There exists a vast literature on numerical methods of linear algebra. In our bibliography list, which is by far not complete, we included some monographs on the subject [46], [15], [32], [39], [11], [21]. The present book is devoted to the theory of algorithms for a single problem of linear algebra, namely, for the problem of solving systems of linear equations with non-full-rank matrix of coefficients. The solution of this problem splits into many steps, the detailed discussion of which are interest� ing problems on their own (bidiagonalization of matrices, computation of singular values and eigenvalues, procedures of deflation of singular values, etc. ). Moreover, the theory of algorithms for solutions of the symmetric eigenvalues problem is closely related to the theory of solv� ing linear systems (Householder's algorithms of bidiagonalization and tridiagonalization, eigenvalues and singular values, etc. ). It should be stressed that in this book we discuss algorithms which to computer programs having the virtue that the accuracy of com� lead putations is guaranteed. As far as the final program product is con� cerned, this means that the user always finds an unambiguous solution of his problem. This solution might be of two kinds: 1. Solution of the problem with an estimate of errors, where abso� lutely all errors of input data and machine round-offs are taken into account. 2.

Synopsis

This volume deals with the theory of algorithms for solving systems of linear algebraic equations having a non-full-rank matrix of co-efficients. This involves a range of 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 an approach for computing reliable eigenvectors from Sturm sequences of a symmetric tridiagonal matrix, and a procedure for characterizing unitary transformations which maintain Hessenberg form. The work should be of interest to researchers whose work involves numerical methods of linear algebra.

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