# Announcement: Fast Spherical Harmonics

**URL:** <https://discourse.julialang.org/t/announcement-fast-spherical-harmonics/3810>\
**Category:** Numerics\
**Created:** [May 19, 2017, 4:29pm UTC](https://discourse.julialang.org/t/announcement-fast-spherical-harmonics/3810 "2017-05-19T16:29:19Z")\
**Posts on this page:** 1\
**Page:** 1

<div class="post-metadata">

**Author:** ![MikaelSlevinsky](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mikaelslevinsky/32/38545_2.png) [@MikaelSlevinsky](https://discourse.julialang.org/u/MikaelSlevinsky)\
**Post date:** [May 19, 2017, 4:29pm UTC](https://discourse.julialang.org/t/announcement-fast-spherical-harmonics/3810/1 "2017-05-19T16:29:19Z")

</div>

I’m excited to announce a major release of [FastTransforms.jl](https://github.com/MikaelSlevinsky/FastTransforms.jl) introducing the first fast and backward stable transform between spherical harmonic expansions and bivariate Fourier series! And it’s in Julia first.

Convert between representations as simply as:

```julia
julia> Pkg.add("FastTransforms")

julia> using FastTransforms

julia> F = sphrandn(Float64, 975, 975);

julia> G = sph2fourier(F);

julia> H = fourier2sph(G);

julia> norm(F-H)
1.9353272933547886e-13

```

The Fortran package [SHTOOLS](https://github.com/SHTOOLS/SHTOOLS) claims to convert spherical harmonic expansions of bandwidths `2,800` to function values on the sphere in around 3 minutes. If we pre-compute a plan, the execution runs roughly 18 times faster:

```julia
julia> F = sphrandn(Float64, 2800, 2800);

julia> P = plan_sph2fourier(F; sketch = :none);
Pre-computing...100%|███████████████████████████████████████████| Time: 0:01:11

julia> @time G = P*F;
  9.834737 seconds (6 allocations: 119.608 MiB, 0.23% gc time)

julia> @time H = P\G;
  8.142422 seconds (6 allocations: 119.608 MiB, 0.08% gc time)

julia> norm(F-H)
7.186118715996615e-12

```

(The two results are essentially related by an FFT, which is negligible in comparison.)

The new algorithms are described in the following pre-print:

R. M. Slevinsky. [Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series](https://arxiv.org/abs/1705.05448), arXiv:1705.05448, 2017.