Consider the Singular Value Decomposition (SVD) of a two-parameter function $A(x)$, $x\in \Omega\subset \R^2$, where $\Omega$ is simply connected and compact, with boundary $\Gamma$. No matter how differentiable the function $A$ is (even analytic), in general the singular values lose all smoothness at points where they coalesce. In thiswork, we propose and implement algorithms which locate points in $\Omega$ where the singular values coalesce. Our algorithms are based on the interplay between coalescing singular values in $\Omega$, and the periodicity of the SVD-factors as one completes a loop along $\Gamma$.

### Singular Values of Two-Parameter Matrices: An Algorithm To Accurately Find Their Intersections

#### Abstract

Consider the Singular Value Decomposition (SVD) of a two-parameter function $A(x)$, $x\in \Omega\subset \R^2$, where $\Omega$ is simply connected and compact, with boundary $\Gamma$. No matter how differentiable the function $A$ is (even analytic), in general the singular values lose all smoothness at points where they coalesce. In thiswork, we propose and implement algorithms which locate points in $\Omega$ where the singular values coalesce. Our algorithms are based on the interplay between coalescing singular values in $\Omega$, and the periodicity of the SVD-factors as one completes a loop along $\Gamma$.
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2008
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11586/13326
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