Two sided Matrix Algorithms

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

• Notes
• 20

This preview shows pages 17–19. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

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.
This is the end of the preview. Sign up to access the rest of the document.
• Fall '07
• qgsdxjhf
• Algorithms, Matrices, Matrix Equations, recursive blocked algorithms, Generalized Sylvester

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern