Formulations and Theorems of Quadratically Convergent Methods for Inverse Symmetric Eigenvalue Problems

Inverse eigenvalue problems arise in a variety of applications, and thus various Newton's methods, which quadratically converge, have been developed both in theory and practice. Among many studies over thirty years, two extremely significant developments are found. Firstly, smooth matrix decompositions have been successfully applied since the 1990s. Secondly, a matrix multiplication based method has been recently proposed. In this paper, such efficient modern solvers are classified in the context of classical Newton's methods according to their mathematical formulations, and then the corresponding convergence theorems and their relationship are surveyed.