# 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