491 research outputs found
A matrix nullspace approach for solving equality-constrained multivariable polynomial least-squares problems
We present an elimination theory-based method for solving equality-constrained multivariable polynomial least-squares problems in system identification. While most algorithms in elimination theory rely upon Groebner bases and symbolic multivariable polynomial division algorithms, we present an algorithm which is based on computing the nullspace of a large sparse matrix and the zeros of a scalar, univariate polynomial
Computing the common zeros of two bivariate functions via Bezout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (� 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions
Equidistribution of zeros of random holomorphic sections
We study asymptotic distribution of zeros of random holomorphic sections of
high powers of positive line bundles defined over projective homogenous
manifolds. We work with a wide class of distributions that includes real and
complex Gaussians. As a special case, we obtain asymptotic zero distribution of
multivariate complex polynomials given by linear combinations of orthogonal
polynomials with i.i.d. random coefficients. Namely, we prove that normalized
zero measures of m i.i.d random polynomials, orthonormalized on a regular
compact set are almost surely asymptotic to the
equilibrium measure of .Comment: Final version incorporates referee comments. To appear in Indiana
Univ. Math.
Analysis of Nonlinear Systems via Bernstein Expansions
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106482/1/AIAA2013-4557.pd
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
- …