ABSTRACT
This monograph concludes by applying the conjugate gradients method, developed in chapter 16 for the minimisation of nonlinear functions, to linear equations, linear least-squares and algebraic eigenvalue problems. The methods suggested may not be the most efficient or effective of their type, since this subject area has not attracted a great deal of careful research. In fact much of the work which has been performed on the sparse algebraic eigenvalue problem has been carried out by those scientists and engineers in search of solutions. Stewart (1976) has prepared an extensive bibliography on the large, sparse, generalised symmetric matrix eigenvalue problem in which it is unfortunately difficult to find many reports that do more than describe a method. Thorough or even perfunctory testing is often omitted and convergence is rarely demonstrated, let alone proved. The work of Professor Axel Ruhe and his co-workers at Umea is a notable exception to this generality. Partly, the lack of testing is due to the sheer size of the matrices that may be involved in real problems and the cost of finding eigensolutions.
