April 1997

This note describes various approachs taken to understanding the position of the roots to a stability equation [Robinson, Rennie, Wright] as given below.

Qe = 1 / (1 + exp(a[Ve - 3])) Qi = 1 / (1 + exp(a[Vi - 3])) Ve = g * {Bee * Qe - Bie * Qi + Mee * Qns + Aee * Qe} Vi = g * {Bei * Qe - Bii * Qi + Mei * Qns + Aei * Qe}

Where

V1 = Aee + Bee V2 = Aei + Bei V3 = Bii / Bie V4 = Mee * Qns V5 = Mei * Qns V6 = Biewhere Q

The problem is to find the 6 dimensional volume(s) where roots exist.
Of particular interest is the volume where there are 4,5 or more roots.
(It turns out that in what follows whenever 4 roots are found there
is another root at Q_{e} = 1, such a value can't be evaluated
using the above formulation due to the ln[1/Q_{e}-1] term)

There are bounds on the range of physically sensible values for
the structural parameters, they are [0,1] for all but V_{3} which
can vary in the range [0,10].

The first exploration simply involved developing an interface which allows the user to interactively view the function while varying any of the 6 parameters.

While it can be useful to "play" with this, the real benefit was the ability to see that a simple zero crossing algorithm will be satisfactory for counting the number of roots. The interface also allows ready evaluation of the range of solutions in one or two variables after fixing the others, for example, testing the sensitivity of the roots to one or more parameters.

The next step was to write software which explores the 6 dimensional space calculating the number of roots for each point and reporting where there are 4 or more roots. This was initially performed exhaustively, varying each parameter in turn by 0.05 over the range [0,1]. On a relatively fast computer (Power Challenge) this took almost 10 hours. Since the processing time for such a technique will increase as the 6th power (the number of parameters) this approach was quickly abandoned. Instead, a random sampling was used.....inefficient, but it gives solutions immediately and for as long as one wishes to run the program. Further this approach can easily be run in parallel.

The first use of this growing database of roots is to look at the distribution of 4 or more roots for each parameter. The distributions after 12,000 successful points are shown below.

Two dimensional versions of the above can be constructed for each pair of variables.

Mean and variance V1 : 0.76491 +- 0.02947 V2 : 0.43775 +- 0.03524 V3 : 0.26655 +- 0.00007 V4 : 0.01080 +- 0.00011 V5 : 0.01276 +- 0.03519 Bie: 0.38126 +- 0.76491 RMS V1 : 0.78393 V2 : 0.47630 V3 : 0.34070 V4 : 0.01381 V5 : 0.01662 Bie: 0.42492 Median V1 : 0.56650 V2 : 0.52700 V3 : 0.49950 V4 : 0.04950 V5 : 0.03300 Bie: 0.49700 Range 0.13400 <= V1 <= 0.99900 0.05500 <= V2 <= 0.99900 0.00000 <= V3 <= 0.99900 0.00000 <= V4 <= 0.09900 0.00000 <= V5 <= 0.06600 0.01700 <= Bie <= 0.97700An application was developed giving an interactive display of the points in V

All the parameters are explored in 0.05 steps, chosen as the finest
resolution for which the solutions could be explored interactively
on an Indigo Max Impact. The user can, in nearly real time,
vary V_{4}, V_{5}, and B_{ie}. The
positions of the solutions in V_{1}, V_{2}, V_{3}
space are drawn as boxes, the colour of which is the number of roots found
at that point.

Another way of viewing the results from the random sampling is to
map V_{1}, V_{2}, V_{3} onto the 3 dimensional
coordinate axes as above,
and map B_{ie} to colour. V_{4} and
V_{5} are collapsed (ignored) in the following diagram.

(Exploring this volume interactively gives more insight than the static views show here)

Generating the samples at a higher resolution doesn't necessarily improve things, at least not until significantly more samples are acquired. The diagrams below show the volume for a 0.01 resolution for all parameters.

An additional attribute of interest is the density of roots in Qe, that is, which solutions have roots that are close together. Considering as before only the solutions where there are 4 or 5 roots, the following shows the distribution of each root.

Using the same mapping to 3D as earlier with
the axes V_{1}, V_{2}, V_{3}
giving the user the ability to vary B_{ie} the standard
deviation of Q_{e} of the first 4 roots was mapped to
the blue to red colour ramp. Example screen shots are shown below.

The behaviour of the function about a point
(v_{1},v_{2},
v_{3},v_{4},
v_{5},B_{ie}) was
examined by graphing a family of curves in Q_{e} varying
one of the 6 variables. In order to get some idea of how the two terms
in the stability equation were related

the colour of the points on the curve were mapped to the ratio A/B.

This project and the above graphics was carried out on a variety of platforms, Macintosh, SGI Indigo Max Impact, 12 Processor SGI Power Challenge. The interactive 3D graphics was handled by GeomView from the Geometry Centre, University of Minnesota. The user interfaces were created using the Forms Library by Mark Overmars. |