High technology industries are in desperate need for adequate tools to assess the validity of simulations produced by ever faster computers for perennial unstable problems. In order to meet these industrial expectations, applied mathematicians are facing a formidable challenge summarized by these words - nonlinearity and coupling. This book is unique as it proposes truly original solutions: (1) Using hypercomputation in quadratic algebras, as opposed to the traditional use of linear vector spaces in the 20th century; (2) complementing the classical linear logic by the complex logic which expresses the creative potential of the complex plane.The book illustrates how qualitative computing has been the driving force behind the evolution of mathematics since Pythagoras presented the first incompleteness result about the irrationality of 2. The celebrated results of Goedel and Turing are but modern versions of the same idea: the classical logic of Aristotle is too limited to capture the dynamics of nonlinear computation. Mathematics provides us with the missing tool, the organic logic, which is aptly tailored to model the dynamics of nonlinearity. This logic will be the core of the "Mathematics for Life" to be developed during this century.

This classic textbook provides a unified treatment of spectral approximation for closed or bounded operators, as well as for matrices. Despite significant changes and advances in the field since it was first published in 1983, the book continues to form the theoretical bedrock for any computational approach to spectral theory over matrices or linear operators. This coverage of classical results is not readily available elsewhere. Spectral Approximation of Linear Operators offers in-depth coverage of properties of various types of operator convergence, the spectral approximation of non-self-adjoint operators, a generalization of classical perturbation theory, and computable error bounds and iterative refinement techniques, along with many exercises (with solutions), making it a valuable textbook for graduate students and reference manual for self-study.

This revised edition of a classic textbook provides a complete guide to the calculation of eigenvalues of matrices. Written at an accessible level, this modern exposition of the subject presents fundamental aspects of the spectral theory of linear operators in finite dimension. Unique features of this book include a treatment of the convergence of eigensolvers based on the notion of the gap between invariant subspaces, and coverage of the impact of the high nonnormality of a matrix on its eigenvalues. Also included is a new chapter uncovering reasons why matrices are fundamental tools for the information processing that takes place in the dynamical evolution of systems. Some of these ideas appear in print for the first time. The book's primary use is as a course text for undergraduate students in mathematics, applied mathematics, physics, and engineering. It is also a useful reference for researchers and engineers in industry.