On the complex Falk–Langemeyer method

Abstract

A new algorithm for the simultaneous diagonalization of two complex Hermitian matrices is derived. It is a proper generalization of the known Falk–Langemeyer algorithm which was originally derived in 1960 for a pair of positive definite matrices. It is proved that the complex Falk–Langemeyer algorithm is defined for a pair of Hermitian matrices which make a definite pair. Special attention is paid to the stability of the formulas for the transformation parameters in the case when the pivot submatrices are almost proportional. The numerical tests show the high relative accuracy of the method if both matrices are definite and if the condition numbers of DAADA and DBBDB are small for some diagonal matrices DA and DB.

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Acknowledgments

The author is thankful to the anonymous referees and the editor N. Higham for very insightful comments which helped him improve the paper.

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Correspondence to Vjeran Hari.

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This paper is dedicated to Professor S. Falk

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This work has been fully supported by Croatian Science Foundation under the project: IP_09_2014_3670.

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Hari, V. On the complex Falk–Langemeyer method. Numer Algor 83, 451–483 (2020). https://doi.org/10.1007/s11075-019-00689-8

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Keywords

  • Generalized eigenvalue problem
  • Complex Hermitian matrices
  • Definite matrix pair
  • Diagonalization method