Certain block size blks which is chosen such that at

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certain block size, blks , which is chosen such that at least a few submatrices involved in the current matrix equation fit in the first-level cache memory. For the solution of the small-sized problems, we apply our new high-performance kernels (see Part I). We present Matlab-style templates for two of the recursive blocked solvers. We remark that all updates with respect to the solution of subproblems in the recursion are GEMM-rich operations of the type C β C + α op( A ) X op( B ) T , where α and β are real scalars, and op(Y) is Y or Y T . A and/or B can be dense or triangular. This is due to the “two-sidedness” of the matrix equations. We refer to Section 4.1 for a discussion of the design of a high-performance implementation of this operation. In the algorithm descriptions, we use the function [ C ] = axb ( A , X , B ), which implements the matrix-product operation C = C + AXB . Other functions, including level-3 BLAS operations are intro- duced the first time they are used in the algorithm descriptions. See Part I for BLAS references. 3.1 Recursive Triangular Generalized Sylvester Solvers Consider the real generalized Sylvester (GSYL) matrix equation AXB T - CXD T = E , (1) where ( A , C ) of size M × M and ( B , D ) of size N × N are in generalized Schur form; that is, A and B are upper quasitriangular and C , B are upper triangular. The right-hand side E and the solution X are of size M × N and, typically, the solution overwrites the right-hand side ( E X ). The GSYL equation (1) has a unique solution if and only if A - λ C and D - λ B are regular and have disjoint spectra [Chu 1987], or equivalently Sep[GSYL] 6= 0. We see that (1) is a generalization of the continuous-time Sylvester equation ( B = I N and C = I M ), as well as the discrete-time Sylvester equation ( C = I M and D = I N ). We consider three alternatives for doing a recursive splitting . Case 1 (1 N M / 2). We split ( A , C ) by rows and columns, and E by rows only: A 11 A 12 A 22 ‚ • X 1 X 2 B T - C 11 C 12 C 22 ‚ • X 1 X 2 D T = E 1 E 2 , or equivalently A 11 X 1 B T - C 11 X 1 D T = E 1 - A 12 X 2 B T + C 12 X 2 D T , A 22 X 2 B T - C 22 X 2 D T = E 2 . The original problem is split in two triangular generalized Sylvester equations. First, we solve for X 2 and after updating E 1 with respect to X 2 , we can solve for X 1 . ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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420 I. Jonsson and B. K˚agstr¨om Case 2 (1 M N / 2). We split ( B , D ) by rows and columns, and E by columns only: A [ X 1 X 2 ] B T 11 B T 12 B T 22 - C [ X 1 X 2 ] D T 11 D T 12 D T 22 = [ E 1 E 2 ], or equivalently AX 1 B T 11 - CX 1 D T 11 = E 1 - AX 2 B T 12 + CX 2 D T 12 , AX 2 B T 22 - CX 2 D T 22 = E 2 . As in Case 1, we first solve for X 2 and then after updating the right-hand side of the first equation, solve for X 1 . Case 3 ( N / 2 < M < 2 N ). We split ( A , C ), ( B , D ), and E by rows and columns: A 11 A 12 A 22 ‚ • X 11 X 12 X 21 X 22 B T 11 B T 12 B T 22 - C 11 C 12 C 22 ‚ • X 11 X 12 X 21 X 22 D T 11 D T 12 D T 22 = E 11 E 12 E 21 E 22 .
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