A method for the e�cient computation of accu� rate reduced�order models of large linear circuits is de� scribed. The method � called MPVL � employs a novel block Lanczos algorithm to compute matrix Pad�e ap� proximations of matrix�valued network transfer func� tions. The reduced�order models � computed to the re� quired level of accuracy � are used tospeed up the anal� ysis of circuits containing large linear blocks. The lin� ear blocks are replaced by their reduced�order models� and the resulting smaller circuit can be analyzed with general�purpose simulators � with signi�cant savings in simulation time and � practically � no loss of accuracy. 1

Abstract. Given a graph with billions of nodes and edges, how can we find patterns and anomalies? Are there nodes that participate in too many or too few triangles? Are there close-knit near-cliques? These questions are expensive to answer unless we have the first several eigenvalues and eigenvectors of the graph adjacency matrix. However, eigensolvers suffer from subtle problems (e.g., convergence) for large sparse matrices, let alone for billion-scale ones. We address this problem with the proposed HEIGEN algorithm, which we carefully design to be accurate, efficient, and able to run on the highly scalable MAPRE-DUCE (HADOOP) environment. This enables HEIGEN to handle matrices more than 1000 × larger than those which can be analyzed by existing algorithms. We implement HEIGEN and run it on the M45 cluster, one of the top 50 supercomputers in the world. We report important discoveries about near-cliques and triangles on several real-world graphs, including a snapshot of the Twitter social network (38Gb, 2 billion edges) and the “YahooWeb ” dataset, one of the largest publicly available graphs (120Gb, 1.4 billion nodes, 6.6 billion edges). 1

Abstract. Dedicated to Professor Ake Bjorck in honor of his 60th birthday We examine two iterative methods for solving rectangular systems of linear equations: LSQR for over-determined systems Ax ~ b, and Craig's method for under-determined systems Ax = b. By including regularization, we extend Craig's method to incompatible systems, and observe that it solves the same damped least-squares problems as LSQR. The methods may therefore be compared on rectangular systems of arbitrary shape. Various methods for symmetric and unsymmetric systems are reviewed to illustrate the parallels. We see that the extension of Craig's method closes a gap in existing theory. However, LSQR is more economical on regularized problems and appears to be more reliable if the residual is not small. In passing, we analyze a scaled "augmented system " associated with regularized problems. A bound on the condition number suggests a promising direct method for sparse equations and least-squares problems, based on indefinite LDL T factors of the augmented matrix.