Root-finding by expansion with independent constraints

Elimination methods are highly effective for the solution of linear and nonlinear systems of equations, but reversal of the elimination principle can be beneficial as well: competent incorporation of additional independent constraints and variables or more generally immersion of the original computational problem into a larger task, defined by a larger number of independent constraints and variables can improve global convergence of iterative algorithms, that is their convergence from the start. A well known example is the dual linear and nonlinear programming, which enhances the power of optimization algorithms. We believe that this is just an ad hoc application of general Principle of Expansion with Independent Constraints; it should be explored systematically for devising iterative algorithms for the solution of equations and systems of equations and for optimization. At the end of this paper we comment on other applications and extensions of this principle.Presently we show it at work for the approximation of a single zero of a univariate polynomial pp of a degree nn. Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all nn zeros, such as Weierstrass–Durand–Kerner’s root-finder, which reduces its root-finding task to Viète’s (Vieta’s) system of nn polynomial equations with nn unknowns. We adjust this root-finder to the approximation of a single zero of pp, preserve its fast global convergence and decrease the number of arithmetic operations per iteration from quadratic to linear. Together with computing a zero of a polynomial pp, the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of pp. Alternatively by using mm processors that exchange no data, one can concurrently approximate up to mm zeros of pp. Our tests confirm the efficiency of the proposed algorithms.Technically our root-finding boils down to computations with structured matrices, polynomials and partial fraction decompositions. Our study of these links can be of independent interest; e.g., as by-product we express the inverse of a Sylvester matrix via its last column, thus extending the celebrated result of Gohberg and Sementsul (1972) [22] from Toeplitz to Sylvester matrix inverses.