22 e t 12 c 11 x 11 rtrglydt a 11 e 11 c 11 blks x x

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22 , E T 12 , C 11 ); X 11 = rtrglydt ( A 11 , E 11 , C 11 , blks ); X = [ X 11 , X 12 ; X 21 , X 22 ]; else X = rtrgsyl ( A , A , E , E , C , blks ); end Algorithm 2. Recursive blocked algorithm for solving the triangular generalized discrete-time Lyapunov equation. Sep[LYDT] 6= 0. If C is (semi)definite and | λ i | < 1 for all i , then a unique (semi)definite solution exists [Hammarling 1982]. The template for the recursive splitting follows the one for the GLYDT equa- tion. If C = C T , then X 21 = X T 12 and the recursive splitting leads to three triangular matrix equations: A 11 X 11 A T 11 - X 11 = C 11 - A 12 X T 12 A T 11 - A 11 X 12 A T 12 - A 12 X 22 A T 12 , A 11 X 12 A T 22 - X 12 = C 12 - A 12 X 22 A T 22 , A 22 X 22 A T 22 - X 22 = C 22 . The first and the third are standard Stein-type equations, while the second is a triangular discrete-time Sylvester equation. The recursive blocked algorithm is named rtrlydt . 3.5 Recursive Triangular Generalized Continuous-Time Lyapunov Solvers Consider the real generalized continuous-time Lyapunov (GLYCT) matrix equation AXE T + EXA T = C , (5) where A and E of size N × N are upper quasitriangular and upper triangular, respectively. If C is symmetric, then X is symmetric as well. We treat this case as a special case of the generalized Sylvester equation with A = C , B = D , and changing the sign in the GSYL equation (1). The right-hand side C and the so- lution X are of size N × N , and typically, the solution overwrites the right-hand ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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Recursive Blocked Algorithms—Part II 425 Table I. Complexity of Standard Algorithms Measured in Flops Matrix Equation Overall Cost in Flops SYDT (2) 2 M 2 N + MN 2 ( M N ) M 2 N + 2 MN 2 ( M > N ) LYDT (4) 4 3 N 3 GSYL (1) 4 M 2 N + 2 MN 2 ( M N ) 2 M 2 N + 4 MN 2 ( M > N ) GLYCT (5) 8 3 N 3 GLYDT (3) 8 3 N 3 side ( C X ). By choosing E = I N in (5), we get the standard continuous-time Lyapunov equation, which is covered in Jonsson and K˚agstr¨om [2001]. The GLYCT equation (5) has a unique symmetric solution if and only if C = C T , and all eigenvalues λ i of A - λ E are finite with λ i + λ j 6= 0 for all i and j , or equivalently Sep[GLYCT] 6= 0. If C is (semi)definite and Re λ i < 0 for all i , then a unique (semi)definite solution exists [Penzl 1998]. Since X is symmetric, the recursive splitting results in two triangular GLYCT equations and one GSYL equation: A 11 X 11 E T 11 + E 11 X 11 A T 11 = C 11 - A 12 X T 12 E T 11 - ( A 11 X 12 + A 12 X 22 ) E T 12 - E 12 X T 12 A T 11 - ( E 11 X 12 + E 12 X 22 ) A T 12 , A 11 X 12 E T 22 + E 11 X 12 A T 22 = C 12 - A 12 X 22 E T 22 - E 12 X 22 A T 22 , A 22 X 22 E T 22 + E 22 X 22 A T 22 = C 22 . We start by solving for X 22 in the third equation. After updating C 12 with respect to X 22 , we can solve for X 12 . Finally, after updating C 11 with respect to X 12 and X 22 , we solve for X 11 . We remark that the update of C 11 includes two SYR2K operations, namely, C 11 = C 11 - ( A 11 X 12 + A 12 X 22 ) E T 12 - E 12 ( A 11 X 12 + A 12 X 22 ) T and C 11 = C 11 - ( E 11 X 12 ) A T 12 - A 12 ( E 11 X 12 ) T , where A 11 X 12 and E 11 X 12 are TRMM operations and A 12 X 22 is a GEMM operation. The recursive blocked algorithm is called rtrglyct .
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