Some terms left unsimplified

January 6, 2017

I had been slowly going through some OEIS entries in an attempt to add or optimize the Mathematica code found there. One thing that struck me with respect to the sequences that are related to the elliptic integrals/elliptic functions was that they were often left in the form that was spit out by a CAS. I had railed about the deficiencies of current software for generating simple expressions for elliptic integrals many times before, so this was but a confluence with the fact that not many people these days have experience in the analytical manipulation of these functions, as classical and important as they are.

I certainly have my work cut out for meā€¦

Yes, I’m still here…

December 21, 2014

…and to those guys who showed concern and worry during my long-ish absence, I’m fine. Two important things, though: the hard drive containing a lot of my unpublished research crashed, including a good amount that I have not been able to back up; and, I have been quite busy with a (relatively) new job over the past few months, details of which I cannot discuss here.

In short, I’m fine, and I hope you guys will still be here when I’m actually back. Happy Holidays!

~ Jan

Using the generator in Mathematica

May 15, 2012

An answer by the user Emre on the Mathematica StackExchange site introduced me to the random number generation service by Mads Haahr. In the spirit of a previous post of mine, I wanted to be able to use the service through the method plug-in framework for the random number generators of Mathematica. I have thus written a barebones set of routines for the purpose:

RandomOrg /: Random`InitializeGenerator[RandomOrg, ___] := RandomOrg[""]

RandomOrg[___]["GeneratesRealsQ"] := True;
RandomOrg[___]["GeneratesIntegersQ"] := True;

RandomOrg[___]["SeedGenerator"[seed_]] := RandomOrg[""]

RandomOrg[url_]["GenerateReals"[n_, {a_, b_}, prec_]] := {a + (b - a) Import[url <> "decimal-fractions/?col=1&format=plain&num=" <> ToString[n] <> "&dec=" <> ToString[Round[prec]] <> "&rnd=new", "List"], RandomOrg[url]}

RandomOrg[url_]["GenerateIntegers"[n_, {a_, b_}]] := {Import[url <> "integers/?col=1&base=10&format=plain" <> "&min=" <> ToString[a] <> "&max=" <> ToString[b] <> "&num=" <> ToString[n] <> "&rnd=new", "List"], RandomOrg[url]}

Here are a few examples of its use:

BlockRandom[SeedRandom[0, Method -> RandomOrg];
RandomReal[{0, 1}, 10, WorkingPrecision -> 20]]

BlockRandom[SeedRandom[0, Method -> RandomOrg];
RandomInteger[{1, 10}, 10]]

As a tiny reminder, the site has a daily quota in place that limits the amount of random numbers you can generate using their service in a single day. See their website for more details.

A short note on Costa’s minimal surface

May 14, 2012

The other day, I finally managed to simplify Alfred Gray’s parametric equations for Costa’s minimal surface. I might edit this post later, with details on how to manipulate the Weierstrass elliptic functions that show up in the equations, but enjoy these for now:

\displaystyle x=\frac{\pi u}{2}-\Re\left(\frac{\zeta(u+iv;g_2,0)}{2}-\frac{\pi\,\wp(u+iv;g_2,0)}{\wp^\prime(u+iv;g_2,0)}\right) \\ y=\frac{\pi v}{2}+\Im\left(\frac{\zeta(u+iv;g_2,0)}{2}+\frac{\pi\,\wp(u+iv;g_2,0)}{\wp^\prime(u+iv;g_2,0)}\right) \\ z=\sqrt{\frac{\pi}{8}}\log\left|\frac1{\frac12+\frac{\wp(u+iv;g_2,0)}{\sqrt{g_2}}}-1\right|

Here, \wp, \wp^\prime and \zeta are respectively the Weierstrass elliptic function, its derivative, and the Weierstrass zeta function, with invariants g_2=\left(\frac12\mathrm{B}\left(\frac14,\frac14\right)\right)^4=\frac{\Gamma(1/4)^8}{16\pi^2} and 0, and \mathrm{B}(x,y) and \Gamma(x) are the usual beta and gamma functions. The invariants are the ones corresponding to the semi-periods \omega_1=\frac12 and \omega_3=\frac{i}{2}. The parameter ranges are 0 < u < 1 and 0 < v < 1.

I was very delighted at my result, that I decided to make a stylized plot of the surface in Mathematica, using Perlin’s simplex noise for the coloring:

Costa's minimal surface

FNV hashing in Mathematica

May 4, 2012

I recently needed to do Fowler/Noll/Vo hashing in Mathematica. I figured other people might be have a need for this hash function as well, so here’s my Mathematica implementation:

(* FNV hash parameters: bit length, FNV prime, FNV offset *)
$FNVData = Append[#, Function[{bi, pr},
Fold[BitXor[Mod[pr #1, 2^bi], #2] &, 0,
ToCharacterCode["chongo /\\../\\"]]] @@ #] & /@ Transpose[{2^Range[5, 10],
2^(8 Quotient[2^Range[5, 10] + 5, 12]) + 2^8 +
Map[FromDigits[#, 16] &,
{"93", "B3", "3B", "63", "57", "8D"}]}];

FNV[dat_String, bi : _Integer : 32, shift : (0 | 1) : 1] := With[{k = IntegerExponent[bi, 2] - 4},
Fold[BitXor[Mod[$FNVData[[k, 2]] #1, 2^bi], #2] &,
shift $FNVData[[k, 3]], ToCharacterCode[dat]]] /;
32 <= bi <= 1024 && BitAnd[bi, bi - 1] == 0

I made sure to provide for both shift-1 and shift-0 versions in the implementation (though as mentioned on the FNV page, FNV-1 is what should be preferred). The FNV page has a few suggestions to extend FNV hashing to bit lengths that are not a power of two, but I did not bother with them. (Maybe somebody else can try!) It should not be too hard to modify the code to do FNV-1a hashing.

Quick-and-dirty numerical inversion of the Laplace transform

May 4, 2012

I present here a set of Mathematica routines implementing a quick-and-dirty method for numerically evaluating the inverse Laplace transform. As it is well-known, the problem is a numerically ill-conditioned one, and thus checking the sensibility of your results is very much important here. The value of the following method, however, compared to more elaborate transform inversion methods, is that if one wants a quick estimate for arguments not too far from the imaginary axis, the approximation derived from this method does not take too much effort to evaluate.

The method, due to Herbert Salzer, is a complex-valued n-point Gaussian quadrature, where the nodes are the (complex!) roots of (a special case of) the Bessel polynomial. Due to this, if you want to use this method, the function you wish to transform should be able to evaluate complex arguments.

The method below does not give an error estimate; what you can try for verifying the accuracy of your results is to evaluate for an initial number of points (the default is ten points in the routine below), and then increase the number of points taken, and then compare the results you have. (You might also wish to use a sufficiently high value of the option WorkingPrecision.)

Here are the the Mathematica routines:

SalzerRuleData[n_Integer, prec_:MachinePrecision] :=
Block[{x}, Transpose[Map[{#, (-1)^(n + 1) # ((2 n - 1)/HypergeometricPFQ[{1 - n, n - 1}, {}, #])^2/n} &,
x /. NSolve[HypergeometricPFQ[{-n, n}, {}, x], x, prec]]]]

Options[NInverseLaplaceTransform] = {Points -> 10,
WorkingPrecision -> MachinePrecision};
NInverseLaplaceTransform[f_, s_, t_, opts:OptionsPattern[]] := Module[{xa, wt},
{xa, wt} = SalzerRuleData[OptionValue[Points], OptionValue[WorkingPrecision]];
Chop[(wt.Function[s, f][1/xa/t])/t]]

As an example, we try the inverse Laplace transform of the function \frac1{(1+s)^2}, which is t\exp(-t):

f[t_] = NInverseLaplaceTransform[(1 + s)^(-2), s, t];

With[{t = N[2]}, {t Exp[-t], f[t], t Exp[-t] - f[t]}]
{0.270671, 0.270671 + 5.68434*10^-14 I, 2.91759*10^-8 - 5.68434*10^-14 I}

(The tiny imaginary part is an artifact of the evaluation; you can use Chop[] if you wish.)

As another example, here is a comparison of the approximate inverse transform of \frac1{\sqrt{1+s^2}} with the true inverse transform, J_0(t) (where J_0(t) is the Bessel function of the first kind):

f[t_] = NInverseLaplaceTransform[1/Sqrt[1 + s^2], s, t, WorkingPrecision -> 25];

GraphicsRow[{Plot[{f[t], BesselJ[0, t]}, {t, 0, 15}, Frame -> True],
Plot[Re[f[t] - BesselJ[0, t]], {t, 0, 15}, Frame -> True, PlotRange -> All]}]

Bessel function via inverse Laplace transformation

The plot on the left shows that the true function and the approximation are almost indistinguishable within the range plotted. The plot on the right shows the difference between the approximate and the true function.

Bunnies, teapots, and noise

March 16, 2012

I should be able to have regular Internet access soon. For now, I’d like to share some of the experiments I did with coloring 3D objects with the help of Ken Perlin’s simplex noise in Mathematica. (Relatedly, I’ve found that the implementation of improved Perlin noise supplied in the Mathematica help file has a few subtle flaws, which I was able to fix. That would be for another blog entry, though.)

Here are some of the results of coloring the Utah teapot with simplex noise:

Utah Teapot with simplex noise coloring

Here are some results of coloring the Stanford bunny with simplex noise:

Stanford Bunny with simplex noise coloring

I never would have thought that Mathematica would ever be able to do graphics as fancy as this. I’ll only note that none of these used the Texture[] construct; both models in fact used simplex noise as solid textures (i.e. an RGBColor[] associated with each point in the model).

I might write a (series of) more descriptive post(s) on these things much later; for now, these pictures, entirely generated in Mathematica (no post-processing at all!) will have to suffice.