Stable Computation of Generalized Polar Decompositions

The QDWH algorithm can compute the polar decomposition of a matrix in a stable and efficient way. We generalize this method in order to compute generalized polar decompositions with respect to signature matrices. Here, the role of the QR decomposition is played by the hyperbolic QR decomposition. However, it doesn't show the same favorable properties concerning stability as its orthogonal counterpart. Remedies are found by exploiting connections to the LDL^T factorization and by employing well-conditioned permuted graph bases. The computed polar decomposition is used to formulate a structure-preserving spectral divide-and-conquer method for pseudosymmetric matrices. Applications of this method are found in computational quantum physics, where eigenvalues and eigenvectors describe optical properties of condensed matter or molecules. Additional properties guarantee fast convergence and a reduction to symmetric definite eigenvalue problems after just one step of spectral divide-and-conquer.