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Numerical analysis in modern scientific computing. An introduction. 2nd revised ed. (English) Zbl 1025.65001

Texts in Applied Mathematics. 43. New York, NY: Springer. xviii, 337 p. (2003).
This textbook contains nine chapters with the following titles:
(1) Linear systems;
(2) Error analysis;
(3) Linear least-squares problems;
(4) Nonlinear systems and least-squares problems;
(5) Linear eigenvalue problems;
(6) Three-term recurrence relations;
(7) Interpolation and approximation;
(8) Large symmetric systems of equations and eigenvalue problems, and finally;
(9) Definite integrals.
In the following we consider the contents of this monograph in more detail.
In Chapter 1, Gaussian elimination and the Cholesky decomposition are introduced, for example, and in Chapter 2, a treatment of the condition of problems and stability of algorithms can be found.
Chapter 3 includes a section on orthogonalization methods, and in Chapter 4, Newton’s method and the Gauss-Newton method as well as fixed point iterations in a general form are considered.
Chapter 5 contains a treatment of the condition of eigenvalue problems as well as the power method, the QR algorithm and the singular value decomposition. As an extension to the first edition (1995; Zbl 0818.65002) which in fact is a translation of the first German edition (1991; Zbl 0734.65001), in this chapter a section on stochastic eigenvalue problems has been included, with a short introduction to the theory of nonnegative matrices. Chapter 7 contains a treatment of polynomial, trigonometric and spline interpolation, and in Chapter 8, the conjugate gradient method, the Lanczos method as well as classical iteration methods are introduced.
Finally, Chapter 9 contains classical material on quadrature formulas as well as sections on Romberg integration and Gauss integration.
For many of the topics considered in this book, the authors present pseudo codes, graphical illustrations, exercises and results on the computational complexity of the corresponding algorithms. The considered material is presented in a clear and instructive form and is well suitable as a textbook for courses on numerical analysis.

MSC:

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65Q05 Numerical methods for functional equations (MSC2000)
65G50 Roundoff error
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Hxx Nonlinear algebraic or transcendental equations
62J05 Linear regression; mixed models
65C30 Numerical solutions to stochastic differential and integral equations
65T50 Numerical methods for discrete and fast Fourier transforms
65Fxx Numerical linear algebra
65Y20 Complexity and performance of numerical algorithms

Software:

ALCON; NewtonLib; LAPACK
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