The Singular Value Decomposition in Product Form

A gain of about $50\% $ in the CP-time required for the calculation of the singular value decomposition of a general matrix can be achieved by not forming the orthogonal factors explicitly, but storing the Householder reflections and Jacobi rotations that compose them. An efficient method for storing the Jacobi rotations is given. The storage required for the resulting decomposition is, for general matrices, about $1\frac{1} {2}$ times what is usual, but it is not larger than usual for matrices arising in ill-posed problems. The storage required for the decomposition of a general matrix can be reduced to what is usual by giving up part of the gain in efficiency and applying an “ultimate shift” strategy.