Iterative Solution of Large Sparse Systems of Equations
In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods with emphasis on the algebraic structure of linear iteration, which is usually ignored in other literature.
The necessary amount of work increases dramatically with the size of systems, so one has to search for algorithms that most efficiently and accurately solve systems of, e.g., several million equations. The choice of algorithms depends on the special properties the matrices in practice have. An important class of large systems arises from the discretization of partial differential equations. In this case, the matrices are sparse (i.e., they contain mostly zeroes) and well-suited to iterative algorithms.
The first edition of this book grew out of a series of lectures given by the author at the Christian-Albrecht University of Kiel to students of mathematics. The second edition includes quite novel approaches.
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2 Iterative Methods
3 Classical Linear Iterations in the Positive Definite Case
4 Analysis of Classical Iterations Under Special Structural Conditions
5 Algebra of Linear Iterations
6 Analysis of Positive Definite Iterations
7 Generation of Iterations
Part II SemiIterations and Krylov Methods
11 Multigrid Iterations
12 Domain Decomposition and Subspace Methods
13 HLU Iteration
14 Tensorbased Methods
Appendix A Facts from Linear Algebra
Appendix B Facts from Normed Spaces
Appendix C Facts from Matrix Theory
Appendix D Hierarchical Matrices
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algorithm applied approximation Assume block structure bound CG method coarse-grid computational condition conjugate gradient method convergence rate corresponding damped defined denoted discretisation domain decomposition method eigenvalues eigenvector equation equivalent error estimate Exercise factor finite element function Gauss–Seidel iteration gradient method grid Hackbusch Hence Hermitian holds implies index set inequality iteration matrix iterative method Jacobi iteration Lemma linear iterations LU decomposition M-matrix matrix norm minimisation multigrid iteration multigrid method normal form obtain optimal parameters Poisson model problem polynomial positive definite iterations positive definite matrix Proof proves recursion regular Remark representation respect Richardson iteration satisfy Schwarz iteration second normal form semi-iterative smoothing solution solving spectral spectral norm spectral radius SSOR statement subdomains subspace symmetric tensor Theorem third normal form two-grid vector xm+1 yields