6 summary and some conclusions the performance

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6. SUMMARY AND SOME CONCLUSIONS The performance results verify that our recursive approach is also very effi- cient for solving two-sided triangular matrix equations on today’s hierarchical memory computer systems. Despite the quite large flops penalties of the re- cursive blocked algorithms they outperform the standard algorithms for large enough problems. For example, solving discrete-time Sylvester and Lyapunov equations with coefficient matrices of size 2000 × 2000 takes approximately one hour using current routines in the SLICOT [2001] library, and the solu- tion times of our recursive blocked algorithms are less than one minute for the same problems. This is partly due to the difference in the data reference pat- terns of the algorithms. Our recursive blocked algorithms automatically match the memory hierarchy of a target machine and provide good data locality. As described in the Part I paper on one-sided matrix equations, we develop new high-performance superscalar kernels for solving the remaining small-sized triangular matrix equations and lightweight GEMM operations, which implies that a larger part of the total execution time is spent in high-performance GEMM operations. Also for the two-sided matrix equations we terminate the recursion with blks = 4 without degrading performance, which in turn means that the implementations are architecture-independent. Moreover, we have de- veloped optimized two-sided matrix product kernels that take any symmetry properties as well as any triangular or trapezoidal structure of the matrices into account. In order to maximize the performance of these two-sided matrix product operations (e.g., AXB T ), we make use of optimized level-3 BLAS, which by default require some temporary storage. Altogether, this leads to simpler and much faster algorithms for solving reduced as well as unreduced two-sided ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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434 I. Jonsson and B. K˚agstr¨om and generalized Sylvester and Lyapunov matrix equations and different associ- ated condition estimation problems. Our intention is to make these algorithms available in the SLICOT [2001] library. For additional references see Part I [Jonsson and K˚agstr¨om 2002]. The concept of recursion is very old. Our invention is new recursive blocked algorithms for efficient solution of triangular one-sided (Part I) and two- sided (Part II) matrix equations including new efficient matrix equation kernel solvers. Our work extends earlier work on level-3 BLAS and matrix factorizations. Furthermore, we want to emphasize that developing a novel set of algorithms for an interesting class of problems is a merit itself. However, in our opinion the real merit comes first when it is proved in practice that the software imple- mentations give a substantial performance boost without sacrificing accuracy for not too ill-conditioned problems. Indeed, both these merits are attained by our work presented in the Part I and II articles. Together they cover all com- mon linear matrix equations. Moreover, since the triangular matrix equations
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