We review recent advances in algorithms for quadrature, transforms, differential equations and singular integral equations using orthogonal polynomials. Quadrature based on asymptotics has facilitated optimal complexity quadrature rules, allowing for efficient computation of quadrature rules with millions of nodes. Transforms based on rank structures in change-of-basis operators allow for quasi-optimal complexity, including in multivariate settings such as on triangles and for spherical harmonics. Ordinary and partial differential equations can be solved via sparse linear algebra when set up using orthogonal polynomials as a basis, provided that care is taken with the weights of orthogonality. A similar idea, together with low-rank approximation, gives an efficient method for solving singular integral equations. These techniques can be combined to produce high-performance codes for a wide range of problems that appear in applications.

中文翻译：

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使用正交多项式的快速算法

我们回顾了使用正交多项式的正交、变换、微分方程和奇异积分方程算法的最新进展。基于渐近的求积促进了最优复杂度求积规则，允许高效计算具有数百万个节点的求积规则。在基变算子中基于秩结构的变换允许准最优复杂性，包括在诸如三角形和球谐函数的多元设置中。当使用正交多项式作为基础建立时，可以通过稀疏线性代数求解常微分方程和偏微分方程，前提是要注意正交性的权重。类似的想法，连同低秩近似，给出了求解奇异积分方程的有效方法。