We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierarchically applying the interpolative decomposition butterfly factorization (IDBF). Experimental evidence suggests that the total running time of our method -- including all necessary precomputations -- is $\mathcal{O}(N^2 \log^3(N))$, where $N$ is the order of the transform. This is nearly asymptotically optimal. Moreover, unlike existing algorithms which are asymptotically optimal or nearly so, the constant in the running time of our algorithm is small enough to make it competitive with state-of-the-art $\mathcal{O}\left(N^3\right)$ methods at relatively small values of $N$. Numerical results are provided to demonstrate the effectiveness and numerical stability of the new framework.
RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures
A Butterfly-Based Direct Integral Equation Solver Using Hierarchical LU Factorization for Analyzing Scattering from Electrically Large Conducting Objects
Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气与计算机工程系）；University of Michigan（密歇根大学）；Department of Microwave Engineering, University of Electronic Science and Technology of China（电子科技大学微波工程系）；University of Electronic Science and Technology of China（电子科技大学）
A Butterfly-Accelerated Manifold Harmonic Transform
Oden Institute, University of Texas at Austin（德克萨斯大学奥斯汀分校奥登研究所）；Robert McNeel & Associates（罗伯特·麦克尼尔与合伙公司）；Courant Institute, New York University（纽约大学库朗数学科学研究所）
Accelerating the Canonical Polyadic Alternating Least Squares Optimization via a Randomized Interpolative Decomposition
PSI Center for Scientific Computing, Theory and Data（PSI科学计算、理论与数据中心）；Institute of Mathematics, EPFL（洛桑联邦理工学院数学研究所）；The University of Maryland, College Park（马里兰大学帕克分校）
Algorithm librla: A library of randomized linear algebra routines
Department of Applied Mathematics, University of Colorado Boulder（科罗拉多大学博尔德分校应用数学系）；National Institute of Standards and Technology (NIST)（美国国家标准与技术研究院）
