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Numerical mathematics. Vol. 2: Eigenvalue problems, linear optimization problems, unconstrained optimization problems. (Numerische Mathematik. Band 2: Eigenwertaufgaben, lineare Optimierungsaufgaben, unrestringierte Optimierungsaufgaben.) (German) Zbl 0746.65005

Vieweg Studium. 33: Aufbaukurs Mathematik. Braunschweig etc.: Friedr. Vieweg & Sohn. viii, 277 p., 8 Abb. u. 122 Aufg. (1992).
[For Vol. 1 see the preceding review.]
In this second volume the author discusses eigenvalue problems, linear optimization and unconstrained optimization.
Chapter 5 (eigenvalue problems) starts with a good theoretical introduction on Gershgorin circles, the Bauer-Fike theorem, the Courant minimum-maximum principle and the Schur decomposition. Section 5.2 describes the QR-method for general (nonsymmetric) matrices. The power method, inverse iteration and the QR-method with shifts are discussed. In Section 5.3 the symmetric case is studied (method of Jacobi, bisection method). The numerical computation of the singular value decomposition is also in this Section.
Chapter 6 (linear optimization) contains the classical simplex method as well as the more recent Karmarkar approach.
Chapter 7 (unconstrained optimization) is extensive (120 pages) and treats topics that are not usually found in textbooks, e.g. local and global convergence of the Broyden-Fletcher-Goldfarb-Shanno method. The Fletcher-Reeves method and trust-region methods are included.
As in the first volume, the presentation is careful, lively and attractive. The reviewer feels that the two volumes could be used for various courses. For example, Chapter 1 (systems of linear equations) and Chapter 5 (eigenvalue problems) would be a good base for a course on numerical linear algebra when completed by an introduction to the available software (nowadays LAPACK). On the other hand, Chapter 1, 2, 6 and 7 together would bring a student near the point where he (she) can start research in optimization theory.
Reviewer: W.Govaerts (Gent)

MSC:

65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65K05 Numerical mathematical programming methods

Citations:

Zbl 0737.65004

Software:

LAPACK
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