C 2002 acm 0098 3500021200 0416 500 acm transactions

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
C 2002 ACM 0098-3500/02/1200-0416 $5.00 ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002, Pages 416–435.
Image of page 1

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

View Full Document Right Arrow Icon
Recursive Blocked Algorithms—Part II 417 1. INTRODUCTION We continue the presentation of recursive blocked algorithms for solving var- ious types of triangular matrix equations. Our goal is to design efficient algo- rithms for today’s HPC systems with multilevel memory hierarchies. The hier- archical recursive blocking promotes good data locality and combined with our highly optimized superscalar kernels, we obtain a notable boost in performance compared to existing algorithms as implemented in state-of-the-art libraries [Anderson et al. 1999; SLICOT 2001]. In Part I [Jonsson and K˚agstr¨om 2002], we introduced and discussed new recursive blocked algorithms for one-sided Sylvester-type equations, including the continuous-time standard Sylvester and Lyapunov equations, and a generalized coupled Sylvester equation. In this contribution (Part II), we introduce and discuss new recursive blocked algorithms for two-sided matrix equations, which include matrix product terms of type op( A ) X op( B ), where op( Y ) can be Y or its transpose Y T . Examples include discrete-time standard and generalized Sylvester and Lyapunov equa- tions. The standard methods for solving two-sided matrix equations are also based on the Bartels–Stewart method [Bartels and Stewart 1972]. As in Part I, we focus on the solution of the two-sided triangular counterparts, which typi- cally are obtained after an initial transformation of matrices (or regular matrix pairs) to Schur (or generalized Schur) form [Anderson et al. 1999; Dackland and K˚agstr¨om 1999]. Before we go into any further details, we outline the contents of the rest of the article. In Section 2, we introduce Sep-functions associated with the two- sided matrix equations and reestablish the relationship between the solution of triangular matrix equations and condition estimation. Section 3 introduces our recursive blocked algorithms for two-sided matrix equations, including the gen- eralized Sylvester equation (Section 3.1), the discrete-time Sylvester equation (Section 3.2), the generalized and standard discrete-time Lyapunov equations (Sections 3.3 and 3.4), and finally the generalized continuous-time Lyapunov equation (Section 3.5). In Section 4, we revisit our discussion about implemen- tation issues, now focusing on the design of optimized two-sided matrix product kernels. Sample performance results of our recursive blocked algorithms are presented and discussed in Section 5. Finally, we give some concluding remarks in Section 6. 2. CONDITION ESTIMATION OF MATRIX EQUATIONS REVISITED The two-sided matrix equations can also be written as a linear system of equa- tions Zx = c , where Z is a Kronecker product matrix representation of the associated matrix equation operator. The solution x and the right-hand side c are represented in vec( · ) notation, where vec( X ) denotes a column vector with the columns of X stacked on top of each other.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern