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Efficient algorithm for simultaneous reduction to the \(m\)-Hessenberg-triangular-triangular form


This paper proposes an efficient algorithm for simultaneous reduction of three matrices by using orthogonal transformations, where \(A\) is reduced to \(m\)-Hessenberg form, and \(B\) and \(E\) to triangular form. The algorithm is a blocked version of the algorithm described by Miminis and Paige (Int J Control 35:341–354, 1982). The \(m\)-Hessenberg-triangular–triangular form of matrices \(A\), \(B\) and \(E\) is specially suitable for solving multiple shifted systems \((\sigma E-A)X=B\). Such shifted systems naturally occur in control theory when evaluating the transfer function of a descriptor system, or in interpolatory model reduction methods. They also arise as a result of discretizing the time-harmonic wave equation in heterogeneous media, or originate from structural dynamics engineering problems. The proposed blocked algorithm for the \(m\)-Hessenberg-triangular-triangular reduction is based on aggregated Givens rotations, and is a generalization of the blocked algorithm for the Hessenberg-triangular reduction proposed by Kågström et al. (BIT 48:563–584, 2008). Numerical tests confirm that the blocked algorithm is much faster than its non-blocked version based on regular Givens rotations only. As an illustration of its efficiency, two applications of the \(m\)-Hessenberg-triangular-triangular reduction from control theory are described: evaluation of the transfer function of a descriptor system at many complex values, and computation of the staircase form used to identify the controllable part of the system.

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The author wishes to thank the referees for giving many helpful suggestions, which inspired the development of the double-blocked algorithm for the \(m\)-Hessenberg-triangular-triangular reduction, and helped to improve the quality of the paper.

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Correspondence to Nela Bosner.

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Communicated by Daniel Kressner.

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Bosner, N. Efficient algorithm for simultaneous reduction to the \(m\)-Hessenberg-triangular-triangular form. Bit Numer Math 55, 677–703 (2015).

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  • \(m\)-Hessenberg-triangular-triangular form
  • Orthogonal transformations
  • Level 3 BLAS
  • Blocked algorithm
  • Solving shifted system
  • Transfer function evaluation
  • Staircase form

Mathematics Subject Classification

  • 15A21
  • 15A06
  • 65F05
  • 65Y20
  • 93B05
  • 93B10
  • 93B17
  • 93B40