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Exercise 1 (4 points) Let A be a matrix with the property that columns 1, 3, 5, 7,. .. are orthogonal to columns 2, 4, 6, 8,. .. In a reduced QR factorization A = ˆ Q ˆ R, what special structure doesˆR possess? You may assume that A has full rank.

We design a general mathematical framework to analyze the properties of nearest neighbor balancing algorithms of the diiusion type. Within this framework we develop a new optimal polynomial scheme (OPS) which we show to terminate within a nite number m of steps, where m only depends on the graph and not on the initial load distribution. We show that all… (More)

Tikhonov–Phillips regularization is one of the best-known regularization methods for inverse problems. A posteriori criteria for determining the regularization parameter α require solving (*) (A * A + αI)x = A * y δ for different values of α. We investigate two methods for accelerating the standard cg-algorithm for solving the family of systems (*). The… (More)

- Andreas Frommer, Bertold Nöckel, Stephan Güsken, Thomas Lippert, Klaus Schilling
- 1995

The computational effort in the calculation of Wilson fermion quark propagators in Lattice Quantum Chromodynamics can be considerably reduced by exploiting the Wilson fermion matrix structure in inversion algorithms based on the non-symmetric Lanczos process. We consider two such methods: QMR (quasi minimal residual) and BCG (biconju-gate gradients). Based… (More)

Convergence results for the restrictive additive Schwarz (RAS) method of Cai and Sarkis [SIAM J. Sci. Comput., 21 (1999), pp. 792–797] for the solution of linear systems of the form Ax = b are provided using an algebraic view of additive Schwarz methods and the theory of multisplittings. The linear systems studied are usually discretizations of partial… (More)

The convergence of multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular M-matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of… (More)

Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M-matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem… (More)

- Numerische Mathematik, Andreas Frommer, Daniel B Szyld
- 1994

Parallel block two-stage iterative methods for the solution of linear systems of algebraic equations are studied. Convergence is shown for monotone matrices and for H-matrices. Two diierent asynchronous versions of these methods are considered and their convergence investigated.