No abstract

I am looking for suggestions for a research project in applied/numerical linear algebra. As far as requirements, there really aren't any except Applied Numerical Linear Algebra-James W. Demmel.djvu Linear Equation Solving. Estimating Condition Numbers. To compute a practical error bound based on a bound like 2.5, we need to estimate A?1. This is Numerical Linear Algebra Numerical linear algebra

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We i n v estigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. To perform most of the numerical computation in dense matrix kernels, we i n troduce the notion of unsymmetric supernodes. To better exploit the memory hierarchy, w e i n troduce unsymmetric supernode-panel updates and two-dimensional data partitioning. To speed up symbolic factorization, we use Gilbert and Peierls's depth-rst search with Eisenstat and Liu's symmetric structural reductions. We h a v e implemented a sparse LU code using all these ideas. We present experiments demonstrating that it is signi cantly faster than earlier partial pivoting codes. We also compare performance with Umfpack, which uses a multifrontal approach; our code is usually faster.Keywords: sparse matrix algorithms; unsymmetric linear systems; supernodes; column elimination tree; partial pivoting. AMSMOS subject classi cations: 65F05, 65F50. Computing Reviews descriptors: G.1.3 Numerical Analysis : Numerical Linear Algebra | Linear systems direct and iterative methods, Sparse and very large systems.

We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR. Our first algorithm, Tall Skinny QR (TSQR), factors m × n matrices in a one-dimensional (1-D) block cyclic row layout, and is optimized for m n. Our second algorithm, CAQR (Communication-Avoiding QR), factors general rectangular matrices distributed in a two-dimensional block cyclic layout. It invokes TSQR for each block column factorization.The new algorithms are superior in both theory and practice. We have extended known lower bounds on communication for sequential and parallel matrix multiplication to provide latency lower bounds, and show these bounds apply to the LU and QR decompositions. We not only show that our QR algorithms attain these lower bounds (up to polylogarithmic factors), but that existing LAPACK and ScaLAPACK algorithms perform asymptotically more communication. We also point out recent LU algorithms in the literature that attain at least some of these lower bounds.Both TSQR and CAQR have asymptotically lower latency cost in the parallel case, and asymptotically lower latency and bandwidth costs in the sequential case. In practice, we have implemented parallel TSQR on several machines, with speedups of up to 6.7× on 16 processors of a Pentium III cluster, and up to 4× on 32 processors of a BlueGene/L. We have also implemented sequential TSQR on a laptop for matrices that do not fit in DRAM, so that slow memory is disk. Our out-of-DRAM implementation was as little as 2× slower than the predicted runtime as though DRAM were infinite.We have also modeled the performance of our parallel CAQR algorithm, yielding predicted speedups over ScaLAPACK's PDGEQRF of up to 9.7× on an IBM Power5, up to 22.9× on a model Petascale machine, and up to 5.3× on a model of the Grid.