observation: high-precision L-function values of ray class field

[Markus Grassl <markus.grassl@ug.edu.pl>]

Hello,

I would like to share an observation that took me some time to identify.
Given ray class fields over real quadratic field, I am computing 
high-precision numerical values of the first derivative of L-functions 
at zero.

The implementation in PARI/GP benefits from multi-threading.

There is a big difference in total running time whether the default 
precision is changed at the very beginning of the program, i.e., before 
constructing the ray class field, or just before calling 'lfun'.

As an example, consider the following (using GP/PARI CALCULATOR Version 
2.13.3)

  default(nbthreads,20)
  default(timer,1)
  default(debug,1)
  default(parisizemax, 100G)
  default(threadsizemax, 4G)

  K=bnfinit(y^2-4493);
  R0=bnrinit(K,[idealprimedec(K,1123)[1],[1,1]]);

  default(realprecision,1000);

  x0=lfun(lfuncreate([R0,[7]]),0,1)


This version reports

  cpu time = 3min, 15,388 ms, real time = 19,554 ms.

When changing the precision before defining K, the reported time is

  cpu time = 4min, 25,952 ms, real time = 1min, 31,090 ms.

So both the total running time and the wall clock time go up. The result 
appears to be identical.
While the speed-up in the first version is about a factor of 10, it 
drops to less than 4 for when starting with high precision.

In my examples, the desired precision is even higher.

I don't have sufficient knowledge of the underlying theory and 
algorithms, so I can only report my observation.
While the results that I have obtained so far appear to have the desired 
precision, I wonder where the result of the L-function calculation is 
independent of the precision that was set when defining the ray class 
field. Apparently, that makes a difference when calling 'lfun' later.


Best

Markus