Proceedings ArticleDOI
Efficient algorithms for computing the nearest polynomial with constrained roots
Markus A. Hitz,Erich Kaltofen +1 more
- pp 236-243
Reads0
Chats0
TLDR
This work gives a polynomial-time algorithm to compute the radius of stability in the Euclidean norm for a variety of stability domains and develops hybrid symbolic-numeric algorithms to constrain one root of a complex or realPolynomial to a curve in the complex plane.Abstract:
The location of polynomial roots is sensitive to perturbations of the coefficients. Continuous changes of the coefficients of a polynomial move the roots continuously. We consider the problem of finding the minimal perturbations to the coefficients to move one or several roots to given loci. We measure minimality in the Euclidean distance to the coefficient vector, as well as the maximal coefficient-wise change in absolute value (infinity norm), and in the Manhattan norm ($l\sp1$-norm). In the Euclidean norm the computational task reduces to a least squares problem, in the infinity norm and the $l\sp1$-norm it can be formulated as a linear program.
We can derive symbolic solutions in closed form for the Euclidean norm in the case of complex coefficients and a single complex root. Our new result is a formula for the minimum change in the infinity norm for the case of real coefficients and a single real root. Based on the principle of parametric minimization we develop hybrid symbolic-numeric algorithms to constrain one root of a complex or real polynomial to a curve in the complex plane.
As an application to robust control, we give a polynomial-time algorithm to compute the radius of stability in the Euclidean norm for a variety of stability domains.read more
Citations
More filters
Proceedings ArticleDOI
Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials
TL;DR: This work presents an algorithm based on a version of the structured total least norm (STLN) method and demonstrates that the algorithm in practice computes globally minimal approximations on a diverse set of benchmark polynomials.
Book ChapterDOI
Structured Low Rank Approximation of a Sylvester Matrix
TL;DR: This work presents iterative algorithms that compute an approximate GCD and that can certify an approximate ∈-GCD when a tolerance ∈ is given on input and demonstrates the practical performance of these algorithms on a diverse set of univariate pairs of polynomials.
Journal ArticleDOI
Challenges of Symbolic Computation
TL;DR: The author presents background to each of his problems and a clear-cut test that evaluates whether a proposed attack has solved one of my problems, and state his favorite eight open problems in symbolic computation.
Journal ArticleDOI
Computation of Approximate Polynomial GCDs and an Extension
TL;DR: Reduction to the approximation of polynomial zeros enabled to obtain a new insight into the GCD problem and to devise effective solution algorithms, and this enables to obtain certified correct solution for a large class of input polynomials.
Proceedings ArticleDOI
Exact certification of global optimality of approximate factorizations via rationalizing sums-of-squares with floating point scalars
TL;DR: This work generalizes the technique by Peyrl and Parillo to computing lower bound certificates for several well-known factorization problems in hybrid symbolic-numeric computation and certifies accurate rational lower bounds near the irrational global optima.
References
More filters
Book
The Art of Computer Programming
TL;DR: The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
A new polynomial-time algorithm for linear programming
TL;DR: It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
Related Papers (5)
Approximate polynomial greatest common divisors and nearest singular polynomials
N. Karmarkar,Y. N. Lakshman +1 more