Accurately Counting Singular Values of Bidiagonal Matrices and Eigenvalues of Skew-Symmetric Tridiagonal Matrices

Abstract

We have developed algorithms to count singular values of a real bidiagonal matrix which are greater than a specified value. This requires the transformation of the singular value problem to an equivalent symmetric eigenvalue problem. The counting of singular values is paramount in the design of bisection- and multisection-type algorithms for computing singular values on serial and parallel machines. The algorithms are based on the eigenvalues of BBt, BtB, and the 2n x 2n zero--diagonal tridiagonal matrix which is permutationally equivalent to the Jordan--Wielandt form [{\scriptsize ${0 \atop B^t}{B \atop 0}$}], where B is an n x n bidiagonal matrix. The two product matrices, which do not have to be formed explicitly, lead to the progressive and stationary qd algorithms of Rutishauser. The algorithm based on the zero--diagonal matrix, which we have named the Golub--Kahan form, may be considered as a combination of both the progressive and stationary qd algorithms. We study important properties such as the backward error analysis, the monotonicity of the inertia count and the scaling of data which guarantee the accuracy and integrity of these algorithms. For high relative accuracy of tiny singular values, the algorithm based on the Golub--Kahan form is the best choice. However, if such accuracy is not required or requested, the differential progressive and differential stationary qd algorithms with certain modifications are adequate and more efficient. We also show how to transform the real skew-symmetric tridiagonal eigenvalue problem to a real bidiagonal singular value problem. Thus, the eigenvalues of a skew-symmetric matrix canbe readily counted using algorithms developed here for bidiagonal matrices.