Accurate complex Jacobi rotations

08/27/2023

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by   Vedran Novaković, et al.

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This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions 𝚌𝚛_𝚑𝚢𝚙𝚘𝚝 and 𝚌𝚛_𝚛𝚜𝚚𝚛𝚝, proposed for standardization in the C programming language as recommended by the IEEE-754 floating-point standard. The rounding to nearest (ties to even) and the non-stop arithmetic are assumed. The numerical examples compare the observed with theoretical bounds on the relative errors in the rotations' elements, and show that the maximal observed departure of the rotations' determinants from unity is smaller than that of the transformations computed by LAPACK.