New homotopic/factorization and symmetrization techniques for Newton’s and Newton/structured iteration

We modify our earlier homotopic (continuation) process for iterative matrix inversion to improve the choice of the initial approximate inverse at every homotopic step. This enables us to control the condition of the auxiliary matrices and to accelerate the convergence of the iteration. The algorithm also extends our earlier approach where we factorized the input matrix into the product of better conditioned factors. We accelerate our new algorithm in the case of a Toeplitz-like input, comment on the extensions to the computations with other structured matrices such as Cauchy–Pick and banded matrices, and propose new techniques for the compression of the displacement of Hermitian matrices.