Extending NextLA.jl with Portable Tile Low-Rank Kernels

This project aims to extend the Julia linear algebra ecosystem by developing vendor-agnostic GPU kernels in NextLA.jl for large-scale structured matrices. It focuses on tile low-rank (TLR) representations, which exploit low-rank structure to reduce computational cost and memory usage. The project will implement truncated SVD routines and core TLR arithmetic kernels, including low-rank GEMM, TRSM, and SYRK with adaptive truncation. These kernels will serve as building blocks for large distributed TLR algorithms, demonstrated through the implementation of a distributed TLR Cholesky factorization. By the end of the project, NextLA.jl will provide a portable, composable foundation for GPU-accelerated distributed linear algebra.