Abstract. As multicore systems continue to gain ground in the High Performance Computing world, linear algebra algorithms have to be reformulated or new algorithms have to be developed in order to take advantage of the architectural features on these new processors. Fine grain parallelism becomes a major requirement and introduces the necessity of loose synchronization in the parallel execution of an operation. This paper presents an algorithm for the Cholesky, LU and QR factorization where the operations can be represented as a sequence of small tasks that operate on square blocks of data. These tasks can be dynamically scheduled for execution based on the dependencies among them and on the availability of computational resources. This may result in an out of order execution of the tasks which will completely hide the presence of intrinsically sequential tasks in the factorization. Performance comparisons are presented with the LAPACK algorithms where parallelism can only be exploited at the level of the BLAS operations and vendor implementations. 1

We present Sequoia, a programming language designed to facilitate the development of memory hierarchy aware parallel programs that remain portable across modern machines featuring different memory hierarchy configurations. Sequoia abstractly exposes hierarchical memory in the programming model and provides language mechanisms to describe communication vertically through the machine and to localize computation to particular memory locations within it. We have implemented a complete programming system, including a compiler and runtime systems for Cell processor-based blade systems and distributed memory clusters, and demonstrate efficient performance running Sequoia programs on both of these platforms.

This paper presents a novel framework for the symbolic bounds analysis of pointers, array indices, and accessed memory regions. Our framework formulates each analysis problem as a system of inequality constraints between symbolic bound polynomials. It then reduces the constraint system to a linear program. The solution to the linear program provides symbolic lower and upper bounds for the values of pointer and array index variables and for the regions of memory that each statement and procedure accesses. This approach eliminates fundamental problems associated with applying standard xed-point approaches to symbolic analysis problems. Experimental results from our implemented compiler show that the analysis can solve several important problems, including static race detection, automatic parallelization, static detection of array bounds violations, elimination of array bounds checks, and reduction of the number of bits used to store computed values.

Matrix computations are both fundamental and ubiquitous in computational science and its vast application areas. Along with the development of more advanced computer systems with complex memory hierarchies, there is a continuing demand for new algorithms and library software that efficiently utilize and adapt to new architecture features. This article reviews and details some of the recent advances made by applying the paradigm of recursion to dense matrix computations on today’s memory-tiered computer systems. Recursion allows for efficient utilization of a memory hierarchy and generalizes existing fixed blocking by introducing automatic variable blocking that has the potential of matching every level of a deep memory hierarchy. Novel recursive blocked algorithms offer new ways to compute factorizations such as Cholesky and QR and to solve matrix equations. In fact, the whole gamut of existing dense linear algebra factorization is beginning to be reexamined in view of the recursive paradigm. Use of recursion has led to using new hybrid data structures and optimized superscalar kernels. The results we survey include new algorithms and library software implementations for level 3 kernels, matrix factorizations, and the solution of general systems of linear equations and several common matrix equations. The software implementations we survey are robust and show impressive performance on today’s high performance computing systems.

We present an approach for synthesizing transformations to enhance locality in imperfectly-nested loops. The key idea is to embed the iteration space of every statement in a loop nest into a special iteration space called the product space. The product space can be viewed as a perfectly-nested loop nest, so embedding generalizes techniques like code sinking and loop fusion that are used in ad hoc ways in current compilers to produce perfectly-nested loops from imperfectly-nested ones. In contrast to these ad hoc techniques however, our embeddings are chosen carefully to enhance locality. The product space is then transformed further to enhance locality, after which fully permutable loops are tiled, and code is generated. We evaluate the effectiveness of this approach for dense numerical linear algebra benchmarks, relaxation codes, and the tomcatv code from the SPEC benchmarks. 1. BACKGROUND AND PREVIOUSWORK Sophisticated algorithms based on polyhedral algebra have been developed for determining good sequences of linear loop transformations (permutation, skewing, reversal and scaling) for enhancing locality in perfectly-nested loops 1. Highlights of this technology are the following. The iterations of the loop nest are modeled as points in an integer lattice, and linear loop transformations are modeled as nonsingular matrices mapping one lattice to another. A sequence of loop transformations is modeled by the product of matrices representing the individual transformations; since the set of nonsingular matrices is closed under matrix product, this means that a sequence of linear loop transformations can be represented by a nonsingular matrix. The problem of finding an optimal sequence of linear loop transformations is thus reduced to the problem of finding an integer matrix that satisfies some desired property, permitting the full machinery of matrix methods and lattice theory to ¢ This work was supported by NSF grants CCR-9720211, EIA-9726388, ACI-9870687,EIA-9972853. £ A perfectly-nested loop is a set of loops in which all assignment statements are contained in the innermost loop.

This paper surveys algorithms that efficiently solve linear equations or compute eigenvalues even when the matrices involved are too large to fit in the main memory of the computer and must be stored on disks. The paper focuses on scheduling techniques that result in mostly sequential data accesses and in data reuse, and on techniques for transforming algorithms that cannot be effectively scheduled. The survey covers out-of-core algorithms for solving dense systems of linear equations, for the direct and iterative solution of sparse systems, for computing eigenvalues, for fast Fourier transforms, and for N-body computations. The paper also discusses reasonable assumptions on memory size, approaches for the analysis of out-of-core algorithms, and relationships between out-of-core, cache-aware, and parallel algorithms.

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Class Figure 5 Simple Array construction operations //Simple 3 x 3 array of integers intArray2D A = new intArray2D(3,3); //This new array has a copy of the data in A, //and the same rank and shape.

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The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional column-major or row-major array layouts to incur high variability in memory system performance as matrix size varies. This paper investigates the use of recursive array layouts to improve performance and reduce variability. Previous work on recursive matrix multiplication is extended to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more complex algorithms of Strassen and Winograd. While recursive layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.2--2.5) for the standard algorithm, they offer little improvement for Strassen's and Winograd's algorithms. For a purely sequential implementation, it is possible to reorder computation to conserve memory space and improve performance between ...