This volume deals with the theory of algorithms for solving systems of linear algebraic equations having a non-full-rank matrix of coefficients. This involves a range of interesting problems, such as the bidiagonalization of matrices, the computation of singular values and eigenvalues, procedures for the deflation of singular values, etc. The algorithms which are discussed in this book lead to computer programs, which guarantee the accuracy of the computations, leading to unambiguous solutions. Some of the algorithms and techniques described are new; for example, the bounds which include underflow effects. Also discussed is a new approach for computing reliable eigenvectors from Sturm sequences of a symmetric tridiagonal matrix, and a procedure for characterizing unitary transformations which maintain Hessenberg form. For researchers whose work involves numerical methods of linear algebra.