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Lineare Gleichungssysteme mit Bandstruktur und ihr asymptotisches Verhalten. (Lizenzausg.). (German) Zbl 0613.15003

München-Wien: Carl Hanser Verlag. 247 S. (Orig. VEB Deutscher Verlag der Wissenschaften, Berlin) (1986).
This is an excellent book. The author writes from his vast experience and personal vantage point on large and infinite banded linear systems of equations. There are 6 chapters, about 200 references, a page of quoted names and an index. The author’s approach is founded on two principles: 1) rather than studying finite subsystems of infinite systems as an approximation to solving the infinite system, the infinite band matrix is studied in its own right and then specialized to give asymptotic approximations to finite such systems; and 2) banded systems of equations (finite or infinite) are studied in their equivalent form as boundary value problems for linear difference equations. The equivalent difference equations are studied using complex analysis for their stability polynomial, continued fractions and differential equations techniques.
Chapter 1 is an introduction to difference equations. Chapter 2 constructs explicit inverses (or generalized inverses) for infinite band matrices of general or specific types like Toeplitz, Jordan matrices, circulants, commuting and other matrices. Chapter 3 investigates stability of linear banded systems in much more detail than the standard matrix condition number approach. The stability polynomial of a Toeplitz matrix together with its roots and multiplicities is again studied to establish the asymptotic behaviour of the inverse (or generalized inverse). A stability analysis of Gauss elimination follows. Chapter 4 treats band matrices whose codiagonal elements converge and who thus behave asymptotically like Toeplitz matrices. Using the theory of maximal and minimal solutions, the asymptotic behaviour of solutions to such systems is studied.
Chapter 5 treats unstable banded systems of equations and methods for solving them stably by regularisation using either the method of a minimal solution with a block decomposition of the system or Tikhonov’s method of regularisation. There is a section on using such regularisation schemes in conjunction with high order and high accuracy but unstable initial or boundary value differential equation solvers. The last chapter describes methods for transforming sparse systems to banded structure. A graph theoretic algorithm for reordering vertices (or columns and rows of A) of the graph g(A) based on the adjacency levels (Stufenzerlegung) is described that will minimize the bandwidth of the resulting matrix. Then the new A can be approximated by the Toeplitz matrix B which has the mean of the codiagonals of A on its codiagonals and the solution of \(Bx=f\) can be used in an ”iterative refinement” type way to solve the original system.
All thoughout the book there are references, examples and applications to partial differential equations, eigenvalue problems, factorisation of polynomials, difference inequations, finite element method and more. The only disadvantage of this beautiful book is that it will not be available to a large enough audience. An english version would be very much appreciated.
Reviewer: F.Uhlig

MSC:

15A06 Linear equations (linear algebraic aspects)
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A12 Conditioning of matrices
15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
15A09 Theory of matrix inversion and generalized inverses
15A15 Determinants, permanents, traces, other special matrix functions
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
40A15 Convergence and divergence of continued fractions
65Q05 Numerical methods for functional equations (MSC2000)
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
65F05 Direct numerical methods for linear systems and matrix inversion
65L20 Stability and convergence of numerical methods for ordinary differential equations
65F50 Computational methods for sparse matrices
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs