We consider the problem of parallelizing restarted backtrack search. With few notable exceptions, most commercial and academic constraint programming solvers do not learn no-goods during search. Depending on the branching heuristics used, this means that there are little to no side-effects between restarts, making them an excellent target for parallelization. We develop a simple technique for p...

The restarted generalized minimum residual method (RGMRES) (Saad and Schultz (1986), Barrett et al. (1994), Dias da Cunha and Hopkins (1994)) starts from the residual r0 1⁄4 b Ax0, where x0 is an initial estimate for the solution (often x0 1⁄4 0). An orthogonal basis for the Krylov subspace spanfAr0g, for k 1⁄4 0; 1; 2; . . ., is generated explicitly: this is referred to as Arnoldi’s method (Ar...

A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular RayleighRitz versions are possible. For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine...

We investigate an acceleration technique for restarted Krylov subspace methods for computing the action of a function of a large sparse matrix on a vector. Its effect is to ultimately deflate a specific invariant subspace of the matrix which most impedes the convergence of the restarted approximation process. An approximation to the subspace to be deflated is successively refined in the course ...

Recent statistical performance surveys of search algorithms in difficult combinatorial problems have demonstrated the benefits of randomising and restarting the search procedure. Specifically, it has been found that if the search cost distribution (SCD) of the non-restarted randomised search exhibits a slower-than-exponential decay (that is, a “heavy tail”), restarts can reduce the search cost ...

The Davidson method is a popular preconditioned variant of the Arnoldi method for solving large eigenvalue problems. For theoretical, as well as practical reasons the two methods are often used with restarting. Frequently, information is saved through approximated eigen-vectors to compensate for the convergence impairment caused by restarting. We call this scheme of retaining more eigenvectors ...