Acm transactions on mathematical software vol 28 no 4

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ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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Recursive Blocked Algorithms—Part II 423 Since all matrices are square, we only have one way of doing a recursive splitting . We split A , E , and C by rows and columns: A 11 A 12 A 22 ‚ • X 11 X 12 X 21 X 22 A T 11 A T 12 A T 22 - E 11 E 12 E 22 ‚ • X 11 X 12 X 21 X 22 E T 11 E T 12 E T 22 = C 11 C 12 C 21 C 22 . Since X 21 = X T 12 , the recursive splitting results in three triangular generalized discrete-time Lyapunov equations: A 11 X 11 A T 11 - E 11 X 11 E T 11 = C 11 - A 12 X T 12 A T 11 - ( A 11 X 12 + A 12 X 22 ) A T 12 + E 12 X T 12 E T 11 + ( E 11 X 12 + E 12 X 22 ) E T 12 , A 11 X 12 A T 22 - E 11 X 12 E T 22 = C 12 - A 12 X 22 A T 22 + E 12 X 22 E T 22 , A 22 X 22 A T 22 - E 22 X 22 E 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 . Four of the two-sided matrix product updates of C 11 can be expressed as two SYR2K operations: C 11 = C 11 - ( A 11 X 12 ) A T 12 - A 12 ( A 11 X 12 ) T , and C 11 = C 11 + ( E 11 X 12 ) E T 12 + E 12 ( E 11 X 12 ) T , where A 11 X 12 and E 11 X 12 are TRMM operations. The GEMM-rich updates C 11 = C 11 - A 12 X 22 A T 12 and C 11 = C 11 + E 12 X 22 E T 12 are performed by the SLICOT MB01RD subroutine; see Section 4.1. In Algorithm 2, a Matlab-style function [ X ] = rtrglydt ( A , E , C , blks ) im- plementing our recursive blocked solver is presented. The function [ X ] = trglydt ( A , E , C ) implements an algorithm for solving triangular generalized discrete-time Lyapunov kernel problems. For solving the triangular general- ized Sylvester equations that appear, we make use of the recursive algorithm rtrgsyl , described in Section 3.1. 3.4 Recursive Triangular Discrete-Time Lyapunov Solvers The real discrete-time Lyapunov (LYDT) or Stein matrix equation is obtained by setting E = I N in (3). We get the equation AXA T - X = C , (4) where A is upper triangular or upper quasitriangular, that is, in real Schur form. This equation is also the result of setting A = B in (2). If C is symmetric, then X is symmetric as well, and the Stein matrix equation corresponds to the discrete-time standard algebraic Lyapunov equation. The LYDT equation (4) has a unique solution if and only if λ i λ j 6= 1 for all i and j , or equivalently ACM Transactions on Mathematical Software, Vol. 28, No. 4, December 2002.
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424 I. Jonsson and B. K˚agstr¨om Algorithm 2: rtrglydt Input: ( A , E ) ( N × N ) in upper generalized Schur form. C ( N × N ) dense matrix. blks , block size that specifies when to switch to a standard algorithm for solving small-sized triangular generalized Lyapunov equations. Output: X ( N × N ), the solution of AXA T - EXE T = C . X is allowed to overwrite C , and is symmetric if C = C T on entry. function [ X ] = rtrglydt ( A , E , C , blks ) if 1 N blks then X = trglydt ( A , E , C ); elseif C is symmetric % Split ( A , E ), and C (all by rows and columns) X 22 = rtrglydt ( A 22 , E 22 , C 22 , blks ); C 12 = axb ( - A 12 , X 22 , A T 22 , C 12 ); C 12 = axb ( E 12 , X 22 , E T 22 , C 12 ); X 12 = rtrgsyl ( A 11 , A 22 , E 11 , E 22 , C 12 , blks ); X 21 = X T 12 ; C 11 = syr2k ( trmm ( A 11 , X 12 ), - A 12 , C 11 ); C 11 = syr2k ( trmm ( E 11 , X 12 ), E 12 , C 11 ); C 11 = axb ( - A 12 , X 22 , A T 12 , C 11 ); C 11 = axb ( E 12 , X
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