More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition

Bidiagonal reduction is the preliminary stage for the fastest stable algorithms for computing the singular value decomposition. However, the best error bounds on bidiagonal reduction methods are of the form A + A = UBV T ; kAk 2 " M f(n)kAk 2 where B is bidiagonal, U and V are orthogonal, " M is machine precision, and f(n) is a modestly growing function of the dimensions of A. A Givens-based bidiagonal reduction procedure is proposed that satisses A + A = U(B + B)V T ; where B is bounded componentwise and A satisses a tighter columnwise bound. Thus the routine obtains more accurate singular values for matrices that have poor column scaling or those arising from rank revealing decompositions.