Computing the R of the QR factorization of tall and skinny matrices using MPI_Reduce

A QR factorization of a tall and skinny matrix with n columns can be represented as a reduction. The operation used along the reduction tree has in input two n-by-n upper triangular matrices and in output an n-by-n upper triangular matrix which is defined as the R factor of the two input matrices stacked the one on top of the other. This operation is binary, associative, and commutative. We can therefore leverage the MPI library capabilities by using user-defined MPI operations and MPI_Reduce to perform this reduction. The resulting code is compact and portable. In this context, the user relies on the MPI library to select a reduction tree appropriate for the underlying architecture

The Problem with the Linpack Benchmark Matrix Generator

We characterize the matrix sizes for which the Linpack Benchmark matrix generator constructs a matrix with identical columns

Algorithmic Based Fault Tolerance Applied to High Performance Computing

We present a new approach to fault tolerance for High Performance Computing system. Our approach is based on a careful adaptation of the Algorithmic Based Fault Tolerance technique (Huang and Abraham, 1984) to the need of parallel distributed computation. We obtain a strongly scalable mechanism for fault tolerance. We can also detect and correct errors (bit-flip) on the fly of a computation. To assess the viability of our approach, we have developed a fault tolerant matrix-matrix multiplication subroutine and we propose some models to predict its running time. Our parallel fault-tolerant matrix-matrix multiplication scores 1.4 TFLOPS on 484 processors (cluster jacquard.nersc.gov) and returns a correct result while one process failure has happened. This represents 65% of the machine peak efficiency and less than 12% overhead with respect to the fastest failure-free implementation. We predict (and have observed) that, as we increase the processor count, the overhead of the fault tolerance drops significantly

Computing the Conditioning of the Components of a Linear Least Squares Solution

In this paper, we address the accuracy of the results for the overdetermined full rank linear least squares problem. We recall theoretical results obtained in Arioli, Baboulin and Gratton, SIMAX 29(2):413--433, 2007, on conditioning of the least squares solution and the components of the solution when the matrix perturbations are measured in Frobenius or spectral norms. Then we define computable estimates for these condition numbers and we interpret them in terms of statistical quantities. In particular, we show that, in the classical linear statistical model, the ratio of the variance of one component of the solution by the variance of the right-hand side is exactly the condition number of this solution component when perturbations on the right-hand side are considered. We also provide fragment codes using LAPACK routines to compute the variance-covariance matrix and the least squares conditioning and we give the corresponding computational cost. Finally we present a small historical numerical example that was used by Laplace in Theorie Analytique des Probabilites, 1820, for computing the mass of Jupiter and experiments from the space industry with real physical data

Tiled QR factorization algorithms

This work revisits existing algorithms for the QR factorization of rectangular matrices composed of p-by-q tiles, where p >= q. Within this framework, we study the critical paths and performance of algorithms such as Sameh and Kuck, Modi and Clarke, Greedy, and those found within PLASMA. Although neither Modi and Clarke nor Greedy is optimal, both are shown to be asymptotically optimal for all matrices of size p = q^2 f(q), where f is any function such that \lim_{+\infty} f= 0. This novel and important complexity result applies to all matrices where p and q are proportional, p = \lambda q, with \lambda >= 1, thereby encompassing many important situations in practice (least squares). We provide an extensive set of experiments that show the superiority of the new algorithms for tall matrices