On the complex Falk–Langemeyer method
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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.
KeywordsGeneralized eigenvalue problem Complex Hermitian matrices Definite matrix pair Diagonalization method
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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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