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.

翻译：本文提出了一种针对二阶一般矩阵奇异值分解（SVD）的增强型Kogbetliantz方法。该方法包含三个阶段：近乎精确的预缩放（该步骤同样有利于LAPACK中针对上三角2x2矩阵SVD的xLASV2例程）、在无非下溢情况下实现高相对精度的三角化，以及一种利用正确舍入的hypot函数计算三角矩阵SVD的替代流程。文中还提出了一种改进左奇异向量数值正交性的启发式策略，并在广泛随机输入矩阵谱上进行了测试。在测试的上三角矩阵中，只要输入元素处于几乎覆盖整个正规范围的安全区间内，所提方法（与xLASV2不同）能以高相对精度同时求得两个奇异值。对于二阶一般矩阵，该方法能保持较小奇异值精确度的安全区间宽度约为正规范围的一半。