Most of the results presented should be quite self

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Most of the results presented should be quite self-explanatory. Therefore, our ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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Recursive Blocked Algorithms—Part II 429 Table III. Performance Results for Triangular Generalized Sylvester Equation: IBM Power3 (Left) and Intel Pentium III (Right) a IBM Power3 Intel Pentium III Time (sec) Speedup Time (sec) Speedup M = N A B/A C/A A B/A C/A 100 1.30e - 02 0.96 1.16 4.30e - 02 0.98 1.26 250 1.31e - 01 0.48 1.26 4.49e - 01 0.96 1.34 500 8.91e - 01 0.67 1.50 3.07e + 00 0.96 1.38 1000 6.42e + 00 1.05 1.91 2.18e + 01 1.05 1.51 1500 2.13e + 01 1.34 2.09 7.10e + 01 1.09 1.54 2000 4.77e + 01 1.48 2.16 1.60e + 02 1.14 1.59 A – rtrgsyl B – A + linking with SMP BLAS C – B + utilizing explicit // in the recursion tree a Labels A–C represent different algorithms and implementations. discussion is restricted to significant or unexpected differences between the implementations executing on different computing platforms. For additional performance results see Jonsson and K˚agstr¨om [2001]. The accuracy of the re- sults computed by our recursive blocked algorithms are overall very good and similar to the accuracy obtained by the corresponding SLICOT routines. For benchmark problems see Kressner et al. [1999a,b] and SLICOT [2001]. 5.1 Triangular Generalized Sylvester Equation In Table III, performance results for different algorithms and implementations, executing on IBM Power3 and Intel Pentium III processor-based systems, for solving the triangular generalized Sylvester equation are displayed. We have not found any library or public software for solving triangular generalized Sylvester equations, so the results presented here are for our recursive blocked algorithms only. 5.2 Triangular Discrete-Time Sylvester Equation In Figure 1, we show performance graphs for different algorithms and imple- mentations, executing on IBM Power3 and Intel Pentium III processor-based systems, for solving triangular discrete-time Sylvester equations. The SLICOT SB04PY implements an explicit Bartels–Stewart solver and is mainly a level-2 routine, which explains its poor performance behavior. Our recursive blocked rtrsydt shows between a 2-fold and a 118-fold speedup with respect to SLICOT SB04PY and an additional speedup up to 2.14 on a four- processor Power3 node for large enough problems. 5.3 Triangular Discrete-Time Lyapunov Equation In Figure 2, performance results for the triangular discrete-time Lyapunov equation are presented, now using IBM PowerPC 604e and SGI MIPS R10000 processor-based systems. We compare our recursive blocked rtrlydt algorithm with the SLICOT SB03MX routine, which mainly is a level-2 implementation ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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430 I. Jonsson and B. K˚agstr¨om Fig. 1. Performance results for the triangular discrete-time Sylvester equation ( M = N ): IBM Power3 (left) and Intel Pentium III (right). as well. As expected, the relative behavior between the different algorithms and implementations follows qualitatively that of the discrete-time Sylvester equation. We remark that the speedup of rtrlydt with respect to the SLICOT SB03MX is between 1.9 and 76 and an additional speedup of 2.6 on a four- processor IBM PowerPC 604e node for large enough problems. The results for
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  • Fall '07
  • qgsdxjhf
  • Algorithms, Matrices, Matrix Equations, recursive blocked algorithms, Generalized Sylvester

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