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.

Triangular matrix equations appear naturally in estimating the condition numbers of matrix equations and different eigenspace computations, including block-diagonalization of matrices and matrix pairs and computation of functions of matrices. To solve a triangular matrix equation is also a major step in the classical Bartels--Stewart method for solving the standard continuous-time Sylvester equation ( AX − XB = C ). We present novel recursive blocked algorithms for solving one-sided triangular matrix equations, including the continuous-time Sylvester and Lyapunov equations, and a generalized coupled Sylvester equation. The main parts of the computations are performed as level-3 general matrix multiply and add (GEMM) operations. In contrast to explicit standard blocking techniques, our recursive approach leads to an automatic variable blocking that has the potential of matching the memory hierarchies of today's HPC systems. Different implementation issues are discussed, including when to terminate the recursion, the design of new optimized superscalar kernels for solving leaf-node triangular matrix equations efficiently, and how parallelism is utilized in our implementations. Uniprocessor and SMP parallel performance results of our recursive blocked algorithms and corresponding routines in the state-of-the-art libraries LAPACK and SLICOT are presented. The performance improvements of our recursive algorithms are remarkable, including 10-fold speedups compared to standard algorithms.

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We continue our study of high-performance algorithms for solving triangular matrix equations. They appear naturally in different condition estimation problems for matrix equations and various eigenspace computations, and as reduced systems in standard algorithms. Building on our successful recursive approach applied to one-sided matrix equations (Part I), we now present novel recursive blocked algorithms for two-sided matrix equations, which include matrix product terms such as AX B T . Examples are the discrete-time standard and generalized Sylvester and Lyapunov equations. The means for achieving high performance is the recursive variable blocking, which has the potential of matching the memory hierarchies of today's high-performance computing systems, and level-3 computations which mainly are performed as GEMM operations. Different implementation issues are discussed, including the design of efficient new algorithms for two-sided matrix products. We present uniprocessor and SMP parallel performance results of recursive blocked algorithms and routines in the state-of-the-art SLICOT library. Although our recursive algorithms with optimized kernels for the two-sided matrix equations perform more operations, the performance improvements are remarkable, including 10-fold speedups or more, compared to standard algorithms.

This paper experimentally addresses the impact of surface roughness on losses and secondary flow in a Turbine Rear Structure (TRS). Experiments were performed in the Chalmers LPT-OGV facility, at an engine representative Reynolds number with a realistic shrouded rotating low-pressure turbine (LPT). Outlet Guide Vanes (OGV) were manufactured to achieve three different surface roughnesses tested at two Reynolds numbers, Re = 235000 and Re = 465000. The experiments were performed at on-design inlet swirl conditions. The inlet and outlet flow of the TRS were measured in 2D planes with a 5-hole probe and 7-hole probe accordingly. The static pressure distributions on the OGVs were measured and boundary layer studies were performed at the OGV midspan on the suction side with a time-resolved total pressure probe. Turbulence decay was measured within the TRS with a single hot-wire. The results showed a surprisingly significant increase in the losses for the high level of surface roughness (25–30 Ra) of the OGVs and Re = 465000. The increased losses were primary revealed as a result of the flow separation on the OGV suction side near the hub. The loss increase was seen but was less substantial for the intermediate roughness case (4–8 Ra). Experimental results presented in this work provide support for the further development of more advanced TRS and data for the validation of new CFD prediction methods for TRS.