New accurate algorithms for singular value decomposition of matrix triplets

This paper presents a new algorithm for accurate floating-point computation of the singular value decomposition (SVD) of the product A = (BSC)-S-tau, where B is an element of R-pxm, C is an element of R-qxn, S is an element o f R-pxq, and p less than or equal to m, q less than or equal to n. The new algorithm uses diagonal scalings, the QR factorization with complete pivoti ng, the QR factorization with column pivoting, and matrix multiplication to replace A by A' = B'(tau) S' C', where A and A have the same singular valu es and the matrix A is computed explicitly. The singular values of A are co mputed using the Jacobi SVD algorithm. It is shown that the accuracy of the new algorithm is determined by (i) the accuracy of the QR factorizations o f B-tau and C; ( ii) the accuracy of the LU factorization with complete piv oting of S; and( iii) the accuracy of the computation of the SVD of a matri x A' with moderate min(D=diag) k(2) (A'D). Theoretical analysis and numeric al evidence show that, in the case of rank(B) = rank(C) = p and full rank S , the accuracy of the new algorithm is unaffected by replacing B, S, C with , respectively, D1B, D2S D-3, D4C, where D-i, i = 1,..., 4, are arbitrary d iagonal matrices. As an application, the paper proposes new accurate algori thms for computing the (H, K)-SVD and (H-1, K)-SVD of S.