Temperament addition: Difference between revisions

Line 873:

===== Addabilization defactoring complications =====

This case was as simple as it can get: we simply needed to add some number of the single linearly dependent vector to the linearly independent vector. However, if there are multiple vectors in the <math>L_{\text{dep}}</math>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic ~~linear~~ solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477

This case was as simple as it can get: we simply needed to add some number of the single linearly dependent vector to the linearly independent vector. However, if there are multiple vectors in the <math>L_{\text{dep}}</math>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477

Another complication is that the greatest factor may be very large, or be a highly composite number. In this case, searching for the linear combination that isolates the greatest factor in its entirety directly may be intractable; it is better to eliminate it piecemeal, i.e., whenever the solver finds a factor of the greatest factor, eliminate it, and repeat until the greatest factor is fully eliminated. The example linked above also does this.

@@ Line 873: / Line 873: @@
 ===== Addabilization defactoring complications =====
-This case was as simple as it can get: we simply needed to add some number of the single linearly dependent vector to the linearly independent vector. However, if there are multiple vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic linear solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477
+This case was as simple as it can get: we simply needed to add some number of the single <span style="color: #3C8031;">linearly dependent vector</span> to the <span style="color: #B6321C;">linearly independent vector</span>. However, if there are multiple vectors in the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span>, the linear combination which surfaces the greatest factor may involve just one or potentially all of those vectors, and the best approach to finding this combination is simply an automatic solver. An example of this approach is demonstrated in the [[RTT library in Wolfram Language]], here: https://github.com/cmloegcmluin/RTT/blob/main/main.m#L477
 Another complication is that the greatest factor may be very large, or be a highly composite number. In this case, searching for the linear combination that isolates the greatest factor in its entirety directly may be intractable; it is better to eliminate it piecemeal, i.e., whenever the solver finds a factor of the greatest factor, eliminate it, and repeat until the greatest factor is fully eliminated. The example linked above also does this.