  An enhanced Kogbetliantz method for the singular value decomposition (SVD) of
general matrices of order two is proposed. The method consists of three phases:
an almost exact prescaling, that can be beneficial to the LAPACK's xLASV2
routine for the SVD of upper triangular 2x2 matrices as well, a highly
relatively accurate triangularization in the absence of underflows, and an
alternative procedure for computing the SVD of triangular matrices, that
employs the correctly rounded hypot function. A heuristic for improving
numerical orthogonality of the left singular vectors is also presented and
tested on a wide spectrum of random input matrices. On upper triangular
matrices under test, the proposed method, unlike xLASV2, finds both singular
values with high relative accuracy as long as the input elements are within a
safe range that is almost as wide as the entire normal range. On general
matrices of order two, the method's safe range for which the smaller singular
values remain accurate is of about half the width of the normal range.
