A Low-Rank Matrix Approach to Compute Polynomial Approximations of Smooth Two-Dimensional Functions

Polynomial approximation of smooth functions is becoming increasingly important in fields like numerical analysis and scientific computing. These approximations are vital in models that rely on spectral methods. To reduce the memory costs for large dimensional problems, various methods to provide data-sparse representations have been proposed, including methods based on singular value decomposition, adaptive cross approximation, and matrices with hierarchical low-rank structures, to mention a few. This work presents implementation details on the polynomial approximation of univariate smooth functions through the Polynomial1 class, and of bivariate smooth functions by low-rank matrix representation via the Polynomial2 class. These approaches are explained within Tau Toolbox, a mathematical software library for solving integro-differential problems by the spectral Tau method.