Randomized Gram--Schmidt Process with Application to GMRES

A randomized Gram--Schmidt algorithm is developed for orthonormalization of high-dimensional vectors or QR factorization. The proposed process can be less computationally expensive than the classical while being at least as numerically stable modified process. Our approach based on random sketching, which a dimension reduction technique consisting in estimation inner products by their small efficiently computable images, so-called sketches. In this way, an approximate orthogonality full obtained orthogonalization provide computational cost any architecture. benefit sketching amplified performing nondominant operations higher precision. case numerical stability guaranteed with working unit roundoff independent problem. applied to Arnoldi iteration and results new Krylov subspace methods solving systems equations eigenvalue problems. Among them we chose GMRES method practical application methodology.