Spherical Harmonic bugs/enhancements

[MikaelSlevinsky]

This issue is a placeholder for spherical harmonic bugs/enhancements that will be part of a patch:

• the way the butterfly algorithm is currently implemented, it only works for dimensions that are powers of two, though nothing stops this from working for general dimensions. This afflicts FastSphericalHarmonicPlan and ThinSphericalHarmonicPlan. For the time being, it's recommended to just pad with zeros to the nextpow2.

• the commands sph2fourier and fourier2sph should work on different band limits in l and m.

• the command plan_sph2fourier is not type-stable as it very crudely dispatches to a different SphericalHarmonicPlan based on the bandlimit. This is negligible compared to pre-computation costs for large degrees (and is somewhat likened to Base's factorize).

• the convenience constructors sphrand, sphrandn, etc... take m and n and create an array with dimensions m and 2n-1 which is arguably misleading.

• it would be great to have an allocation-free FFTW override for converting the bivariate Fourier array to function values on the sphere at tensor product grids equispaced in angle. (Added starting with 67a6b56)

• remove segmentation fault when using an even number of columns, thanks to @meggart for finding this. (Added starting with c0e4a54)

• add methods for complex data,

• and, NUFFT variants for nonuniform point.

[MikaelSlevinsky]

As this eventually plugs into ApproxFun's multivariate side, @dlfivefifty, do you have a recommendation regarding the dimensions of the convenience constructors?

[dlfivefifty]

What's the usage of the convenience constructors?

[marcusdavidwebb]

What separates a convenience constructor from a constructor? Marcus

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On 31 May 2017, at 12:16, Sheehan Olver ***@***.***> wrote: What's the usage of the convenience constructors? — You are receiving this because you are subscribed to this thread. Reply to this email directly, view it on GitHub <#16 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/ADbXaPhKnr1NRN_bqJY2csB9F3w_uejEks5r_T3mgaJpZM4NfY0a>.

[dlfivefifty]

I think what this package really needs is some inconvenience constructors 😜

I think he just means "convenience" as in they are not necessary for the package, but included for convenience.

[marcusdavidwebb]

Ah ok. I would have called them “developer constructors” as I imagine only developers of the software will actually use them. Marcus

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On 31 May 2017, at 12:25, Sheehan Olver ***@***.***> wrote: I think what this package really needs is some inconvenience constructors 😜 I think he just means "convenience" as in they are not necessary for the package, but included for convenience. — You are receiving this because you commented. Reply to this email directly, view it on GitHub <#16 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/ADbXaOkTTXlrv--ayZWH1wkOYZLxbiDHks5r_UAigaJpZM4NfY0a>.

[MikaelSlevinsky]

Well, no constructor is more inconvenient than a useless constructor! They stay as is for now.

[MikaelSlevinsky]

Another potential improvement to add to the list is to transpose the data. Givens rotations have better memory localization in row-major ordering, and the 1D row FFTs wouldn't need to transpose if they were 1D column FFTs.

[meggart]

Is it possible to allow Spherical harmonics analysis for arrays with an even number of columns? This currently kills my Julia session:

using FastTransforms
x=rand(100,100)

PA = FastTransforms.plan_analysis(x)
O=zeros(x)

A_mul_B!(O,PA,x)

while it works fine for an uneven number of columns, e.g. x=rand(100,99)

[MikaelSlevinsky]

In principle, yes, it just requires choosing a convention. The problem is equivalent to creating a Fourier interpolant through an even number of points. For three points, we return an expansion in the basis {1, sin(θ), cos(θ)}, but for four points should the expansion be in the basis {1, sin(θ), cos(θ), sin(2θ)}, {1, sin(θ), cos(θ), cos(2θ)} or a weighted average of the two? (In the code, the natural choice would be the first option.)

The current segfault behaviour is undeniably undesirable, apologies.

[meggart]

No problem. I think since there is no obvious reason for picking one or the other option, choosing the the one that is most natural to code should be ok? Thanks for implementing this in Julia, so far I had to call into SHTOOLS, and having a Julia-native solution helps a lot.