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

We present a new algorithm for floating-point computation of the singular value decomposition (SVD) of the product BτC, 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'τC'. In this way, each nonzero singular value σ is approximated with some σ+δσ, where the relative error |δσ|/σ is, up to a factor of the dimensions, of order ε min_Δ∈ Dκ₂(Δ B)+ min_Δ∈ D κ₂(Δ C)\, where D denotes the set of diagonal nonsingular matrices, κ₂(·) denotes the spectral condition number, and ε\ is the roundoff unit of floating-point arithmetic. The new algorithm is applied to the eigenvalue problem HM x = λ x with symmetric positive definite H and M. It is shown that each eigenvalue λ is computed with high relative accuracy and that the relative error |δλ|/λ of the computed approximation λ+δλ is, up to a factor of the dimension, of order εmin_Δ∈ Dκ₂(Δ HΔ) + min_Δ∈ Dκ₂(Δ MΔ)\. The new algorithm can also be used for accurate SVD computation of a single matrix G that admits an accurate factorization G=BτC.