
of differential equations that is not solvable. What can I do? 
I thank you in advance.
Delphine Wolfersberger.

In article  Dik T. Winter, dik@cwi.nl
writes:
>In article <58k2dj$ntl@frazier.backbone.ou.edu> ws@aix1.ucok.edu (Bill Stockwell) writes:
> > : One can 'conceive' of a complex as a 2x2 matrix, and have it inherit all
> > : of the usual matrix operations (whatever they may be).
>...
> > I think you need to elaborate a bit more.  After all, representing a complex
> > by a 2x2 matrix uses twice as much space as it needs to; moreover, the normal
> > matrix multiplication it inherits would NOT correspond to complex multiplication;
> > I fail to see ANY advantage to doing this.
>
>Represent a + bi as
>    [ a   -b ]
>    [ b    a ]
>and show in what way matrix multiplication does NOT correspond to complex
>multiplication.
>
it makes fine sense to me. Multiplication by i corresponds
to a rotation of 90 DEG in the complex plane.
Multiplication by a complex number corresponds to a
rotation and a scaling in the complex plane. A complex
number can be written  r * EXP(i*theta). Similarly, the
number
| a  -b |
| b   a  |
can be written as
| cos(theta)   -sin(theta)  |
| sin(theta)     cos(theta) |    * R
where R = SQRT(a^2+b^2), or SQRT(Determinant), and
Theta = arctan(b/a)
So R scales, and the matrix rotates.
Mike Yukish
Applied Research Lab
may106@psu.edu
http://elvis.arl.psu.edu/~may106/

thomas delbert wilkinson wrote:
> 
> > I'm looking for a routine that can create a cubic spline fit from an
> > arbitrary set of points in 3 dimensions, represented by $(x,y,z,v)_i$
> > (preferrably in a weighted least squares sense).
> >
> > For 2 dimensions (i.e. for surfaces with points $(x,y,v)_i$ )these
> > routines already exist. For example such a routine is given by NAGs
> > E02DAF.
> 
> Is it possible that you can call the 2-D function to calculate $(x,y,v)$
> and call it again for $(z,0,v)$?  Mathematically, it makes sense because
> you are dealing with linearly independant functions, ie. the value of
> the spline for $(z,0,v)$ should have ZERO effect on the values for
> $(x,y,v)$.
> 
> The only problems I can see for using this idea is if the functions
> require that the known values of $(x,y,v)$ are stored in a
> two-dimensional array instead of three one-dimensional array.
> 
> A better idea is if there is code to produce a one-dimensional spline
> $(x,v)$, call it three times, for $(x,v)$, $(y,v)$, $(z,v)$.  This is
> also valid because x, y, and z are linearly independant, but it is a
> saivngs in work done by the computer because calculating $(z,0,v)$
> involves a waste of work because the function will calculate a spline
> for $(0, v)$
> 
> > Secondly, has anybody experience with these kind of representations.
> 
> I've been toying with it, but I don't have any code that works
> completely, because I have been trying to code a spline funtion by
> myself.
> 
> --
> _____________________________________________________________________
> thomas delbert wilkinson 038 henday lister hall university of alberta
> If god  were perfect,  why  did He  create  discontinuous  functions?
> http://ugweb.cs.ualberta.ca/~wilkinso/
	You may want to try using smoothing splines. The free version of
O-Matrix has a subrotuine that will fit smoothing splines of any order
in any dimension (search for "smoothing splines" in its help index).
To obtain a copy of the free version of O-Matrix see:
	http://world.std.com/~harmonic
The method they use is based on the article
	"Surface fitting with scattered noisy data
	on Euclidean d-space and on the sphere", 
	by G. Wahba,
	Rocky Mountain Journal of Mathematics, 
	Volume 14, 
	Number 1, 
	1984.