New Accurate Algorithms for Singular Value Decomposition of Matrix Triplets
Zlatko Drmač · SIAM Journal on Matrix Analysis and Applications · 2000
This paper presents a new algorithm for accurate floating-point computation of the singular value decomposition (SVD) of the product $A = B^{\tau} S C,$ where $ B\in {\hbox{{\bf R}$^{p\times m}$}},$ $C\in {\hbox{{\bf R}$^{q\times n}$}},$ $S\in {\hbox{{\bf R}$^{p\times q}$}},$ and $p\leq m,$ $q\leq n$.\ The new algorithm uses diagonal scalings, the LU factorization with complete pivoting, 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 values and the matrix A' is computed explicitly. The singular values of A' are computed 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 of $B^{\tau}$ and $C^{\tau}$; (ii) the accuracy of the LU factorization with complete pivoting of S; and (iii) the accuracy of the computation of the SVD of a matrix A'$ with moderate $\min_{D=\diag}\kappa_2(A'D)$.\ Theoretical analysis and numerical 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, D 1 B , D 2 SD 3 ,D 4 C , where D i , i=1, . . .,4, are arbitrary diagonal matrices. As an application, the paper proposes new accurate algorithms for computing the (H,K)--SVD and (H -1 ,K)--SVD of S.