Highly Accurate Hyperbolic Svd and Symmetric Eigenvalue Decomposition

The hyperbolic singular value decomposition (HSVD) of the pair (), where has full column rank and is diagonal matrix of signs, is defined as = Σ, where is orthogonal, Σ is positive definite diagonal, and is -orthogonal matrix, = . We analyze when it is possible to compute the HSVD with high relative accuracy. This means that each computed hyperbolic singular value is guaranteed to have some correct digits, even if they have widely varying magnitudes. We show that one-sided -orthogonal Jacobi method method computes the HSVD with high relative accuracy. Essentially, we show that the computed singular values will have log(/()) correct decimal digits, where is machine precision and is the matrix whose columns are the normalized columns of . We give the necessary relative perturbation bounds and error analysis of the algorithm. Our numerical tests confirmed all theoretical results.For the symmetric non-singular eigenvalue problem = , we analyze the two-step algorithm which consists of factorization = followed by the computation of the HSVD of the pair (). Essentially, we show that the computed eigenvalues will have correct decimal digits, where is the matrix whose rows are the normalized rows of . This accuracy can be much higher then the one obtained by the classical QR and Jacobi methods, where the accuracy depends on the spectral condition number of , particularly if the matrices and are well conditioned, and we are interested in the accurate computation of tiny eigenvalues. Again, we give the perturbation and error bounds, and all our theoretical predictions are confirmed by a series of numerical experiments.We also give the corresponding results for hyperbolic singular vectors and eigenvectors