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The BR algorithm, a new method for calculating the eigenvalues of an upper Hes- senberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrow-band, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30–60 in computing time and a factor of over 100 in matrix storage space. [ABSTRACT FROM AUTHOR]

In this paper we describe methods for mitigating the degradation in performance caused by high latencies in parallel and distributed networks. For example, given any "dataflow" type of algorithm that runs in T steps on an n-node ring with unit link delays, we show how to run the algorithm in O(T) steps on any n-node bounded-degree connected network with average link delay O(1). This is a significant improvement over prior approaches to latency hiding, which require slowdowns proportional to the maximum link delay. In the case when the network has average link delay dave, our simulation runs in O(√daveT) steps using n/√dave processors, thereby preserving efficiency. We also show how to efficiently simulate an n × n array with unit link delays using slowdown Õ (d&frac23ave) on a two-dimensional array with average link delay dave. Last, we present results for the case in which large local databases are involved in the computation. [ABSTRACT FROM AUTHOR]

Tree-structured computations are relatively easy to process in parallel. As leaf processes are recursively spawned they can be assigned to independent processors in a multicomputer network. However, to achieve good performance the on-line mapping algorithm must maintain load balance, i.e., distribute processes equitably among processors. Additionally, the algorithm itself must be distributed in nature, and process allocation must be completed via message-passing with minimal communication overhead. This paper investigates bounds on the performance of deterministic and randomized algorithms for on-line tree embeddings. In particular, we study trade-offs between computation overhead (load imbalance) and communication overhead (message congestion). We give a simple technique to derive lower bounds on the congestion that any on-line allocation algorithm must incur in order to guarantee load balance. This technique works for both randomized and deterministic algorithms. We prove that the advantage of randomization is limited. Optimal bounds are achieved for several networks, including multidimensional grids and butterflies. [ABSTRACT FROM AUTHOR]

In this paper, we study the additive Schwarz method for the h-p version of the finite element method in three dimensions. The main idea is to treat separately the h-version (linear) components and the p-version (high-order) components by a vertex-based method. It can also be viewed as a three-level method with the level being the linear finite element approximation on the coarse mesh, the linear finite element approximation on the fine mesh, and the high-order finite element approximation on the fine mesh, respectively. The resulting algorithm can be implemented in parallel on the subdomain level for the h-version components and on the element level for the p-version components. The condition number is of order [This symbol cannot be presented in ASCII format](1 + ln Hipi/hi)⇔2, where Hi stands for the diameter of the subdomain Ωi, hi is the diameter of the elements in Ωi, and pi is the maximum of the polynomial degrees used in Ωi. [ABSTRACT FROM AUTHOR]

In this paper the problem of computing bifurcation diagrams for large-scale nonlinear parameter-dependent steady state systems which arise following the spatial discretization of semilinear PDEs is investigated. A continuation algorithm which employs a preconditioned version of the recursive projection method (RPM) is presented. The RPM is often expensive when it is used in conjunction with the numerical method of lines. Preconditioning the Jacobian of the underlying fixed point operator results in an algorithm (the preconditioned recursive projection method (PRPM)) which is capable of efficiently computing equilibrium solution diagrams of large stiff systems. For many PDE problems the PRPM is a fast and effective means of detecting both steady state and Hopf bifurcation along a branch of solutions. A description of the performance of the PRPM when applied to two numerical examples is given. [ABSTRACT FROM AUTHOR]