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A factored variant of the Newton iteration for the solution of algebraic Riccati equations via the matrix sign function

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Abstract

In this paper we introduce a variant of the Newton iteration for the matrix sign function that results in an efficient numerical solver for a certain class of algebraic Riccati equations (AREs). In particular, when the Hamiltonian matrix associated with the ARE can be composed as \(\left [\begin {array}{llll}{A}&{BB^{T}}\\{C^{T}C}&{-A^{T}}\end {array}\right ]\), with B and \(C^{T}\) having a much larger number of rows than columns, the new algorithm exploits the special structure of the off-diagonal blocks to yield an alternative factored Newton iteration which reduces the cost per iteration by a factor of up to 8 (16 in case A is symmetric negative definite) w.r.t. the conventional iterative scheme. Experiments with a large collection of benchmark examples show that the factored iteration attains numerical accuracy similar to that of the conventional Newton iteration as well as the structure-preserving doubling algorithm. High-performance implementations of these methods, making heavy use of LAPACK linked to a multi-threaded implementation of BLAS, demonstrate the clear advantage of the new iteration on a 48-core AMD-based platform.

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Authors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, D-39106, Magdeburg, Germany

    Peter Benner & Alfredo Remón

  2. Instituto de Computación, Universidad de la República, 11.300, Montevideo, Uruguay

    Pablo Ezzatti

  3. Dpto. de Ingeniería y Ciencia de Computadores, Universidad Jaime I, 12.071, Castellón, Spain

    Enrique S. Quintana-Ortí

Authors
  1. Peter Benner
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  2. Pablo Ezzatti
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  3. Enrique S. Quintana-Ortí
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  4. Alfredo Remón
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Correspondence to Alfredo Remón.

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Benner, P., Ezzatti, P., Quintana-Ortí, E. et al. A factored variant of the Newton iteration for the solution of algebraic Riccati equations via the matrix sign function. Numer Algor 66, 363–377 (2014). https://doi.org/10.1007/s11075-013-9739-2

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  • DOI: https://doi.org/10.1007/s11075-013-9739-2

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