Subject: Re: numerical integration
From: mathar@qtp.ufl.edu (Richard Mathar)
Date: Sat, 16 Nov 1996 16:53:40 EST
In article <328B495B.5020@eleceng.ucl.ac.uk>, Channing Walton writes:
|> Hi,
|> I developed a method of of integration which seems to be very quick and accurate and
|> was wondering what problems may occur with it or if it has been done before (very
|> likely).
|> It goes like this:
|> You wish to integrate a function between A and B,
|> fit a polynomial (5th order seems good) to the function and you have your first
|> approximation.
|> then repeat by integrating between A and (B-A)/2 + integral between (B-A)/2 to B
|> if the result is within a suitable tolerance to the first attempt then end
|> if it is not then continue dividing and checking.
|>
|> the point been that incoding this you can use a simple recursive procedure.
|>
|> Now there are obvious problems with particular functions but if you ensure that the
|> integral is chopped up into sufficient pieces then all is well.
|>
|> I have found this to be very quick.
|>
If you replace your 5th order polynomial with
a 2nd order one, you have the Simpson rule of
integration. Using higher order polynomials than
3 (the famous cubic spline) on parts of integration
intervals is usually avoided, because at the endings
of the interval higher order polynomials tend to form
large oscillations. The important property of
the Simpson rule is that you can re-use all function
values of one integration step also for the next, because
the discrete points for the calculation with 2N
abscissa points in the next refinement contain the N
points of the previous plus N new points in the middle
of the previous intervals. If you have a interpolation
with a polynomial of odd order, the division of the
original interval by two means you cannot use any of
the previous abscissa values but those two at the endings...
--
mathar@qtp.ufl.edu
Subject: Re: DFT, Hilbert Transform, and causality
From: mathar@qtp.ufl.edu (Richard Mathar)
Date: Sat, 16 Nov 1996 16:43:29 EST
Patrice Koehl writes:
|>
|> Suppose I have a continous signal x(t), causal. Its Fourier transform,
|> X(f) is continuous, and verify the Kramers Kronig relations, i.e.
|> Re(X(f)) and Im(X(f)) are related through a Hilbert Transform. This
|> I understand.
|>
|> Now let suppose I have a discrete signal, xn, which is non zero for
|> n = 0, 1, ..., N-1. I also suppose xn to be complex, in which case
|> the discrete signal I have contains 2*N experimental information.
|> Its Discrete Fourier Transform, Xn, is also complex, and is calculated
|> over N frequency. However, according to the Kramers Kronig relationship,
|> the imaginary parts and the real parts of Xn are not independant, and
|> can be derived from each other using a discrete Hilbert Transform.
|> That would mean that Xn is composed of N independant information,
|> while xn had 2*N information. Where did the N other values go, knowing
|> that DFT is linear and invertible ?
|> Furthermore, if I throw away the complex part of Xn and recalculate it
|> using the discrete Hilbert Transform of Xn, I don't come back to the
|> original values. Am I missing something ? What is wrong in my
|> reasoning ?
If we have a discrete signal xn with n=0,...,N-1
nonzero complex values, which is in addition supposed
to be causal, the input to the DFT are n=0,....,2N-1
complex values, the last N of which are zero. They are
needed to represent the values for negative times (and are
mapped to the end of the data stream by the circular
property of the DFT). These 2N-1 complex values xn carry
N nonzero complex values, i.e. 2N real "pieces" of information.
After the DFT of the 2N ("zero-padded") input values we get
2N complex values Xn, the last N of which represent
the negative frequencies (again by the circular/periodic
property, in the frequency region now). These 2N values Xn
(n=0,..,2N-1) should be conform with the Kramers-Kronig
transform (i.e. real and imaginary part be a Hilbert pair).
The 2N complex values Xn have in principal 4N real values,
but the Hilbert relationship reduces this to be actually 2N
real values of information. So there's nothing lost.
--
mathar@qtp.ufl.edu
Subject: Re: complex Newton's method
From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Date: 16 Nov 1996 21:42:15 -0500
In article <56l5f5$noa@newstand.syr.edu>, Tom Scavo wrote:
:In article <56ahjg$ku4@mcmail.CIS.McMaster.CA>,
:Zdislav V. Kovarik wrote:
:>In article <567nn3$4s1@newstand.syr.edu>, Tom Scavo
:wrote: >>>
:>>Anyone who has studied Newton's method on the real line knows its
:>>geometric interpretation, that is, a sequence of tangent lines whose
:>>zeros converge to a root of a function. What is the corresponding
:>>geometric interpretation of Newton's method in the complex plane?
:>
:> Before it becomes tangled in the intricacies of fractals, let me sum up
:>the "good news":
:> [...]
:
:Nice analysis, but I didn't see anything that looked like
:geometry. So if one considers the complex Newton method as the
:iteration of a function from R^2 into R^2, does a geometric
:interpretation emerge? I'd be keenly interested in hearing
:about that.
[...]
(1) I'm not sure you want to see, as you wrote,
"the complex Newton method as the iteration of a function
from R^2 into R^2"
because, although possible, such an illustration is not the popular one:
the popular one is a neatly drawn graph of a convex (i.e. concave up
according to recent textbooks) increasing function which crosses the
horizontal axis, with the familiar segments of tangent lines at the
appropriate points of the graph. (That's what you indicated earlier.)
The desired extension to complex Newton's method takes place in a
4-dimensional space, considered as spanned by two perpendicular complex
lines (z-line and w-line, to give them names).
Then the graph of an analytic function f is a complex curve, and at a
point z_0 in the domain of f, we draw a complex line which is the tangent
to the graph, and has equation w = f(z_0) + f'(z_0) * (z - z_0).
(It is, by the way, made of the first two terms of the Taylor expansion
of f, and analytic functions have such expansions.)
Then we intersect this complex line with the complex line w=0 and obtain
a complex point z_1 = z_0 - f(z_0)/f'(z_0). The we continue creating a
complex point z_2 using z_1 etc.
If you have four-dimensional imagination (which I don't), everything will
be immediately clear. And before you explode with anger over such a
disappointing answer, let me remark that I read in a book about the theory
of singularities ("catastrophe theory") that a blind mathematician who
worked on this research had quite a clear para-visual idea about
singularities in spaces of dimension higher than three. I wish I
remembered which book it was (Arnold?).
Cheers, ZVK (Slavek).
Subject: Re: fortran77 compiler for sun/sparc 20
From: John Turner
Date: 16 Nov 1996 23:54:03 -0600
nmm1@cus.cam.ac.uk (Nick Maclaren) writes:
> Yes. NAG Fortran 90 is available for Suns, though Sun also have their
> own Fortran 90 (which, I believe, uses NAG technology under licence).
No, SunSoft F90 is based on CraySoft F90.
> You could probably also port g77, if you are adventurous :-)
No porting necessary. Should build on Suns quite easily.
--
John A. Turner
Los Alamos National Laboratory, MS B226, Los Alamos, NM 87545
Group: XTM (Radiation Transport Methods)
Location: TA-3, Bldg. 43, Rm. D150
Phone: 505-665-1303 e-mail: turner@lanl.gov
