Parallel computing (2nd ed.): theory and practice
Parallel computing (2nd ed.): theory and practice
Polynomial Root-Finding Algorithmsand Branched Covers
SIAM Journal on Computing
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
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
A Jacobi--Davidson Iteration Method for Linear EigenvalueProblems
SIAM Journal on Matrix Analysis and Applications
Quasi-Laguerre Iteration in Solving Symmetric Tridiagonal Eigenvalue Problems
SIAM Journal on Scientific Computing
Graeffe's, Chebyshev-like, and Cardinal's processes for splitting a polynomial into factors
Journal of Complexity - Special issue for the Foundations of Computational Mathematics conference, Rio de Janeiro, Brazil, Jan. 1997
Matrix computations (3rd ed.)
A unifying convergence analysis of second-order methods for secular equations
Mathematics of Computation
Mathematics of Computation
The symmetric eigenvalue problem
The symmetric eigenvalue problem
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
Multivariate polynomials, duality, and structured matrices
Journal of Complexity
Templates for the solution of algebraic eigenvalue problems: a practical guide
Templates for the solution of algebraic eigenvalue problems: a practical guide
Computation of approximate polynomial GCDs and an extension
Information and Computation
Structured matrices and polynomials: unified superfast algorithms
Structured matrices and polynomials: unified superfast algorithms
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
On the geometry of Graeffe Iteration
Journal of Complexity
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
Time Series Analysis, Forecasting and Control
Time Series Analysis, Forecasting and Control
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
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
Probabilistic algorithms for sparse polynomials
EUROSAM '79 Proceedings of the International Symposiumon on Symbolic and Algebraic Computation
Modern Computer Algebra
Solving Polynomials with Small Leading Coefficients
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
Schur aggregation for linear systems and determinants
Theoretical Computer Science
Experimental evaluation and cross-benchmarking of univariate real solvers
Proceedings of the 2009 conference on Symbolic numeric computation
The amended DSeSC power method for polynomial root-finding
Computers & Mathematics with Applications
Real and complex polynomial root-finding with eigen-solving and preprocessing
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Random polynomials and expected complexity of bisection methods for real solving
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
The DMM bound: multivariate (aggregate) separation bounds
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Root-finding by expansion with independent constraints
Computers & Mathematics with Applications
Efficient polynomial root-refiners: A survey and new record efficiency estimates
Computers & Mathematics with Applications
Univariate polynomial root-finding by arming with constraints
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Real and complex polynomial root-finding by means of eigen-solving
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Root-refining for a polynomial equation
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
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Matrix methods are increasingly popular for polynomial root-finding. The idea is to approximate the roots as the eigenvalues of the companion or generalized companion matrix associated with an input polynomial. The algorithms also solve secular equation. QR algorithm is the most customary method for eigen-solving, but we explore the inverse Rayleigh quotient iteration instead, which turns out to be competitive with the most popular root-finders because of its excellence in exploiting matrix structure. To advance the iteration we preprocess the matrix and incorporate Newton's linearization, repeated squaring, homotopy continuation techniques, and some heuristics. The resulting algorithms accelerate the known numerical root-finders for univariate polynomial and secular equations, and are particularly well suited for the acceleration by using parallel processing. Furthermore, even on serial computers the acceleration is dramatic for numerical approximation of the real roots in the typical case where they are much less numerous than all complex roots.