bad Mathieu functions
in [Mathematica]
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From: DrMajorBob on 13 Mar 2010 07:57 A fourth option is to set $MinPrecision. Help for it has the following examples: Block[{$MinPrecision = $MachinePrecision}, N[Pi, 50] - 265099323460521503743/84383735478118508040] 4.066722320572635*10^-41 Block[{$MinPrecision = 20, $MaxPrecision = 20}, 1 - Sqrt[1 - Exp[-10`20]]] 0.000022700222531293910900 In both cases, you start with an "arbitrary precision" number, and $MinPrecision prevents calculations from dropping to zero precision. If you start with a machine precision number, you lose that advantage, so that $MinPrecision has no effect in this code: Block[{$MinPrecision = $MachinePrecision}, N(a)Pi - 265099323460521503743/84383735478118508040] 4.44089*10^-16 Block[{$MinPrecision = 50}, N(a)Pi - 265099323460521503743/84383735478118508040] 4.44089*10^-16 This throws an error, too, because you're starting with too little precision to satisfy $MinPrecision: Block[{$MinPrecision = 50}, N[Pi, 20] - 265099323460521503743/84383735478118508040] 4.0667223205726345135327471876400850811855821022270*10^-41 (Yet it seems to have worked.) Bobby On Fri, 12 Mar 2010 06:12:26 -0600, gekko <pfalloon(a)gmail.com> wrote: > On Mar 11, 10:35 pm, becko BECKO <becko...(a)hotmail.com> wrote: >> I am no expert in Mathieu functions, but I don't think this gives the >> right result: >> >> ce1[r_, q_, z_] := MathieuCPrime[MathieuCharacteristicA[r, q], q, z] >> >> Plot[I ce1[3, q, I], {q, 0, 1000}] >> >> In another system you get a smooth graph, making very small >> oscillations about zer o as q increases. I've read that Mathieu >> functions are difficult to deal with. I guess that Mathematica's >> implementation doesn't support arbitrary values of the parameters. I >> made a bug report in the WRI site. Maybe someone here has some comments >> to share? It would be nice if there were a package or something with >> more robust implementations of Mathieu functions. > > > Au contraire, Mathematica is one of the few systems that does support > arbitrary precision evaluation of all special for arbitrary values of > parameters. However, there is no getting around the fact that some > regions of parameter space are just hard to compute! > > In the case of Mathieu functions, the series expansion by which they > are computed converges more slowly as the parameter q becomes larger. > As a consequence, to get meaningful values for large q you need to > work with higher precision. > > Compare the following three cases (using your ce1 function): > > In[179]:= {ce1[3, 1000., I], ce1[3, 1000`20, I], ce1[3, 1000`60, I]} > > Out[179]= {0. + 1608.56 I, 0.*10^2 I, 7.293885910335973*10^-22 I} > > In order, what's going on here is: > > 1. Machine-precision evaluation: all internal computations are done to > machine precision, which is insufficient in this case, and hence the > final answer is (machine-precision) garbage. > > 2. High-precision evaluation, using 20 significant digits: here > Mathematica pays attention to significant figures and correctly > returns a zero-precision result, since 20 digits is not enough to > compute the underlying sum. > > 3. High-precision evaluation, using 60 significant digits: here 60 > digits is high enough to compute the sum and hence return a meaningful > answer. Notice that the final answer has far fewer significant digits > than the input, since precision is lost evaluating the sum. > > > Now, you could legitimately argue that Mathematica should be "smarter" > in cases 1 and 2 above. For case 1, I would say that the current > behaviour reasonable (and somewhat "expected") since it is a general > design decision that machine precision computation should be fast at > the expense of complete control over precision. > > For case 2 I would tend to agree that Mathematica *could* be smarter: > there is no reason why the internal algorithm couldn't recognize that > the output has insufficient precision, and thus increase the internal > working precision. > > > In any case, to return to your original example: if you want to get > your plot "working", you can simply force it to use higher precision: > > LogPlot[Abs[I ce1[3, SetPrecision[q,100], I]], {q, 0, 1000}, PlotRange- >> All] > > Hope this helps. > > Cheers, P. > -- DrMajorBob(a)yahoo.com
From: DrMajorBob on 13 Mar 2010 07:58 Or, a trifle simpler: ce1[r_, q_, z_] := MathieuCPrime[MathieuCharacteristicA[r, q], q, z] Plot[I ce1[3, SetPrecision[qq, 123], I], {qq, 750, 1000}, Frame -> True, Axes -> None] or simpler yet: Plot[I ce1[3, qq, I], {qq, 750, 1000}, Frame -> True, Axes -> None, WorkingPrecision -> 123] Note that this doesn't work at all, and it's much slower: ce1[r_, q_, z_] := MathieuCPrime[MathieuCharacteristicA[r, SetPrecision[q, 123]], q, z] Plot[I ce1[3, qq, I], {qq, 750, 1000}, Frame -> True, Axes -> None] Does anyone have a clue why that is? Bobby On Fri, 12 Mar 2010 06:14:16 -0600, Peter Pein <petsie(a)dordos.net> wrote: > Am 11.03.2010 12:35, schrieb becko BECKO: >> I am no expert in Mathieu functions, but I don't think this gives the >> right result: >> >> ce1[r_, q_, z_] := MathieuCPrime[MathieuCharacteristicA[r, q], q, z] >> >> Plot[I ce1[3, q, I], {q, 0, 1000}] >> >> >> In another system you get a smooth graph, making very small >> oscillations about zer o as q increases. > ... > > Oh yes, these oscillations are really tiny as you can see with > Mathematica using: > > Plot[I ce1[3,q,I] /. q->SetPrecision[qq,123], {qq, 750, 1000}, > Frame->True, Axes->None] > > -- DrMajorBob(a)yahoo.com
From: becko BECKO on 13 Mar 2010 08:00 Nevermind. I think I'm getting the point (thanks gekko for detailed reply). With MachinePrecision, Mathematica doesn't keep track of precision (in the interest of speed), so there is no way to issue a warning since Mathematica doesn't know the precision of the result. If I specify the precision xplicitly (using tick marks) Mathematica will keep track of precision (making it a bit slower), and then I'll know the precision of the result. > Date: Fri, 12 Mar 2010 07:12:04 -0500 > From: becko565(a)hotmail.com > Subject: Re: bad Mathieu functions > To: mathgroup(a)smc.vnet.net > > Thanks Carl, Peter and Daniel. > > Guess I was too hasty to send that bug report... Though it would be useful > to issue a warning or something (perhaps putting it in the Possible Issues > section of the help on Mathieu functions). I'm kind of falling in love with > Mathematica, so it's reassuring to know that this was my mistake and not > Mathematica's. > > Another question. How do I know that WorkingPrecision->50 is enough? What if I wanted to Plot all the way to q=100000?
From: David Park on 13 Mar 2010 08:09 Use greater WorkingPrecision. Plot[I ce1[3, q, I], {q, 0, 100}, WorkingPrecision -> 30, PlotRange -> {-1, 1} 5] David Park djmpark(a)comcast.net http://home.comcast.net/~djmpark/ From: becko BECKO [mailto:becko565(a)hotmail.com] I am no expert in Mathieu functions, but I don't think this gives the right result: ce1[r_, q_, z_] := MathieuCPrime[MathieuCharacteristicA[r, q], q, z] Plot[I ce1[3, q, I], {q, 0, 1000}] In another system you get a smooth graph, making very small oscillations about zer o as q increases. I've read that Mathieu functions are difficult to deal with. I guess that Mathematica's implementation doesn't support arbitrary values of the parameters. I made a bug report in the WRI site. Maybe someone here has some comments to share? It would be nice if there were a package or something with more robust implementations of Mathieu functions.
From: DrMajorBob on 14 Mar 2010 06:13 > If I specify the precision xplicitly (using tick marks) Mathematica will > keep track of precision (making it a bit slower), and then I'll know the > precision of the result No, because Mathematica's precision calculations generally err on the pessimistic side. That's better than if they were too optimistic, of course. Bobby On Sat, 13 Mar 2010 07:00:09 -0600, becko BECKO <becko565(a)hotmail.com> wrote: > > Nevermind. I think I'm getting the point (thanks gekko for detailed > reply). > With MachinePrecision, Mathematica doesn't keep track of precision (in > the interest of speed), so there is no way to issue a warning since > Mathematica doesn't know the precision of the result. If I specify the > precision xplicitly (using tick marks) Mathematica will keep track of > precision (making it a bit slower), and then I'll know the precision of > the result. > > >> Date: Fri, 12 Mar 2010 07:12:04 -0500 >> From: becko565(a)hotmail.com >> Subject: Re: bad Mathieu functions >> To: mathgroup(a)smc.vnet.net >> >> Thanks Carl, Peter and Daniel. >> >> Guess I was too hasty to send that bug report... Though it would be >> useful >> to issue a warning or something (perhaps putting it in the Possible >> Issues >> section of the help on Mathieu functions). I'm kind of falling in love >> with >> Mathematica, so it's reassuring to know that this was my mistake and >> not >> Mathematica's. >> >> Another question. How do I know that WorkingPrecision->50 is enough? >> What if I wanted to Plot all the way to q=100000? > > -- DrMajorBob(a)yahoo.com
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