Magnitude Analysis

The following is a graph of the maximum(blue) and minimum(pink) magnitude ratios up to sign 100.
This was built by generating 1000 random pairs of polysigned values p1 and p2.
the ratio of :

    | p1 p2 | / ( | p1 | | p2 | ) 

is then taken. The max and min are the lines drawn here.
As you can see the magnitude changes drastically from unity beyond sign 3 but appear to settle toward unity at very large sign.

Grahph of Magnitude Factors

Taking more samples does alter the graph slightly but the basic shape remains the same.
Here are some values based on 50000 random sample pairs:

   sign:4   max:1.718   min:0.010
   sign:5   max:1.412   min:0.053
   sign:6   max:2.106   min:0.142
   sign:7   max:1.694   min:0.149
   sign:8   max:2.123   min:0.147
   sign:9   max:1.819   min:0.185
   sign:10   max:2.397   min:0.190
   sign:11   max:1.912   min:0.256
   sign:12   max:2.175   min:0.176
   sign:13   max:1.947   min:0.332
   sign:14   max:2.766   min:0.273
   sign:15   max:1.984   min:0.297
   sign:16   max:2.185   min:0.340
   sign:17   max:2.004   min:0.361
   sign:18   max:2.094   min:0.403
   sign:19   max:2.004   min:0.389
   sign:20   max:2.031   min:0.384

The magnitude values are generated by dimensional analysis of polysigned numbers.
each polysigned value is converted to a cartesian value then its magnitude taken.

The following is a histogram plot of four-signed product magnitude ratios.
The peak is at unity.

histogram of 4-signed products

The following is a histogram of squared value magnitude ratios.
This perhaps indicates a source of stability in three dimensions.

4Signed Squareing Behavior

More Histograms

Back to Polysigned Numbers