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Numerical analysis: A second course. (English) Zbl 0701.65002
Classics in Applied Mathematics, 3. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xiii, 201 p. $ 25.50 (1990).
The book is a long expected reprint of the meanwhile classic work which had been published by Academic press in 1972. Despite the reviews concerning this former edition (cf. e.g. Zbl 0248.65001) the importance and the brilliance of the book justify to review it briefly again.
It central subjects are errors and (in)stabilities handled in four major parts.
After an introductory chapter on linear algebra, in Part I, the mathematical stability and ill conditioning are considered for systems of linear equations (Ch. 2), for eigenvalues and eigenvectors (Ch. 3) and for differential and difference equations (Ch. 4). Perturbation theorems in the first two cases and stability theorems in the last one characterize this part. Continuity results for eigenpairs can be found as well as Gershgorin-like theorems and an error estimate for the Rayleigh quotient.
In Part II, the discretization error is studied for initial value problems (Ch. 5) and for boundary value problems (Ch. 6). The concept of local and global discretization error is introduced for one step and multistep methods. Consistency, convergence and stability of these methods are discussed, the order of consistency is defined and is computed for particular cases. Starting with the rather general class of boundary value problems \[ \left\{\begin{gathered} y''(x)=f(x,y'(x),y(x)), a\leq x\leq b,\\ y(a)=\alpha, y(b)=\beta, \end{gathered}\right.\tag{*} \] bounds for the discretization error are derived when restricting (*) to linear problems and when applying the difference method.
In part III, the convergence of iterative methods is examined which are used to solve systems of linear (Ch. 7) and nonlinear equations (Ch. 8). The Jacobi iteration, the Gauss-Seidel iteration and the SOR iteration are presented as special methods based on splittings \(A=B-C\) of the coefficient matrix A. Convergence results are derived, connections to the asymptotic convergence factor are established and applications to discrete analogues of some subclasses of (*) are given. Regular and P- regular splittings are introduced to build up more general classes of iterations. For systems of nonlinear equations, iterations of the form \(x^{k+1}=G(x^ k)\), \(k=0,1,...\), are studied, specialized later on to Newton’s method and to the Newton-Gauss-Seidel method. Theorems on their convergence including speed and errors follow.
In part IV, rounding errors are illuminated by the example of the Gaussian elimination process (Ch. 9). The method is presented, pivoting strategies and iterative refinement are discussed, Wilkinson’s backward error analysis is applied.
All sections contain exercises which are very instructive; the chapters end with hints for further reading. The selection of the material, its presentation with many examples and, in particular, counterexamples make this book extremely valuable - not only for students.
Reviewer: G.Mayer

65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Fxx Numerical linear algebra
65Hxx Nonlinear algebraic or transcendental equations
65G50 Roundoff error
65L10 Numerical solution of boundary value problems involving ordinary differential equations