On the skew choice for SNFS

Batalov · 2009-06-07, 23:27

It is relatively well known that the skew parameter in SNFS is not a dummy one, not to be completely neglected. Usually, though, it needs very little attention and is most frequently chosen to be
skew = |c₀/c_n|^1/n
especially if linear polynomial is simply x-m.

Here's a bit of an additional twist when one uses a reciprocal polynomial.
If the substitution for the reciprocal was t = x +- 1/x, then nothing changes. However, e.g. 2L/M Aurifeuillian cofactors with an exponent divisible by 3, 5 or 7 a substitution may be t = 2x +- 1/x
Then the optimal skew turns out to be
skew = 2 |c₀/c_n|^1/n

In hindsight, the reasons for this seem to be transparent and are left as an excercise to the reader.
Check it out, if you will have an encounter with a number of similar form.

Batalov · 2009-11-17, 21:18

A much later P.S.:

For larger problems, always run simulations (a.k.a. sims) (i.e. staggered -f points covering your intended sieving range and -c 2000, then do some statistical analysis*). All seasoned people already do. When I wrote the skew trifle above, I meant that the described starting value may be closer to your final optimized skew. It is not necessarily precisely that value. The sim will show. For larger problems, do away with the stock script, too; use your own parallelization scripts and then, manual filtering, BL and sqrt.

For small problems (up to snfs-200 but not quartics, and up to gnfs-150), good starting guess is probably all that one needs. Much shorter sims, if any. Then factMsieve.pl.

____
* Note, that -a side sim results are more difficult to interpret. The level of yield noise is much higher there, so don't make rush decisions like "sieved a bit with this skew, then with that, and look at this huge change, this must be much better". Not so fast. Instead, at the very least, do average yield rates and yields per se in at least 4-5 points, and decide.

2009-06-07, 23:27	#1
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 9,497 Posts	On the skew choice for SNFS It is relatively well known that the skew parameter in SNFS is not a dummy one, not to be completely neglected. Usually, though, it needs very little attention and is most frequently chosen to be skew = \|c₀/c_n\|^1/n especially if linear polynomial is simply x-m. Here's a bit of an additional twist when one uses a reciprocal polynomial. If the substitution for the reciprocal was t = x +- 1/x, then nothing changes. However, e.g. 2L/M Aurifeuillian cofactors with an exponent divisible by 3, 5 or 7 a substitution may be t = 2x +- 1/x Then the optimal skew turns out to be skew = 2 \|c₀/c_n\|^1/n In hindsight, the reasons for this seem to be transparent and are left as an excercise to the reader. Check it out, if you will have an encounter with a number of similar form.

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2009-11-17, 21:18	#2
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 9,497 Posts	A much later P.S.: For larger problems, always run simulations (a.k.a. sims) (i.e. staggered -f points covering your intended sieving range and -c 2000, then do some statistical analysis). All seasoned people already do. When I wrote the skew trifle above, I meant that the described starting value may be closer to your final optimized skew. It is not necessarily precisely that value. The sim will show. For larger problems, do away with the stock script, too; use your own parallelization scripts and then, manual filtering, BL and sqrt. For small problems (up to snfs-200 but not quartics, and up to gnfs-150), good starting guess is probably all that one needs. Much shorter sims, if any. Then factMsieve.pl. ____ Note, that -a side sim results are more difficult to interpret. The level of yield noise is much higher there, so don't make rush decisions like "sieved a bit with this skew, then with that, and look at this huge change, this must be much better". Not so fast. Instead, at the very least, do average yield rates and yields per se in at least 4-5 points, and decide.