A Lanczos Algorithm for Computing Singular Values and Vectors of Large Matrices

Abstract

Any real matrix A has associated with it the real symmetric matrix \[ B \equiv \left(\begin{array}{*{20}c} 0 \\ {A^T } \\ \end{array} \begin{array}{*{20}c} A \\ 0 \\ \end{array} \right) \] whose positive eigenvalues are the nonzero singular values of A. Using B and our Lanczos algorithms for computing eigenvalues and eigenvectors of very large real symmetric matrices, we obtain an algorithm for computing singular values and singular vectors of large sparse real matrices. This algorithm provides a means for computing the largest and the smallest or even all of the distinct singular values of many matrices