Accurate Computation of the Product-Induced Singular Value Decomposition with Applications

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We present a new algorithm for floating-point computation of the singular value decomposition (SVD) of the product $B^{\tau}C$, where $B$ and $C$ are full row rank matrices. The algorithm replaces the pair $(B,C)$ with an equivalent pair $(B',C')$ and then it uses the Jacobi SVD algorithm to compute the SVD of the explicitly computed matrix $B'^{\tau}C'$. In this way, each nonzero singular value $\sigma$ is approximated with some $\sigma+\delta\sigma$, where the relative error $|\delta\sigma|/\sigma$ is, up to a factor of the dimensions, of order $\{\min_{\Delta\in{\cal D}}\kappa_2(\Delta B)+ \min_{\Delta\in{\cal D}} \kappa_2(\Delta C)\}$, where ${\cal D}$ denotes the set of diagonal nonsingular matrices, $\kappa_2(\cdot)$ denotes the spectral condition number, and \boldmath${\varepsilon}$\unboldmath is the roundoff unit of floating-point arithmetic. The new algorithm is applied to the eigenvalue problem $HM x = \lambda x$ with symmetric positive definite $H$ and $M$. It is shown that each eigenvalue $\lambda$ is computed with high relative accuracy and that the relative error $|\delta\lambda|/\lambda$ of the computed approximation $\lambda+\delta\lambda$ is, up to a factor of the dimension, of order \boldmath${\varepsilon}$\unboldmath$\{\min_{\Delta\in{\cal D}}\kappa_2(\Delta H\Delta) + \min_{\Delta\in{\cal D}}\kappa_2(\Delta M\Delta)\}$. The new algorithm can also be used for accurate SVD computation of a single matrix $G$ that admits an accurate factorization $G=B^{\tau}C$.