On the complex Falk–Langemeyer method

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 D A A D A and D B B D B are small for some diagonal matrices D A and D B .

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