Just Stuff

Just another WordPress.com weblog

Archive for April 2009

More on Bounds on the Roots of Polynomials

leave a comment »

In an earlier article/post (reference 1) I showed the Cauchy upper bound for the absolute cvalue of the roots of a monic polynomial. The restriction to a monic polynomial was purly for convienience, but here we will require an arbitary polynomial  of degree n so the Cauchy lower bound is: If  x is a root of the polynomial:

P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0

then:

|x| \le 1+ \frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }

Now I will extend this to give a lower bound for the absolute value of the non-zero roots of P(x) Without any real loss of generality we can assume that both a_n and a_0 \ne 0 (if the lowest power of  x appearing in p(x) is \rho \ne 0  we divide through by x^{\rho} to get a polynomial of the required form and whos non-zero roots are the same as the polynomial we started with).

Let B(a_0, a_1, ..., a_n) be an an upper bound on the ansolute value of roota of the ploynomial: P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0 where a_n and a_0 are both non-zero.  Now let x be a root of P(x), so:

P(x)=a_nx^n+a_{n-1}x^{n-1}+ ... + a_0=0

Now divide through by x^n and put y=1/x, so we have:

Q(y)=a_0 y^n + a_1 y^{n-1}+ ... + a_n=0

Hence:

|y| \le B(a_n, a_1, ..., a_0)

or:

|x| \ge 1/B(a_n, a_1, ..., a_0) .

So combinining this lower bound with the upper bound we have for any n-th degree polynomial with only non-zero roots for any root x:

\frac{1}{B(a_n, a_{n-1}, ..., a_0) } \le |x| \le B(a_0, a_1, ..., a_n)

and for the Cauchy bound we have:

B(a_0, a_1, ... , a_n) =1+\frac{\max_{i=0, .. , n-1} |a_i|}{|a_n| }

So:

\frac{|a_0|}{|a_0|+\max_{i=1,..,n}|a_i|}\le |x| \le \frac{|a_n|+\max_{i=0,..,n-1}|a_i|}{|a_n|}

References:

1. https://captainblack.wordpress.com/2009/03/08/cauchys-upper-bound-for-the-roots-of-a-polynomial/

Written by CaptainBlack

April 27, 2009 at 07:10

Posted in Maths and Stuff

Tagged with

More Domino Wave Speed Stuff

leave a comment »

Discussions in Backscatter Column of Citizen Scientist (ref 4) of my paper and experiments on measuring the speed of the Domino Effect (ref 1 and 2) and attempts to repeat the experiments also reported in CS seem to miss the point of using DEMON spectra (ref 3).

I may be partially responsible for this since I have in the past commented that the spikes of noise which I presume to be the clicks of domino impacts on closer examination appear to be tonals modulated by a narrow envelope. I now believe that this was partially a misinterpretation of the data and partially a result of over fierce band pass filtering on my part.

Whatever the truth of the above we can observe directly from a plot (fig 1) of the auto-correlation of a recording of the domino wave, that there is no clear signal at the expected delay (of about 0.04 s for this data file):

Auto-Correlation, data file d9p2.wav

Fig 1: Auto-Correlation, data file d9p2.wav

Fig 2 shows the DEMON spectrum for this data file and here we see a “Sore Thumb” signature at about 24.5 Hz, which would correspond to a delay of about 0.04 seconds in the auto-correlation for this data set, where there is no clear feature. (the 24.5 Hz frequency corresponds to a wave speed close to 1 m/s on an 0.04 m pitch domino array, or a normalised wave speed of ~1.4 for dominoes of height ~0.052 m and thickness ~-.008 m (the width does not affect the wave speed but here its is ~0.025m)

Fig 2: DEMON Spectrum,  data file d9p2.wav

Note: The datafile d9p2.wav is the recording of one of the repeat runs, not reported in refs 1 and 2, that I have  done to look at the variability in speed measurements. In this case the speed measured is within about 2% of that previously reported.

References:

1.  Larham R., Measuring the Speed of the Domino Effect Using the Windows Sound Recorder, Part 1, Citizen Scientist, Nov 2007, http://www.sas.org/tcs/weeklyIssues_2007/2007-11-02/project2/index.html

2.  Larham R., Measuring the Speed of the Domino Effect Using the Windows Sound Recorder, Part 2, Citizen Scientist, Dec 2007, http://www.sas.org/tcs/weeklyIssues_2007/2007-12-07/project2/index.html

3. Smith R., Attempted Recreation of Ron Larham’s Domino Wave Experiment, Citizen Scientist, March 2009, http://www.sas.org/tcs/weeklyIssues_2009/2009-03-06/project1/index.html

4. Hannon J., Still More About Domino Waves, Backscatter Column, Citizen Scientist, April 2009, http://www.sas.org/tcs/weeklyIssues_2009/2009-04-03/backscatter/index.html

Written by CaptainBlack

April 8, 2009 at 13:50

Posted in DSP, Maths and Stuff

Tagged with , ,