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Nearest common root of a set of polynomials: A structured singular value approach
The paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients. In particular, we seek the minimum-magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root. It is shown that the problem is equivalent to the solution of a structured singular value (μ) problem arising in robust control for which numerous techniques are available. It is also shown that the method can be extended to the calculation of an “approximate GCD” of fixed degree by introducing the notion of the generalized structured singular value of a matrix. The work generalizes previous results by the authors involving the calculation of the “approximate GCD” of two polynomials, although the general case considered here is considerably harder and relies on a matrix-dilation approach and several preliminary transformations
Computing GCRDs of Approximate Differential Polynomials
Differential (Ore) type polynomials with approximate polynomial coefficients
are introduced. These provide a useful representation of approximate
differential operators with a strong algebraic structure, which has been used
successfully in the exact, symbolic, setting. We then present an algorithm for
the approximate Greatest Common Right Divisor (GCRD) of two approximate
differential polynomials, which intuitively is the differential operator whose
solutions are those common to the two inputs operators. More formally, given
approximate differential polynomials and , we show how to find "nearby"
polynomials and which have a non-trivial GCRD.
Here "nearby" is under a suitably defined norm. The algorithm is a
generalization of the SVD-based method of Corless et al. (1995) for the
approximate GCD of regular polynomials. We work on an appropriately
"linearized" differential Sylvester matrix, to which we apply a block SVD. The
algorithm has been implemented in Maple and a demonstration of its robustness
is presented.Comment: To appear, Workshop on Symbolic-Numeric Computing (SNC'14) July 201
Phase Diagram for Anderson Disorder: beyond Single-Parameter Scaling
The Anderson model for independent electrons in a disordered potential is
transformed analytically and exactly to a basis of random extended states
leading to a variant of augmented space. In addition to the widely-accepted
phase diagrams in all physical dimensions, a plethora of additional, weaker
Anderson transitions are found, characterized by the long-distance behavior of
states. Critical disorders are found for Anderson transitions at which the
asymptotically dominant sector of augmented space changes for all states at the
same disorder. At fixed disorder, critical energies are also found at which the
localization properties of states are singular. Under the approximation of
single-parameter scaling, this phase diagram reduces to the widely-accepted one
in 1, 2 and 3 dimensions. In two dimensions, in addition to the Anderson
transition at infinitesimal disorder, there is a transition between two
localized states, characterized by a change in the nature of wave function
decay.Comment: 51 pages including 4 figures, revised 30 November 200
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