We analyze the convergence of randomized trace estimators. Starting at 1989, several algorithms have been proposed for estimating the trace of a matrix by 1. These algorithms are useful in applications in which there is no explicit representation of A but rather an efficient method compute z T Az given z. Existing results only analyze the variance of the different estimators. In contrast, we analyze the number of samples M required to guarantee that with probability at least 1 − δ, the relative error in the estimate is at most . We argue that such bounds are much more useful in applications than the variance. We found that these bounds rank the estimators differently than the variance; this suggests that minimum-variance estimators may not be the best.We also make two additional contributions to this area. The first is a specialized bound for projection matrices, whose trace (rank) needs to be computed in electronic structure calculations. The second is a new estimator that uses less randomness than all the existing estimators.

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Flash memory is a type of electrically-erasable programmable read-only memory (EEPROM). Because flash memories are nonvolatile and relatively dense, they are now used to store files and other persistent objects in handheld computers, mobile phones, digital cameras, portable music players, and many other computer systems in which magnetic disks are inappropriate. Flash, like earlier EEPROM devices, suffers from two limitations. First, bits can only be cleared by erasing a large block of memory. Second, each block can only sustain a limited number of erasures, after which it can no longer reliably store data. Due to these limitations, sophisticated data structures and algorithms are required to effectively use flash memories. These algorithms and data structures support efficient not-in-place updates of data, reduce the number of erasures, and level the wear of the blocks in the device. This survey presents these algorithms and data structures, many of which have only been described in patents until now.

Several innovative random-sampling and random-mixing techniques for solving problems in linear algebra have been proposed in the last decade, but they have not yet made a significant impact on numerical linear algebra. We show that by using a high-quality implementation of one of these techniques, we obtain a solver that performs extremely well in the traditional yardsticks of numerical linear algebra: it is significantly faster than high-performance implementations of existing state-of-the-art algorithms, and it is numerically backward stable. More specifically, we describe a least-squares solver for dense highly overdetermined systems that achieves residuals similar to those of direct QR factorization-based solvers (lapack), outperforms lapack by large factors, and scales significantly better than any QR-based solver.

Understanding animal movement is essential to elucidate how animals interact, survive, and thrive in a changing world. Recent technological advances in data collection and management have transformed our understanding of animal “movement ecology” (the integrated study of organismal movement), creating a big-data discipline that benefits from rapid, cost-effective generation of large amounts of data on movements of animals in the wild. These high-throughput wildlife tracking systems now allow more thorough investigation of variation among individuals and species across space and time, the nature of biological interactions, and behavioral responses to the environment. Movement ecology is rapidly expanding scientific frontiers through large interdisciplinary and collaborative frameworks, providing improved opportunities for conservation and insights into the movements of wild animals, and their causes and consequences.

Seven decades of research on the “cognitive map,” the allocentric representation of space, have yielded key neurobiological insights, yet field evidence from free-ranging wild animals is still lacking. Using a system capable of tracking dozens of animals simultaneously at high accuracy and resolution, we assembled a large dataset of 172 foraging Egyptian fruit bats comprising >18 million localizations collected over 3449 bat-nights across 4 years. Detailed track analysis, combined with translocation experiments and exhaustive mapping of fruit trees, revealed that wild bats seldom exhibit random search but instead repeatedly forage in goal-directed, long, and straight flights that include frequent shortcuts. Alternative, non–map-based strategies were ruled out by simulations, time-lag embedding, and other trajectory analyses. Our results are consistent with expectations from cognitive map–like navigation and support previous neurobiological evidence from captive bats.

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Sparse matrix-vector multiplication is an important kernel that often runs ine ciently on superscalar RISC processors. This paper describes techniques that increase instruction-level parallelism and improve performance. The techniques include reordering to reduce cache misses originally due to Das et al., blocking to reduce load instructions, and prefetching to prevent multiple load-store units from stalling simultaneously. The techniques improve performance from about 40 M ops (on a well-ordered matrix) to over 100 M ops on a 266 M ops machine.