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Numerische Algorithmen in Softwaresystemen unter besonderer Berücksichtigung der NAG-Bibliothek. (Numerical algorithms in software systems under special regard of the NAG library). (German) Zbl 0708.65001

Stuttgart: B.G. Teubner. xiv, 394 S. DM 58.00 (mit Diskette) (1990).
This is not a standard book on numerical analysis. It rather describes part of the software that is currently available for implementing the algorithms that one studies in a numerical analysis course. It may therefore be of interest to both reseachers and lecturers.
The choice of the material is good and organized in a logical way. Chapter 1 is on systems of linear equations. The usual direct methods are discussed but no iterative methods except for a remark on the conjugate gradient method. (Gauss-Seidel is mentioned in Chapter 9 in connection with multigrid methods).
Chapter 2 deals with linear optimization (the simplex method). Chapter 3 gives the basic material on interpolation and approximation. Among other things, the fast Fourier transform, cubic and bicubic splines are considered. Chapter 4 on nonlinear equations includes Brent’s method and Broyden’s continuation method. In the case of systems of nonlinear equations it gives Newton’s method and Powell’s hybrid method. Chapter 5 on eigenvalue problems gives a good survey of the basic ideas of both the special problem \(Ax=\lambda x\) and the general problem \(Ax=\lambda Bx\). The QR and QZ-algorithms are described. The singular value decomposition method is also treated. In the section on sparse matrices the power method and subspace iteration are considered. Other methods (the Arnoldi method, acceleration with Chebyshev polynomials) are not considered.
Chapter 6 (numerical integration) and Chapter 7 (initial value problems) are fairly complete (e.g. backward differentiation formulae for stiff differential equations). Chapter 8 (boundary value and eigenvalue problems) deals mainly with Sturm-Liouville problems, the finite difference method and simple and multiple shooting. Chapter 9 (partial differential equations) gives the basic ideas of finite difference methods, finite element methods and multigrid methods. The packages PLTMG and QHARM are recommended.
In all cases the use of the NAG-routines is discussed. The IMSL-routines are in many cases also described though in less detail. MATLAB is mentioned but used only in an illustrative example. The competing package GAUSS is not mentioned. Packages like LINPACK, EISPACK, QUADPACK etc. are mentioned with good references. Systems for symbolic manipulation are mentioned but not used.
The mathematical description of the algorithms is usually sufficient for the purposes of the book. We noted, however, that Theorem 1.4 of Chapter 1 is misleading. A matrix with only positive real eigenvalues can, of course, be nonsymmetric. As another example, the discussion in Section 5.3 suggests that singular values of a nonsquare matrix are mainly computed to derive the eigenvalues of the associated normal matrix.
The book contains many examples. As a special feature, it contains in Appendix A the heads of the calling Fortran programs of the NAG-routines described in the book. The complete programs are provided separately on a floppy disk. Appendix B discusses the NAG-GS Graphics Library. Appendix C describes the problem solving environments PAN and SPADE, which were developed by the author’s students to help implement the algorithms. PAN works with the graphics of a SUN3 workstation and X Window System. SPADE (partial differential equations) now works only under SUNVIEW. The floppy disk contains a demonstration program of PAN for IBM PC.
The value of the work is enhanced by a good introduction, an extensive list of references and subject index.
A book like this is necessarily rather “zeitgebunden” but is strongly recommended to both lectures in numerical analysis and non-mathematicians who are interested in good quality numerical software for their problems. In their interest we hope that the author will find time to continue his work and provide regular updates.
Reviewer: W.Govaerts

MSC:

65-04 Software, source code, etc. for problems pertaining to numerical analysis
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65Fxx Numerical linear algebra
65Hxx Nonlinear algebraic or transcendental equations
65K05 Numerical mathematical programming methods
65Dxx Numerical approximation and computational geometry (primarily algorithms)
65Lxx Numerical methods for ordinary differential equations
65Nxx Numerical methods for partial differential equations, boundary value problems
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
15-04 Software, source code, etc. for problems pertaining to linear algebra
90-04 Software, source code, etc. for problems pertaining to operations research and mathematical programming
35-04 Software, source code, etc. for problems pertaining to partial differential equations
12-04 Software, source code, etc. for problems pertaining to field theory
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
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