<title>Efficient and accurate computation of generalized singular-value decompositions</title>

Abstract

We present a new family of algorithms for accurate floating--point computation of the singular value decomposition (SVD) of various forms of products (quotients) of two or three matrices. The main goal of such an algorithm is to compute all singular values to high relative accuracy. This means that we are seeking guaranteed number of accurate digits even in the smallest singular values. We also want to achieve computational efficiency, while maintaining high accuracy. To illustrate, consider the SVD of the product A=B<SUP>T</SUP>SC. The new algorithm uses certain preconditioning (based on diagonal scalings, the LU and QR factorizations) to replace A with A'=(B')<SUP>T</SUP>S'C', where A and A' have the same singular values and the matrix A' is computed explicitly. Theoretical analysis and numerical evidence show that, in the case of full rank B, C, S, the accuracy of the new algorithm is unaffected by replacing B, S, C with, respectively, D<SUB>1</SUB>B, D<SUB>2</SUB>SD<SUB>3</SUB>, D<SUB>4</SUB>C, where D<SUB>i</SUB>, i=1,...,4 are arbitrary diagonal matrices. As an application, the paper proposes new accurate algorithms for computing the (H,K)-SVD and (H<SUB>1</SUB>,K)-SVD of S.