## Abstract

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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## Acknowledgments

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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## Additional information

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). https://doi.org/10.1007/s10543-014-0516-y

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DOI: https://doi.org/10.1007/s10543-014-0516-y

### Keywords

- \(m\)-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