Newton-like iteration for general and structured matrices
Gianni Codevico, Victor Ya. Pan, Marc Van Barel, Xinmao Wang, Ai-Long Zheng · 2003
We review and amend the known applications of Newton’s iteration computing the inverse or Moore–Penrose generalized inverse of a matrix. Then we specialize this approach to the case of structured (and particularly Toeplitz or Toeplitz-like) matrices where all input, output and intermediate auxiliary matrices are represented in a compressed form, via their short displacement generators. We briefly recall the known policies of compression via the truncation of the smallest singular values of the displacement and via substitution and elaborate upon the very recent policy of the least-squares compression. Numerical experiments demonstrate the effectiveness of our new Newton-like iteration based on a cubic polynomial. Keywords: inverse of a matrix, Moore-Penrose generalized inverse, Newtonlike iteration, structured matrices, displacement structure.