**Purpose**

To solve the Sylvester equation -AX + XB = C, where A and B are M-by-M and N-by-N matrices, respectively, in real Schur form. This routine is intended to be called only by SLICOT Library routine MB03RD. For efficiency purposes, the computations are aborted when the infinity norm of an elementary submatrix of X is greater than a given value PMAX.

SUBROUTINE MB03RY( M, N, PMAX, A, LDA, B, LDB, C, LDC, INFO ) C .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LDC, M, N DOUBLE PRECISION PMAX C .. Array Arguments .. DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*)

**Input/Output Parameters**

M (input) INTEGER The order of the matrix A and the number of rows of the matrices C and X. M >= 0. N (input) INTEGER The order of the matrix B and the number of columns of the matrices C and X. N >= 0. PMAX (input) DOUBLE PRECISION An upper bound for the infinity norm of an elementary submatrix of X (see METHOD). A (input) DOUBLE PRECISION array, dimension (LDA,M) The leading M-by-M part of this array must contain the matrix A of the Sylvester equation, in real Schur form. The elements below the real Schur form are not referenced. LDA INTEGER The leading dimension of array A. LDA >= MAX(1,M). B (input) DOUBLE PRECISION array, dimension (LDB,N) The leading N-by-N part of this array must contain the matrix B of the Sylvester equation, in real Schur form. The elements below the real Schur form are not referenced. LDB INTEGER The leading dimension of array B. LDB >= MAX(1,N). C (input/output) DOUBLE PRECISION array, dimension (LDC,N) On entry, the leading M-by-N part of this array must contain the matrix C of the Sylvester equation. On exit, if INFO = 0, the leading M-by-N part of this array contains the solution matrix X of the Sylvester equation, and each elementary submatrix of X (see METHOD) has the infinity norm less than or equal to PMAX. On exit, if INFO = 1, the solution matrix X has not been computed completely, because an elementary submatrix of X had the infinity norm greater than PMAX. Part of the matrix C has possibly been overwritten with the corresponding part of X. LDC INTEGER The leading dimension of array C. LDC >= MAX(1,M).

INFO INTEGER = 0: successful exit; = 1: an elementary submatrix of X had the infinity norm greater than the given value PMAX.

The routine uses an adaptation of the standard method for solving Sylvester equations [1], which controls the magnitude of the individual elements of the computed solution [2]. The equation -AX + XB = C can be rewritten as p l-1 -A X + X B = C + sum A X - sum X B kk kl kl ll kl i=k+1 ki il j=1 kj jl for l = 1:q, and k = p:-1:1, where A , B , C , and X , are kk ll kl kl block submatrices defined by the partitioning induced by the Schur form of A and B, and p and q are the numbers of the diagonal blocks of A and B, respectively. So, the elementary submatrices of X are found block column by block column, starting from the bottom. If any such elementary submatrix has the infinity norm greater than the given value PMAX, the calculations are ended.

[1] Bartels, R.H. and Stewart, G.W. T Solution of the matrix equation A X + XB = C. Comm. A.C.M., 15, pp. 820-826, 1972. [2] Bavely, C. and Stewart, G.W. An Algorithm for Computing Reducing Subspaces by Block Diagonalization. SIAM J. Numer. Anal., 16, pp. 359-367, 1979.

2 2 The algorithm requires 0(M N + MN ) operations.

Let ( A C ) ( I X ) M = ( ), Y = ( ). ( 0 B ) ( 0 I ) Then -1 ( A 0 ) Y M Y = ( ), ( 0 B ) hence Y is an non-orthogonal transformation matrix which performs the reduction of M to a block-diagonal form. Bounding a norm of X is equivalent to setting an upper bound to the condition number of the transformation matrix Y.

**Program Text**

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