  In this article, we present a new Nested Cross Approximation (NNCA) for
constructing H2 matrices. It differs from the existing
NCAs~\cite{bebendorf2012constructing, zhao2019fast} 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 H2 matrix-vector product,
that uses NNCA for the appropriate low-rank approximations. We illustrate the
timing profiles and the accuracy of NNCA based H2 matrix-vector product. We
also provide a comparison of NNCA based H2 matrix-vector product with the
existing NCA based H2 matrix-vector products. A key observation is that NNCA
performs better than the existing NCAs. In addition, using the NNCA based H2
matrix-vector product, we accelerate i) solving an integral equation in 3D and
ii) Support Vector Machine (SVM). In the spirit of reproducible computational
science, the implementation of the algorithm developed in this article is made
available at \url{https://github.com/SAFRAN-LAB/NNCA}.
