Unified Newton’s method for matrix decompositions

We propose a way to unify the construction of Newton’s methods for matrix decompositions. This construction is based on a perturbation analysis of a suitable system associated with matrix decompositions. Then it appears that the resolution of a linear Sylvester equation permits defining Newton’s method. We give a general result to analyze the quadratic convergence of this Newton’s method. We apply it to classical matrix decompositions: LU decomposition, QR decomposition, eigenproblem in the diagonalizable case, singular value decomposition, and Schur decomposition. Finally, we propose for future work a generalization of this construction to approximate matrix decomposition by high-order methods.