Alessandro Pugliese

Two-Parameter SVD: Coalescing Singular Values and Periodicity

We consider matrix valued functions of two parameters in a simply connected region

Singular values of two-parameter matrices: an algorithm to accurately find their intersections

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Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

. We propose a new criterion to detect when such functions have coalescing singular values. For generic coalescings, the singular values come together in a “double cone”-like intersection. We relate the existence of any such singularity to the periodic structure of the orthogonal factors in the singular value decomposition of the one-parameter matrix function obtained restricting to closed loops in

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

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Locating coalescing singular values of large two-parameter matrices

. Our theoretical result is very amenable to approximate numerically the location of the singularities.

Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

Consider the singular value decomposition (SVD) of a two-parameter function A(x), x@?@W@?R^2, where @W is simply connected and compact, with boundary @C. No matter how differentiable the function A is (even analytic), in general the singular values lose all smoothness at points where they coalesce. In this work, we propose and implement algorithms which locate points in @W where the singular values coalesce. Our algorithms are based on the interplay between coalescing singular values in @W, and the periodicity of the SVD-factors as one completes a loop along @C.

Numerical methods for computing SVD in the D-orthogonal group

We consider a Hermitian matrix valued function

Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

A(x)\in \mathbb{C}^{n\times n}

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

, smoothly depending on parameters

Hermitian matrices of three parameters: perturbing coalescing eigenvalues and a numerical method

x\in \Omega\subset \mathbb{R}^3