In this thesis, a replacement for the QZ algorithm is developed. The design of the new spectral divide and conquer algorithms is oriented towards the capabilities of current computer architectures, including the support for accelerator devices. The thesis describes the co-design of the underlying mathematical ideas and the hardware aspects.
Closely connected with the generalized eigenvalue value problem, the solution of Sylvester-like matrix equations is the concern of the second part of this work. Following the co-design approach, introduced in the first part of this thesis, a flexible framework covering (generalized) Sylvester, Lyapunov, and Stein equations is developed. The combination of the new algorithms for the generalized eigenvalue problem and the Sylvester-like equation solves problems within an hour, whose solution took several days incorporating the QZ and the Bartels-Stewart algorithm.
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