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Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters

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Current Challenges in Stability Issues for Numerical Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2082))

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Abstract

This contribution is a summary of four lectures delivered by the first author at the CIME Summer school in June 2011 at Cetraro (Italy). Preparation of those lectures was greatly aided by the other authors of these lecture notes. Our goal is to present some classical, as well as some new, results related to decompositions of matrices depending on one or more parameters, with particular emphasis being paid to the case of coalescing eigenvalues (or singular values) for matrices depending on two or three parameters. There is an extensive literature on this subject, but a systematic collection of relevant results is lacking, and this provided the impetus for writing the lecture notes. During the last 15 years, Dieci has had several collaborators on the topics under scrutiny. Besides the coauthors of these lectures, the collaboration with the following people is gratefully acknowledged: Timo Eirola (Helsinki University of Technology, Finland), Jann-Long Chern (National Central University, Taiwan), Mark Friedman (University of Alabama, Huntsville) and Maria Grazia Gasparo (University of Florence, Italy).

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Notes

  1. 1.

    No sub-diagonal element is 0.

  2. 2.

    Recall that a matrix M is irreducible if there does not exist a permutation matrix E for which E T ME is in block upper triangular form: \({E}^{T}ME = \left [\begin{array}{cc} M_{11} & M_{12}\\ 0 & M_{ 22} \end{array} \right ]\). In particular, a permutation matrix P is irreducible if there does not exist another permutation matrix E for which \({E}^{T}PE = \left [\begin{array}{cc} P_{11} & 0\\ 0 & P_{ 22} \end{array} \right ]\) with P 11 and P 22 permutation matrices of smaller size.

  3. 3.

    They are also called “diabolical points”, with reference to the popular toy “diablo”.

  4. 4.

    See Remark 2.8.

  5. 5.

    The same would be true if we assumed transversal intersection of the pair Γ3 and Γ 4 .

  6. 6.

    Recall that the occurrence of a coalescing point along a curve is non generic. However, in case it happens, then it is actually detected by our algorithm (since the stepsize is pushed below the minimum allowed stepsize). This is actually a somewhat pleasant occurrence, since after all we are trying to detect coalescing points!

  7. 7.

    Typical values used are tolls = 10−1, h max = L∕10, h min = 10−14.

  8. 8.

    Typical values used are tollp = π∕6, which corresponds to restricting a phase variation to be at most π∕4, Δ max to be 1∕10, Δ min = 10−14, Δ 0 = Δ max.

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Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Luca Dieci

  2. Dip. di Ingegneria Industriale, Universitá di Firenze, Florence, Italy

    Alessandra Papini

  3. Dip. di Matematica, Universitá degli studi di Bari “Aldo Moro”, Bari, Italy

    Alessandro Pugliese

  4. Laboratory of Wave Mechanics and Multi-field Interactions, EPFL, Lausanne, Switzerland

    Alessandro Spadoni

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  1. Luca Dieci
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  2. Alessandra Papini
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  3. Alessandro Pugliese
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  4. Alessandro Spadoni
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Correspondence to Luca Dieci .

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Dieci, L., Papini, A., Pugliese, A., Spadoni, A. (2014). Continuous Decompositions and Coalescing Eigenvalues for Matrices Depending on Parameters. In: Current Challenges in Stability Issues for Numerical Differential Equations. Lecture Notes in Mathematics(), vol 2082. Springer, Cham. https://doi.org/10.1007/978-3-319-01300-8_4

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