Function: lfuncost
Section: l_functions
C-Name: lfuncost0
Prototype: GDGD0,L,b
Help: lfuncost(L,{sdom},{der=0}): estimate the cost of running
lfuninit(L,sdom,der) at current bit precision. Returns [t,b], to indicate
that t coefficients a_n will be computed at bit accuracy b. Subsequent
evaluation of lfun at s evaluates a polynomial of degree t at exp(h s).
If L is already an Linit, then sdom and der are ignored.
Doc: estimate the cost of running
\kbd{lfuninit(L,sdom,der)} at current bit precision. Returns $[t,b]$, to
indicate that $t$ coefficients $a_n$ will be computed, as well as $t$ values of
\tet{gammamellininv}, all at bit accuracy $b$.
A subsequent call to \kbd{lfun} at $s$ evaluates a polynomial of degree $t$
at $\exp(h s)$ for some real parameter $h$, at the same bit accuracy $b$.
If $L$ is already an \kbd{Linit}, then \var{sdom} and \var{der} are ignored
and are best left omitted; the bit accuracy is also inferred from $L$: in
short we get an estimate of the cost of using that particular \kbd{Linit}.
\bprog
? \pb 128
? lfuncost(1, [100]) \\ for zeta(1/2+I*t), |t| < 100
%1 = [7, 242] \\ 7 coefficients, 242 bits
? lfuncost(1, [1/2, 100]) \\ for zeta(s) in the critical strip, |Im s| < 100
%2 = [7, 246] \\ now 246 bits
? lfuncost(1, [100], 10) \\ for zeta(1/2+I*t), |t| < 100
%3 = [8, 263] \\ 10th derivative increases the cost by a small amount
? lfuncost(1, [10^5])
%3 = [158, 113438] \\ larger imaginary part: huge accuracy increase
? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5)
? lfuncost(L, [100]) \\ at s = 1/2+I*t), |t| < 100
%5 = [11457, 582]
? lfuncost(L, [200]) \\ twice higher
%6 = [36294, 1035]
? lfuncost(L, [10^4]) \\ much higher: very costly !
%7 = [70256473, 45452]
? \pb 256
? lfuncost(L, [100]); \\ doubling bit accuracy
%8 = [17080, 710]
@eprog\noindent In fact, some $L$ functions can be factorized algebraically
by the \kbd{lfuninit} call, e.g. the Dedekind zeta function of abelian
fields, leading to much faster evaluations than the above upper bounds.
In that case, the function returns a vector of costs as above for each
individual function in the product actually evaluated:
\bprog
? L = lfuncreate(polcyclo(5)); \\ Dedekind zeta for Q(zeta_5)
? lfuncost(L, [100]) \\ a priori cost
%2 = [11457, 582]
? L = lfuninit(L, [100]); \\ actually perform all initializations
? lfuncost(L)
%4 = [[16, 242], [16, 242], [7, 242]]
@eprog\noindent The Dedekind function of this abelian quartic field
is the product of four Dirichlet $L$-functions attached to the trivial
character, a non-trivial real character and two complex conjugate
characters. The non-trivial characters happen to have the same conductor
(hence same evaluation costs), and correspond to two evaluations only
since the two conjugate characters are evaluated simultaneously.
For a total of three $L$-functions evaluations, which explains the three
components above. Note that the actual cost is much lower than the a priori
cost in this case.
Variant: Also available is
\fun{GEN}{lfuncost}{GEN L, GEN dom, long der, long bitprec}
when $L$ is \emph{not} an \kbd{Linit}; the return value is a \typ{VECSMALL}
in this case.