# QRP Variation of Cross--Approximation Iterations for Low Rank Approximation

## TL;DR

This paper explores a variation of cross-approximation iterations for low rank matrix approximation, emphasizing superfast algorithms that significantly reduce computational resources, with applications to accelerating the Fast Multipole Method.

## Contribution

It introduces a QRP variation of cross-approximation iterations, enhancing superfast low rank approximation techniques for large-scale matrices.

## Key findings

- Superfast LRA reduces computational complexity.
- Application to accelerate the Fast Multipole Method.
- Extensive testing confirms efficiency improvements.

## Abstract

We call matrix algorithms superfast if they use much fewer flops and memory cells than the input matrix has entries. Using such algorithms is indispensable for Big Data Mining and Analysis, where the input matrices are so immense that one can only access a small fraction of all their entries. A natural remedy is Low Rank Approximation (LRA) of these matrices, which is routinely computed by means of Cross-Approximation iterations for more than a decade of worldwide application in computational practice. We point out and extensively test an important application of superfast LRA to significant acceleration of the celebrated Fast Multipole Method, which turns it into Superfast Multipole Method.