This invaluable textbook creates a general framework for the study of optimal iterative procedures for problems that are solved approximately. Emphasis is given to the efficiency of numerical methods. For generality the setting is abstract, but the book presents many applications to practical problems, allowing the reader to take advantage of the most modern high speed calculating devices, and provides examples to illustrate concepts and major theorems. The examples are selected from astrophysics (radiative transfer and the kinetic theory of gases), mechanics (elasticity), economics , the n-dimensional Euclidean space and other applied areas. At the end of the textbook, contemporary numerical algorithms to be used for high speed computations have been included.The book will benefit not only senior undergraduates, graduate students and researchers in the field but also those who wish to obtain information about specific results or techniques that take into account the particular nature of the equation.