A new Nested Cross Approximation

In this article, we present a new Nested Cross Approximation (NCA) for $\mathcal{H}^{2}$ matrices. It differs from the existing NCAs in the technique of choosing pivots, a key part of the approximation. Our technique of choosing pivots is purely algebraic and involves only a single tree traversal. We demonstrate its applicability by developing a fast $\mathcal{H}^{2}$ matrix-vector product, that uses the new NCA for the appropriate low-rank approximations. We perform various numerical experiments to illustrate the timing profiles and the accuracy of our method. We also provide a comparison of the proposed NCA with the existing NCAs. A key observation is that the proposed NCA performs better than the existing NCAs. In the spirit of reproducible computational science, the implementation of the algorithm developed in this article is made available at https://github.com/vaishna77/nNCA2D.