Highly accurate symmetric eigenvalue decomposition and hyperbolic SVD

Abstract

Let G be a m×n real matrix with full column rank and let J be a n×n diagonal matrix of signs, J ii∈{−1,1} . The hyperbolic singular value decomposition (HSVD) of the pair (G,J) is defined as G=UΣV −1 , where U is orthogonal, Σ is positive definite diagonal, and V is J-orthogonal matrix, V T JV=J . We analyze when it is possible to compute the HSVD with high relative accuracy. This essentially 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 J-orthogonal Jacobi method method computes the HSVD with high relative accuracy. More precisely, let B=GD −1 , where D is diagonal such that the columns of B have unit norms. Essentially, we show that the computed hyperbolic singular values of the pair (G,J) will have log 10(ε/σ min (B)) correct decimal digits, where ε is machine precision. 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 Hx=λx, we analyze the two-step algorithm which consists of factorization H=GJG T followed by the computation of the HSVD of the pair (G,J). Here G is square and non-singular. Let B ̂ = D ̂ G , where D ̂ is diagonal such that the rows of B ̂ have unit norms, and let B be defined as above. Essentially, we show that the computed eigenvalues of H will have log 10(ε/σ 2 min ( B ̂ )+ε/σ min (B)) correct decimal digits. This accuracy can be much higher then the one obtained by the classical QR and Jacobi methods applied to H, where the accuracy depends on the spectral condition number of H, particularly if the matrices B and B ̂ are well conditioned, and we are interested in the accurate computation of tiny eigenvalues. Again, we give the perturbation and error bounds, and our theoretical predictions are confirmed by a series of numerical experiments.We also give the corresponding results for eigenvectors and hyperbolic singular vectors.