Algorithms for extending arithmetic precision through compensated summation or arithmetics like double-double rely on operations commonly called twoSum and twoProduct. The current draft of the IEEE 754 standard specifies these operations under the names augmentedAddition and augmentedMultiplication. These operations were included after three decades of experience because of a motivating new use: bitwise reproducible arithmetic. Standardizing the operations provides a hardware acceleration target that can provide at least a 33 % speed improvements in reproducible dot product, placing reproducible dot product almost within a factor of two of common dot product. This paper provides history and motivation for standardizing these operations. We also define the operations, explain the rationale for all the specific choices, and provide parameterized test cases for new boundary behaviors.

Applications in computer network security, social media analysis, health care, and other areas rely on analyzing a changing environment. The data is rich in relationships and lends itself to graph analysis. Traditional static graph analysis cannot keep pace with network security applications analyzing nearly one million events per second. Streaming frameworks like STINGER support ingesting up three million of edge changes per second but there are few streaming analysis kernels that keep up with these rates. We present a new computational model along with promising novel architectures to tackle massive data rates.

Applications in computer network security, social media analysis, and other areas rely on analyzing a changing environment. The data is rich in relationships and lends itself to graph analysis. Traditional static graph analysis cannot keep pace with network security applications analyzing nearly one million events per second and social networks like Facebook collecting 500 thousand comments per second. Streaming frameworks like STINGER support ingesting up three million of edge changes per second but there are few streaming analysis kernels that keep up with these rates. Here we introduce a new, non-stop model and use it to decouple the analysis from the data ingest.

The digital world has given rise to massive quantities of data that include rich semantic and complex networks. A social graph, for example, containing hundreds of millions of actors and tens of billions of relationships is not uncommon. Analyzing these large data sets, even to answer simple analytic queries, often pushes the limits of algorithms and machine architectures. We present GraphCT, a scalable framework for graph analysis using parallel and multithreaded algorithms on shared memory platforms. Utilizing the unique characteristics of the Cray XMT, GraphCT enables fast network analysis at unprecedented scales on a variety of input data sets. On a synthetic power law graph with 2 billion vertices and 17 billion edges, we can find the connected components in 2 minutes. We can estimate the betweenness centrality of a similar graph with 537 million vertices and over 8 billion edges in under 1 hour. GraphCT is built for portability and performance.

Analyzing static snapshots of massive, graph-structured data cannot keep pace with the growth of social networks, financial transactions, and other valuable data sources. We introduce a framework, STING (Spatio-Temporal Interaction Networks and Graphs), and evaluate its performance on multicore, multisocket Intel®-based platforms. STING achieves rates of around 100 000 edge updates per second on large, dynamic graphs with a single, general data structure. We achieve speedups of up to 1000× over parallel static computation, improve monitoring a dynamic graph's connected components, and show an exact algorithm for maintaining local clustering coefficients performs better on Intel-based platforms than our earlier approximate algorithm.

Social networks produce an enormous quantity of data. Facebook consists of over 400 million active users sharing over 5 billion pieces of information each month. Analyzing this vast quantity of unstructured data presents challenges for software and hardware. We present GraphCT, a Graph Characterization Toolkit for massive graphs representing social network data. On a 128-processor Cray XMT, GraphCT estimates the betweenness centrality of an artificially generated (R-MAT) 537 million vertex, 8.6 billion edge graph in 55 minutes and a real-world graph (Kwak, et al.) with 61.6 million vertices and 1.47 billion edges in 105 minutes. We use GraphCT to analyze public data from Twitter, a microblogging network. Twitter's message connections appear primarily tree-structured as a news dissemination system. Within the public data, however, are clusters of conversations. Using GraphCT, we can rank actors within these conversations and help analysts focus attention on a much smaller data subset.

We present the algorithm, error bounds, and numerical results for extra-precise iterative refinement applied to overdetermined linear least squares (LLS) problems. We apply our linear system refinement algorithm to Björck’s augmented linear system formulation of an LLS problem. Our algorithm reduces the forward normwise and componentwise errors to O ( ε w ), where ε w is the working precision, unless the system is too ill conditioned. In contrast to linear systems, we provide two separate error bounds for the solution x and the residual r . The refinement algorithm requires only limited use of extra precision and adds only O ( mn ) work to the O ( mn 2 ) cost of QR factorization for problems of size m -by- n . The extra precision calculation is facilitated by the new extended-precision BLAS standard in a portable way, and the refinement algorithm will be included in a future release of LAPACK and can be extended to the other types of least squares problems.

We present the design and testing of an algorithm for iterative refinement of the solution of linear equations where the residual is computed with extra precision. This algorithm was originally proposed in 1948 and analyzed in the 1960s as a means to compute very accurate solutions to all but the most ill-conditioned linear systems. However, two obstacles have until now prevented its adoption in standard subroutine libraries like LAPACK: (1) There was no standard way to access the higher precision arithmetic needed to compute residuals, and (2) it was unclear how to compute a reliable error bound for the computed solution. The completion of the new BLAS Technical Forum Standard has essentially removed the first obstacle. To overcome the second obstacle, we show how the application of iterative refinement can be used to compute an error bound in any norm at small cost and use this to compute both an error bound in the usual infinity norm, and a componentwise relative error bound.We report extensive test results on over 6.2 million matrices of dimensions 5, 10, 100, and 1000. As long as a normwise (componentwise) condition number computed by the algorithm is less than 1/max{10,√n}εw, the computed normwise (componentwise) error bound is at most 2 max{10,√n} · εw, and indeed bounds the true error. Here,nis the matrix dimension and εw= 2-24is the working precision. Residuals were computed in double precision (53 bits of precision). In other words, the algorithm always computed a tiny error at negligible extra cost for most linear systems. For worse conditioned problems (which we can detect using condition estimation), we obtained small correct error bounds in over 90% of cases.