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A rapid algorithm is derived for the Helmholtz--Hodge decomposition on thesurface of the sphere in spherical coordinates. The algorithm uncouples modesof spherical harmonics with different absolute order, writes the conversion asbarely-overdetermined banded linear systems, and solves them with banded $QR$decompositions that factor and execute in optimal complexity. Rigorous upperbounds on the $2$-norm relative condition number of the banded linear systemssupport the observable low error growth with respect to truncation degree.

A rapid and well-conditioned algorithm for the Helmholtz--Hodge decomposition of vector fields on the sphere