Polynomial roots from companion matrix eigenvalues
Mathematics of Computation
Fast Gaussian elimination with partial pivoting for matrices with displacement structure
Mathematics of Computation
Displacement structure: theory and applications
SIAM Review
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Approximate polynomial greatest common divisors and nearest singular polynomials
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Stable and Efficient Algorithms for Structured Systems of Linear Equations
SIAM Journal on Matrix Analysis and Applications
On approximate GCDs of univariate polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
When are two numerical polynomials relatively prime?
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Polynomials and Linear Control Systems
Polynomials and Linear Control Systems
An iterated eigenvalue algorithm for approximating roots of univariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Relations between roots and coefficients, interpolation and application to system solving
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
A fast solver for linear systems with displacement structure
Numerical Algorithms
QR factoring to compute the GCD of univariate approximate polynomials
IEEE Transactions on Signal Processing
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We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]/(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f, g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.