A Brief Introduction to Numerical Analysis

This is an advanced textbook based on lectures given at the Moscow Physico-Technical Institute. The lectures are characterized by brevity, logical organization, and occasionally a lighthearted approach. It aims to involve the reader by asking questions, hinting, giving recommendations, comparing different methods, and discussing optomistic and pessemistic approaches to numerical analysis. Since matrix analysis underlies numerical analysis, this text emphasizes methods and algorithms of matrix analysis. Function approximations, methods of solving nonlinear equations, and minimization methods are also considered. As well as considering classical methods, new methods and approaches are discussed - such as spectral distribution theory and what it offers for design and proof of modern preconditioning strategies for large-scale linear algebra problems.