8 597 x 750 1871 682 x 1000 4289 672 x 50

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41.8 59.7 17.3 2.9 2.41 X 750 187.1 68.2 61.1 2.7 3.06 X 1000 428.9 67.2 144.1 2.5 2.98 X 50 0.120 85.4 0.0285 39.1 4.19 X + Sep 100 0.741 83.5 0.166 26.2 4.47 X + Sep 250 10.3 82.1 2.69 31.0 3.83 X + Sep 500 161.3 89.5 20.0 15.3 8.06 X + Sep 750 807.7 92.6 67.9 12.1 11.89 X + Sep 1000 1857.4 92.4 174.0 18.9 10.68 X + Sep For illustration, we first use the SLICOT routine SG03AD which solves unre- duced generalized discrete-time Lyapunov equations. SG03AD also has an op- tion to compute an estimate of the separation Sep[GLYDT] (see Section 2). In Table V(a), timings for the SG03AD routine are displayed for problem sizes ranging from 50 to 1000 using two different triangular matrix equation solvers. These solvers are SG03AX provided in SLICOT, which implements a variant of the Bartels–Stewart method calling BLAS [Penzl 1998], and our recursive blocked rtrglydt algorithm. In the second column, the total times for solving an unreduced system with SG03AX as the triangular solver are displayed. This includes the time for the generalized Schur factorization and backtransformation of the solution. In the fourth and fifth columns, similar results are displayed when SG03AX is replaced by the rtrglydt routine. We see up to 90% speedup for the problem sizes considered. The numbers in the lower part of Table V(a) also include the time for computing a 1-norm-based estimate for Sep[GLYDT]. The condition estimation process includes repeated ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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Recursive Blocked Algorithms—Part II 433 calls (typically five) to the generalized triangular solver, and as expected we see a four- to fivefold speedup when using our recursive solver. In Table V(b), the corresponding results are displayed when we have replaced the LAPACK routines DGGHRD and DHGEQZ by Dackland–K˚agstr¨om’s blocked Hessenberg-triangular reduction and Q Z algorithms [Dackland and K˚agstr¨om 1999] for transforming the regular pair ( A , E ) to generalized Schur form. For large enough problems, this gives another factor of two speedup. We have also compared the routine SYLG [Gardiner et al. 1992a,b] which implements a variant of the Hessenberg–Schur method [Golub et al. 1979] for solving unreduced generalized Sylvester equations (1). Also for this ma- trix equation, we see a substantial impact in using our recursive blocked algo- rithm (almost a factor 3 speedup for N = 1000). For detailed results we refer to Jonsson and K˚agstr¨om [2001]. We remark that we only see small benefits (or no benefits at all) in using our recursive algorithm for small problem sizes or when M = 10 N . This result is not due to the properties of the recursive solvers, but rather to the properties of the Hessenberg–Schur method, where only one of the matrices of A and D is reduced to Schur form, and the other is reduced to Hessenberg form. The Hessenberg–Schur algorithm is best suited for cases when M N or M ¿ N , when only the smaller of the matrix pairs is reduced to upper quasitriangular Schur form.
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