9
$\begingroup$

When trying to understand better the question Eisenstein Series in Mathematica?, I stumbled on the following: issuing

Derivative[1][DedekindEta][.11 I]

gives

error messages

This by itself I cannot argue is strange (except that there is nothing about EllipticReducedHalfPeriods in the documentation) - there seemingly is some internal computation mechanism that fails on this value. What I find strange is this:

Table[Derivative[1][DedekindEta][I t], {t, .01, .2, .01}] // TableForm

produces

derivative failure

so that this internal mechanism somehow oscillates between failing and not failing. It seems to exhibit quite complicated behavior: sort of zooming in,

Table[Derivative[1][DedekindEta][I t], {t, .099, .101, .0001}] // TableForm

results in

derivative failure

What is happening? How to compute the derivative of the Dedekind $\eta$ function for values with small imaginary part?

Update

Using the answer by Michael E2, I made a plot with

ContourPlot[Log[Abs[N@N[Derivative[1][DedekindEta][x + I y], 3 $MachinePrecision]]],
    {x, -.5, .5}, {y, 0, .5},
    PlotPoints -> 50, Contours -> 50, ImageSize -> Full, AspectRatio -> Automatic]

and it reveals a strange blind spot:

plot with blind spot

The white area along the real line is understandable - $\eta$ has singularities there - but where the white disk comes from, I cannot figure out.

What is also very strange (although this probably belongs to another question), I only managed to produce this picture by cropping a screenshot: when I tried to save the plot as an image (right-clicking and choosing "Save graphic as..." png, I got the following message:

message

(sorry for unreadable text, it can be enlarged one way or another).

Strange.

$\endgroup$
  • 1
    $\begingroup$ For reference: EllipticReducedHalfPeriods[] effectively performs a modular transformation to return half-periods that are effectively equivalent to the input ones, for subsequent use with the Weierstrass and modular functions. $\endgroup$ – J. M. will be back soon Jan 6 '17 at 7:49
7
$\begingroup$

There is some issue with internal numerics. As the argument gets smaller, the more precision is needed. For t == 0.01, three times $MachinePrecision is sufficient.

Table[
 N@N[
   Derivative[1][DedekindEta][I Round[t, 1/100]],
   3 $MachinePrecision],
 {t, .01, .2, .01}]
(*
{0. - 1.09595*10^-7 I, 0. - 0.00919556 I, 0. - 0.256807 I, 
 0. - 1.08606 I, 0. - 2.25421 I, 0. - 3.34786 I, 0. - 4.15565 I, 
 0. - 4.64526 I, 0. - 4.86564 I, 0. - 4.8859 I, 0. - 4.76924 I, 
 0. - 4.56545 I, 0. - 4.31087 I, 0. - 4.03087 I, 0. - 3.74249 I, 
 0. - 3.45684 I, 0. - 3.18087 I, 0. - 2.91867 I, 0. - 2.67239 I, 
 0. - 2.44293 I}
*)

Note that the results are cached, so that the error messages are seen only once if this is computed first (use Quit[] to restart):

Table[
 N@N[
   Derivative[1][DedekindEta][I Round[t, 1/100]],
   $MachinePrecision],
 {t, .01, .2, .01}]

Note also, that Mathematica still computes the derivatives (if arbitrary precision is used). The error messages seem to be leaks. I suspect they are internal errors that N catches and adjusts the internal precision.


Update:

The problem with the ContourPlot is that ContourPlot passed machine reals to the plotted function. This means N has no effect and some of the derivatives return DedekindEta'[..] unevaluated. These are plotted as blank, white space in the contour plot. So one has to convert the input to the plotted function to an infinite precision (exact) number, using either SetPrecision or Rationalize. It also helps, again from a numerics point of view, to not go all the way down to the real axis. It takes a lot of computational time to do so. (After a few hours while we made and ate dinner, the plot had still not finished computing the graphics when the domain for y was {y, 0.001, .5}. Then I aborted it.)

Block[{f},
 f[z_?NumericQ] := Block[{res},
   res = Log[Abs[N@ Quiet@ N[Derivative[1][DedekindEta][Rationalize[z, 0]], 6]]];
   res /; NumericQ[res]];
 ContourPlot[f[x + I y], {x, -.5, .5}, {y, 0.001, .5}, Contours -> 50,
   ImageSize -> Full, AspectRatio -> Automatic]
 ]

(* numerics-related errors omitted -- I guess Quiet didn't work *)

Mathematica graphics


Update 2

Per J.M.'s comment, one might use a high WorkingPrecision instead of N. (The difference is that one has to start with a high enough WorkingPrecision, whereas N will figure that out automatically and adjust as needed.) To close up the gaps as above, the WorkingPrecision needs to be around 150 or more, which was determined by trial and error. Each method takes about 30 sec. on my machine, and the plot below takes a couple of seconds longer than the one above.

Quiet@ContourPlot[
  Log[Abs[Derivative[1][DedekindEta][x + I y]]], {x, -.5, .5}, {y, 
   0.001, .5}, Contours -> 50, ImageSize -> Full, 
  AspectRatio -> Automatic, WorkingPrecision -> 150]

Some references to the documentation:

$\endgroup$
  • $\begingroup$ Great, thanks! Not that I understand the reason, but it works perfectly :D $\endgroup$ – მამუკა ჯიბლაძე Nov 15 '15 at 20:10
  • $\begingroup$ @მამუკაჯიბლაძე I suppose some numerical algorithm is used that starts with insufficient precision; only at some point in the middle is it discovered that more precision is needed. $\endgroup$ – Michael E2 Nov 15 '15 at 20:13
  • $\begingroup$ Trying to plot it I am puzzled even more. Let me add this to the question... $\endgroup$ – მამუკა ჯიბლაძე Nov 15 '15 at 20:50
  • 1
    $\begingroup$ @მამუკაჯიბლაძე The problem is ContourPlot is feeding machine precision numbers to the function, so N[..., 3 $MachinePrecision] has no effect. The circle is where DedekindEta'[..] is being returned. $\endgroup$ – Michael E2 Nov 15 '15 at 22:42
  • 2
    $\begingroup$ What if you use an appropriate setting of WorkingPrecision in ContourPlot[]? $\endgroup$ – J. M. will be back soon Nov 16 '15 at 7:34
4
$\begingroup$

The derivative of $\eta(\tau)$ satisfies the following identity:

$$\frac{\eta^\prime(\tau)}{\eta(\tau)}=\zeta(-\pi i\mid-\pi i,-\pi i \tau)$$

where $\zeta$ is the Weierstrass zeta function, which is built-in as WeierstrassZeta[].

Now, it is actually a bit wasteful to call WeierstrassZeta[] for this evaluation, since it takes a list of the invariants $g_2,g_3$ as its second argument instead of the half-periods $\omega_1,\omega_3$. One can use WeierstrassInvariants[] to convert the invariants to half-periods, but that is a wasted round-trip since the invariants are again internally converted to half-periods for the evaluation of WeierstrassZeta[].

Thus, after spelunking the code for WeierstrassZeta[] and taking needed pieces from it, I came up with this routine for $\eta^\prime(\tau)$:

etaPrime[τ_?InexactNumberQ] := Module[{eta1, f, id, p1, p2, q, tn, w, zr},
         {p1, p2} = EllipticReducedHalfPeriods[-I π {1, τ}];
         {zr, id} = System`EllipticDump`WPred[-I π, {p1, p2}];
         tn = p2/p1; q = Exp[I π tn]; f = π/(2 p1); w = f zr;
         eta1 = System`EllipticDump`eta1Value[p1, q];
         DedekindEta[τ] (eta1 zr/p1 + 2 id.{eta1, eta1 tn - I f} + 
                         f EllipticThetaPrime[1, w, q]/EllipticTheta[1, w, q])]

Now, a plot:

ContourPlot[Log[Abs[etaPrime[x + I y]]], {x, -1, 1}, {y, 1/100, 1}, 
            AspectRatio -> Automatic, Contours -> 50]

logarithm of magnitude of the derivative of the Dedekind function

The routine does fine for arguments far away from the real axis (recall that these modular functions are only defined for $\Im\tau>0$); otherwise, the numerical evaluation becomes unstable, and arbitrary precision is needed.

(See the edit history for the previous version of this answer.)

$\endgroup$
  • $\begingroup$ I see. How about using q Derivative[1][Function[x, Log[x QPochhammer[x^2, x^2]^12]]][q]? You can check that q D[Series[Log[q QPochhammer[q^2, q^2]^12], {q, 0, 12}], q] coincides with the series for $E_2$ you had in that question. $\endgroup$ – მამუკა ჯიბლაძე Nov 16 '15 at 9:36
  • $\begingroup$ (The reason I keep seeking alternatives is the hope to find a version which would analytically continue as far as possible the branch containing the origin of your $E_2$ from that question) $\endgroup$ – მამუკა ჯიბლაძე Nov 16 '15 at 9:45
  • $\begingroup$ The avenue I said that I haven't yet been able to pursue is the only possibility I see now; FWIW, if you look through the Wolfram functions site, the long and inelegant formula for the derivative of the Dedekind function listed there was the starting point for the formula for the Eisenstein series in my other answer. $\endgroup$ – J. M. will be back soon Nov 16 '15 at 10:15
  • 2
    $\begingroup$ At least it's a small improvement over my naive precision approach. It works a little faster and fills in the gap with WorkingPrecision -> 100. And with tau instead of z in etaPrime, of course. :) $\endgroup$ – Michael E2 Nov 16 '15 at 21:17
  • 1
    $\begingroup$ @J.M. Done. Feel free to edit according to your own style. (BTW, this took 28.7 sec., instead of 75.4 sec. that your original code took.) $\endgroup$ – Michael E2 Nov 17 '15 at 2:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.