Like the other volume “Afternotes on numerical analysis” by the same author (1996; Zbl 0844.65002), this new volume is the result of writing down these notes immediately after each lecture was given. Here the topics treated are not elementary and require a deeper mathematical understanding. This well-written book is an ideal source for graduate courses on advanced numerical analysis or seminars selecting the corresponding material.
The contents of the volume cover some topics of classical and modern numerical analysis. The first part, based on 9 lectures, deals with approximation including the approximation in normed linear spaces, existence and uniqueness of best approximation in the space \(C[0,1]\), Chebyshev approximation, discrete, continuous and weighted least squares approximation and the QR factorization, Householder transformation and orthogonal triangularization. A second short part consists of 2 lectures on linear and cubic splines and is followed by a third part of 7 lectures devoted to the eigensystems and containing the standard techniques of matrix computations, the Schur decomposition, the block diagonalization, Jordan canonical form, the Rayleigh quotient and the power method, the QR algorithm and the Hessenberg QR algorithm. The fourth part consisting of 6 lectures on Krylov sequence methods is devoted to Arnoldi decompositions, the Lanczos algorithm and the conjugate gradient methods. A fifth part consists of 2 lectures concerning with some classical linear and nonlinear iterations and a multivariate version of the Newton method.