61 research outputs found
Implicit QR with compression
AbstractIn this paper, we elaborate on the implicit shifted QR eigenvalue algorithm given in [D.A. Bini, P. Boito, Y. Eidelman, L. Gemignani, I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl. 432 (2010), 2006–2031]. The algorithm is substantially simplified and speeded up while preserving its numerical robustness. This allows us to obtain a potentially important advance towards a proof of its backward stability together with both cost reductions and implementative benefits
Algebraic construction of the Darboux matrix revisited
We present algebraic construction of Darboux matrices for 1+1-dimensional
integrable systems of nonlinear partial differential equations with a special
stress on the nonisospectral case. We discuss different approaches to the
Darboux-Backlund transformation, based on different lambda-dependencies of the
Darboux matrix: polynomial, sum of partial fractions, or the transfer matrix
form. We derive symmetric N-soliton formulas in the general case. The matrix
spectral parameter and dressing actions in loop groups are also discussed. We
describe reductions to twisted loop groups, unitary reductions, the matrix Lax
pair for the KdV equation and reductions of chiral models (harmonic maps) to
SU(n) and to Grassmann spaces. We show that in the KdV case the nilpotent
Darboux matrix generates the binary Darboux transformation. The paper is
intended as a review of known results (usually presented in a novel context)
but some new results are included as well, e.g., general compact formulas for
N-soliton surfaces and linear and bilinear constraints on the nonisospectral
Lax pair matrices which are preserved by Darboux transformations.Comment: Review paper (61 pages). To be published in the Special Issue
"Nonlinearity and Geometry: Connections with Integrability" of J. Phys. A:
Math. Theor. (2009), devoted to the subject of the Second Workshop on
Nonlinearity and Geometry ("Darboux Days"), Bedlewo, Poland (April 2008
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