ECE
Two sided Matrix Algorithms

# We introduce the following z matrices z sydt z axb t

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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.

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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.
Recursive Blocked Algorithms—Part II 419 the recursion when the new problem sizes ( M and/or N ) are smaller than a

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• Fall '07
• qgsdxjhf
• Algorithms, Matrices, Matrix Equations, recursive blocked algorithms, Generalized Sylvester

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