Re: Searching zeros of complex function

From: Gerry Thomas (gfthomas_at_sympatico.ca)
Date: 05/15/04 

Date: Fri, 14 May 2004 19:01:51 -0400

"Steven G. Kargl" <kargl@troutmask.apl.washington.edu> wrote in message
news:c83ghc$amk$1@nntp6.u.washington.edu...
> In article <3Zapc.81451$FH5.1890949@news20.bellglobal.com>,
> "Gerry Thomas" <gfthomas@sympatico.ca> writes:
> >
> > "Steven G. Kargl" <kargl@troutmask.apl.washington.edu> wrote in message
> > news:c83c4b$jb8$1@nntp6.u.washington.edu...
> >>
> >> I understand. The appendix in my paper describes exactly
> >> what you want to do. You need to be able to compute the
> >> contour integral. Let f(b) = det(A). You want to find a
> >> value B such that f(B) = 0. The winding number theorem
> >> and a Cauchy integral will allow you to do this. Briefly,
> >> the winding number theorem gives
> >>
> >> 1 / f'(b)
> >> N - P = ------ | ------- db
> >> i*2*pi / f(b)
> >>
> >> The integral is a contour integral around a simply closed contour.
> >> N is the number of zeros within the contour and P is the number of
> >> poles. You need to be able to compute f(b) and f'(b) where f'(b)
> >> is the derivative of f(b) with respect to b.
> >>
> >> If there are no poles, then P = 0. If N = 1, you have a single
> >> isolated zero. You can refine the contour to guarantee N=1, P=0.
> >> Once you have N=1, P=0, then
> >>
> >> 1 / b f'(b)
> >> B = ------ | -------- db
> >> i*2*pi / f(b)
> >>
> >> The trick is to choose a circular contour around a candidate location
> >> of for B. In my application B was not very sensitive to the radius
> >> of contour, and one can evaluate a good estimate of B with a fairly
> >> small number of evaluations of f(b).
> >>
> >
> > This sounds interesting. What's the reference on your www? Is there any
> > connection with pseudospectra techniques?
> >
>
> Here's the complete citation
>
> Steven G. Kargl and Philip L. Marston, ``Observations and modeling of
> the backscattering of short tone bursts from a spherical shell: Lamb
> wave echoes, glory, and axial reverberations,'' Journal Acoustical
> Society America, vol 85, pp 1014-1028 (1989).
>
> I don't have a PDF version of this paper. The code to implement the
> above two integrals is on the order of 40 lines.
>
> I don't know of any connection with pseudospectra techniques. It
> is a straight forward application of contour integrals in the
> complex plane.
>

Thanks, I'll take a peek. 

-- 
E&OE
Ciao,
Gerry T.

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