Newton's Iteration for Matrix Inversion, Advances and Extensions

• Published in: Matrix Methods: Theory, Algorithms And Applications pp. 364 - 381
• Main Author: Pan, Victor Y.
• Format: Book Chapter
• Language: English
• Published: WORLD SCIENTIFIC 01.04.2010
• Subjects: Matrices and Algorithms; matrix inversion; null spaces; Newton's iteration; continuation (homotopy); matrix structure; divide-and-conquer algorithms
• ISBN: 9812836020; 9789812836014; 9812836012; 9814469556; 9789812836021; 9789814469555
• DOI: 10.1142/9789812836021_0024

We first cover Newton's iteration for generalized matrix inversion, its ameliorations, recursive compression of its iterates in the case of structured inputs, some techniques of continuation via factorization, and extension to splitting the Singular Value Decomposition. We combine the latter extension with our recent fast algorithms for the null space bases (prompted by our progress in randomized preconditioning). We applied these combinations to compute the respective spaces of singular vectors and to arrive at divide-and-conquer algorithms for matrix inversion and computing determinants. Our techniques promise to be effective for computing other matrix functions in the case of ill conditioned inputs.