This presentation delves into the latest advancements in Krylov methods developed during my thesis, including BiLQ, BiLQR, TriLQR, TriCG, TriMR, GPMR, MinAres, and CAR.  It discusses the advantages of these new methods and their applications in solving linear systems.

Additionally, I will describe how the implementation of these methods supports GPUs architectures within the Julia package Krylov.jl, focusing on their efficiency and versatility across various computational platforms.