discreteSylvesterReturn solution of the discrete-time Sylvester equation A*X*B + sgn*X = C |
This information is part of the Modelica Standard Library maintained by the Modelica Association.
X = Matrices.discreteSylvester(A, B, C); X = Matrices.discreteSylvester(A, B, C, AisHess, BTisSchur, sgn, eps);
Function discreteSylvester computes the solution X of the discrete-time Sylvester equation
A*X*B + sgn*X = C.
where sgn = 1 or sgn = -1. The algorithm applies the Hessenberg-Schur method proposed by Golub et al [1]. For sgn = -1, the discrete Sylvester equation is also known as Stein equation:
A*X*B - X + Q = 0.
In a nutshell, the problem is reduced to the corresponding problem
H*Y*S' + sgn*Y = F.
with H=U'*A*U is the Hessenberg form of A and S=V'*B'*V is the real Schur form of B',
F=U'*C*V and Y=U*X*V'
are appropriate transformations of C and X. This problem is solved sequentially by exploiting the specific forms of S and H.
Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisHess" and "BTisSchur" indicate to omit one or both of the transformation to Hessenberg form or Schur form respectively in the case that A and/or B have already Hessenberg form or Schur respectively.
[1] Golub, G.H., Nash, S. and Van Loan, C.F. A Hessenberg-Schur method for the problem AX + XB = C. IEEE Transaction on Automatic Control, AC-24, no. 6, pp. 909-913, 1979.
A = [1.0, 2.0, 3.0; 6.0, 7.0, 8.0; 9.0, 2.0, 3.0]; B = [7.0, 2.0, 3.0; 2.0, 1.0, 2.0; 3.0, 4.0, 1.0]; C = [271.0, 135.0, 147.0; 923.0, 494.0, 482.0; 578.0, 383.0, 287.0]; X = discreteSylvester(A, B, C); results in: X = [2.0, 3.0, 6.0; 4.0, 7.0, 1.0; 5.0, 3.0, 2.0];
A |
Type: Real[:,size(A, 1)] Description: Square matrix A in A*X*B + sgn*X = C |
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B |
Type: Real[:,size(B, 1)] Description: Square matrix B in A*X*B + sgn*X = C |
C |
Type: Real[size(A, 2),size(B, 1)] Description: Rectangular matrix C in A*X*B + sgn*X = C |
AisHess |
Default Value: false Type: Boolean Description: = true, if A has already Hessenberg form |
BTisSchur |
Default Value: false Type: Boolean Description: = true, if B' has already real Schur form |
sgn |
Default Value: 1 Type: Integer Description: Specifies the sign in A*X*B + sgn*X = C |
eps |
Default Value: Matrices.norm(A, 1) * 10 * Modelica.Constants.eps Type: Real Description: Tolerance |
X |
Type: Real[size(A, 2),size(B, 1)] Description: solution of the discrete Sylvester equation A*X*B + sgn*X = C |
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