Implementation of Jacobi Rotations for Accurate Singular Value Computation in Floating Point Arithmetic

Abstract

In this paper we con if \begin{displaymath} 1 - m\r value decomposition (SVD) $A=U\Sigma V^{\tau}$ of $A=[a_1,a_2]\in\R^{m\times 2}$ accurately in floating point arithmetic. It is shown how the rotation angacobi rotation V (the right singular vector matrix) and how to compute $AV=U\Sigma$ even if the floating point representation of V is the identity matrix. In the case $\ns{a_1}\gg\ns{a_2}$, underflow can produce the identity matrix as the floating point value of V, even for $a_1$, $a_2$ that are far from being mutually orthogonal. This can cause loss of accuracy and failure of convergence of the floating point implementation of the Jacobi method for computing the SVD. The modified Jacobi method recommended in this paper can be implemented as a reliable and highly accurate procedure for computing the SVD of general real matrices whenever the exact singular values do not exceed the underflow or overflow limits.