• 科學研究
    報告題目:

    Sensitivity Analysis of Repeated Eigenvalues and Corresponding Eigenvectors of Quadratic Eigenvalue Problems

    報告人:

    Prof.Roger C E Tan (National University of Singapore)

    報告時間:

    報告地點:

    理學院東北樓四樓報告廳(404))

    報告摘要:

    We consider the variation, with ρ, of the eigenvalues, λ(ρ), and corresponding right and left eigenvectors, x(ρ) (≠ 0) and y(ρ) (≠ 0), of the parameter-dependent quadratic eigenvalue problem

    2M + λC + K)(ρ)x(ρ) = 0,  yT(ρ)(λ2M + λC + K)(ρ) = 0,     (1)

    where M, C and K are mappings from the real line to the space of (real or complex) n×n matrices. We assume that (i) (1) has a semisimple eigenvalue of multiplicity r > 1 when ρ = ρ0, and (ii) M, C and K are analytic in some open neighborhood D0 of ρ0, and (iii) λ, x and y are su?ciently di?erentiable at ρ0. We develop new algorithms for computing derivatives of λ, x and y at ρ0. In practical numerical computation, there is no clear distinction between exact and approximate equality. Standard methods for computing derivatives of λ, x and y in (1) break down when eigenvalues are very close, and methods requiring eigenvalue derivatives to be distinct break down when these derivatives are also very close. By designing algorithms that allow eigenvalues and their derivatives to be multiple, we obtain methods that are much more accurate than classical methods for tightly clustered eigenvalues. In most applications, M, C and K are known in closed form, and hence accurate values of their derivatives at ρ0 are available. Extending ideas of an earlier work of [Andrew & Tan], we use these derivatives to convert what would otherwise be an ill-conditioned problem into a well-conditioned one.


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