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The solution of characteristic value-vector problems by Newton's method

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Abstract

LetT(λ) be a bounded linear operator in a Banach spaceX for eachλ in the scalar fieldS. The characteristic value-vector problemT(λ)x = 0 with a normalization conditionφ x = 1, whereφ ε X *, is formulated as a nonlinear problem inX xS:P(y) ≡ (T(λ)x, φ x - 1) = 0,y= (X, A). Newton's method and the Kantorovič theorem are applied. For this purpose, representations and criteria for existence ofP′(y)−1 are obtained. The continuous dependence onT of characteristic values and vectors is investigated. A numerical example withT(λ) =A +λB +λ 2 C is presented.

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Authors and Affiliations

  1. Oregon State University, 97330, Corvallis, Oregon, USA

    P. M. Anselone

  2. Mathematics Research Center United States Center, University of Wisconsin, 53706, Madison, Wisconsin, USA

    L. B. Rall

Authors
  1. P. M. Anselone
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  2. L. B. Rall
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Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.

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Anselone, P.M., Rall, L.B. The solution of characteristic value-vector problems by Newton's method. Numer. Math. 11, 38–45 (1968). https://doi.org/10.1007/BF02165469

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  • DOI: https://doi.org/10.1007/BF02165469

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