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.
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.
The following is a histogram of squared value magnitude ratios.
This perhaps indicates a source of stability in three dimensions.
More Histograms
Back to Polysigned Numbers