13,112 research outputs found
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
GPGCD: An iterative method for calculating approximate GCD of univariate polynomials
We present an iterative algorithm for calculating approximate greatest common
divisor (GCD) of univariate polynomials with the real or the complex
coefficients. For a given pair of polynomials and a degree, our algorithm finds
a pair of polynomials which has a GCD of the given degree and whose
coefficients are perturbed from those in the original inputs, making the
perturbations as small as possible, along with the GCD. The problem of
approximate GCD is transfered to a constrained minimization problem, then
solved with the so-called modified Newton method, which is a generalization of
the gradient-projection method, by searching the solution iteratively. We
demonstrate that, in some test cases, our algorithm calculates approximate GCD
with perturbations as small as those calculated by a method based on the
structured total least norm (STLN) method and the UVGCD method, while our
method runs significantly faster than theirs by approximately up to 30 or 10
times, respectively, compared with their implementation. We also show that our
algorithm properly handles some ill-conditioned polynomials which have a GCD
with small or large leading coefficient.Comment: Preliminary versions have been presented as
doi:10.1145/1576702.1576750 and arXiv:1007.183
A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models
We consider the estimation of smoothing parameters and variance components in
models with a regular log likelihood subject to quadratic penalization of the
model coefficients, via a generalization of the method of Fellner (1986) and
Schall (1991). In particular: (i) we generalize the original method to the case
of penalties that are linear in several smoothing parameters, thereby covering
the important cases of tensor product and adaptive smoothers; (ii) we show why
the method's steps increase the restricted marginal likelihood of the model,
that it tends to converge faster than the EM algorithm, or obvious
accelerations of this, and investigate its relation to Newton optimization;
(iii) we generalize the method to any Fisher regular likelihood. The method
represents a considerable simplification over existing methods of estimating
smoothing parameters in the context of regular likelihoods, without sacrificing
generality: for example, it is only necessary to compute with the same first
and second derivatives of the log-likelihood required for coefficient
estimation, and not with the third or fourth order derivatives required by
alternative approaches. Examples are provided which would have been impossible
or impractical with pre-existing Fellner-Schall methods, along with an example
of a Tweedie location, scale and shape model which would be a challenge for
alternative methods
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