An improved Newton interaction for the generalized inverse of a Matrix, with applications
SIAM Journal on Scientific and Statistical Computing
Decreasing the displacement rank of a matrix
SIAM Journal on Matrix Analysis and Applications
Parallel solution of Toeplitzlike linear systems
Journal of Complexity
A bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Parallel computing (2nd ed.): theory and practice
Parallel computing (2nd ed.): theory and practice
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Computing xm mod p(x) and an application to splitting a polynomial into factors over a fixed disc
Journal of Symbolic Computation
Computation of approximate polynomial GCDs and an extension
Information and Computation
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Time Series Analysis, Forecasting and Control
Time Series Analysis, Forecasting and Control
Journal of Symbolic Computation
An iterated eigenvalue algorithm for approximating roots of univariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Concurrent Iterative Algorithm for Toeplitz-like Linear Systems
IEEE Transactions on Parallel and Distributed Systems
A 2002 update of the supplementary bibliography on roots of polynomials
Journal of Computational and Applied Mathematics
Iterative inversion of structured matrices
Theoretical Computer Science - Algebraic and numerical algorithm
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Root-refining for a polynomial equation
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
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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 p of a degree n. Empirical global convergence of the known algorithms for this task is much weaker than that of the algorithms for all n zeros, such as Weierstrass-Durand-Kerner's root-finder, which reduces its root-finding task to Viete's (Vieta's) system of n polynomial equations with n unknowns. We adjust this root-finder to the approximation of a single zero of p, 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 p, the algorithm deflates this polynomial as by-product, and then could be reapplied to the quotient to approximate the next zero of p. Alternatively by using m processors that exchange no data, one can concurrently approximate up to m zeros of p. 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.