An Optimal In-Situ Multipole Algorithm for the Isotropic Three-Point Correlation Functions

We present an optimised multipole algorithm for computing the three-point correlation function (3PCF), tailored for application to large-scale cosmological datasets. The algorithm builds on a $in\, situ$ interpretation of correlation functions, wherein spatial displacements are implemented via translation window functions. In Fourier space, these translations correspond to plane waves, whose decomposition into spherical harmonics naturally leads to a multipole expansion framework for the 3PCF. To accelerate computation, we incorporate density field reconstruction within the framework of multiresolution analysis, enabling efficient summation using either grid-based or particle-based schemes. In addition to the shared computational cost of reconstructing the multipole-decomposed density fields - scaling as $\mathcal{O}(L^2_{\text{trun}} N_g \log N_g)$ (where $N_g$ is the number of grids and $L_{\text{trun}}$ is the truncation order of the multipole expansion) - the final summation step achieves a complexity of $\mathcal{O}(D^6_{\text{sup}} N_g)$ for the grid-based approach and $\mathcal{O}(D^3_{\text{sup}} N_p)$ for the particle-based scheme (where $D_{\text{sup}}$ is the support of the basis function and $N_p$ is the number of particles). The proposed $in\, situ$ multipole algorithm is fully GPU-accelerated and implemented in the open-source $Hermes$ toolkit for cosmic statistics. This development enables fast, scalable higher-order clustering analyses for large-volume datasets from current and upcoming cosmological surveys such as Euclid, DESI, LSST, and CSST.