We introduce the following z matrices z sydt z axb t

Info icon This preview shows pages 2–5. Sign up to view the full content.

View Full Document Right Arrow Icon
We introduce the following Z -matrices. Z SYDT = Z AXB T - X = B A - I n I m , Z LYDT = Z AXA T - X = A A - I n I n , Z GSYL = Z AXB T - CXD T = B A - D C , ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
Image of page 2

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

View Full Document Right Arrow Icon
418 I. Jonsson and B. K˚agstr¨om Z GLYCT = Z AXE T + EXA T = E A + A E , Z GLYDT = Z AXA T - EXE T = A A - E E , where from top to bottom they represent the matrix operators of the discrete-time Sylvester and Lyapunov equations, the generalized (discrete/ continuous-time) Sylvester equation, and the generalized continuous-time and discrete-time Lyapunov equations. Similar techniques to those described in the Part I article [Jonsson and K˚agstr¨om 2002] can be used for estimating the conditioning of two-sided ma- trix equations. This leads to solving triangular two-sided matrix equations for estimating the Sep-function: Sep[ · ] = k Z - 1 k - 1 2 = σ min ( Z ), where Z is any of the Kronecker product matrices above. The Sep-functions associated with the two-sided matrix equations considered are: Sep[SYDT] = inf k X k F = 1 k AXB T - X k F = σ min ( Z SYDT ), Sep[LYDT] = inf k X k F = 1 k AXA T - X k F = σ min ( Z LYDT ), Sep[GSYL] = inf k X k F = 1 k AXB T - CXD T k F = σ min ( Z GSYL ), Sep[GLYCT] = inf k X k F = 1 k AXE T - EX ( - A T ) k F = σ min ( Z GLYCT ), Sep[GLYDT] = inf k X k F = 1 k AXA T - EXE T k F = σ min ( Z GLYDT ) . One-norm Sep - 1 -estimators based on LAPACK techniques [Hager 1984; Higham 1988; Anderson et al. 1999] are implemented in the SLICOT [2001] library. Each estimator involves solving several (around five) triangular ma- trix equations. Therefore, it is important that these triangular matrix equation problems can be solved efficiently on today’s memory tiered systems. The underlying perturbation theory for standard and generalized matrix equations is presented in Higham [1993] and K˚agstr¨om [1994], respectively. 3. RECURSIVE BLOCKED ALGORITHMS FOR TWO-SIDED MATRIX EQUATIONS There is a quite extensive literature on the solution of matrix equations, and we refer to the selection of fundamental papers by Bartels and Stewart [1972], Golub et al. [1979], Hammarling [1982], Chu [1987], K˚agstr¨om and Westin [1989], Gardiner et al. [1992a], K˚agstr¨om and Poromaa [1996], and Penzl [1998] that all present reliable algorithms. Our recursive approach also applies to triangular two-sided matrix equations, which is the topic of this section. Some of the matrix equations considered can be seen as special cases of other formulations. However, this is mainly of theoretical interest, since either these equivalences include matrix inversion (when transforming a generalized matrix equation to a standard counterpart), or the matrix equations have symmetry structure that we want to utilize in the algorithms. As in Part I, we define recursive splittings for each matrix equation which in turn lead to a few similar subproblems to be solved. These splittings are re- cursively applied to all “half-sized” triangular matrix equations. We terminate ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
Image of page 3
Recursive Blocked Algorithms—Part II 419 the recursion when the new problem sizes ( M and/or N ) are smaller than a
Image of page 4

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

View Full Document Right Arrow Icon
Image of page 5
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