Regularization and Matrix Computation in Numerical Polynomial Algebra

The Numerical Factorization of Polynomials

“…, p r ). Detailed discussion on the Gauss-Newton iteration and conditions for its convergence can be found in [3,5,14,32]. In a nutshell, the iteration (26) locally converges to the least squares solution z * that is the point satisfying f , 1, . .…”

“…Letf be a polynomial with a factorization structure N with θf be given in (14). For any empirical data f off satisfying f −f < 1 2 θf , the factorization structure N off is uniquely identifiable using the data f by…”

“…The formulation of the numerical factorization follows the same "three-strikes" principles that have been effectively applied to the regularization of other ill-posed algebraic problems [14]: The numerical factorization of f is the exact factorization of a nearby polynomialf within a backward error tolerance (backward nearness principle). The nearby polynomialf is of the highest singularity among all the polynomials in the -neighborhood of f (maximum singularity principle).…”

“…Letf ∈ R m be a polynomial with its critical gap θf as in (14) and an irreducible factorizationαf 1f2 . .…”

“…In a seminal technical report [8], Kahan is the first to discover the hidden continuity on manifolds for generally discontinuous solutions of ill-posed algebraic problems. Recent works such as [6,14] made progress along this directions. This work provides a complete regularization of a typical ill-posed algebraic problem in numerical polynomial factorization by establishing its existence, uniqueness, Lipschitz continuity, convergence and condition number.…”