.Modelica_LinearSystems2.Math.Matrices.dsylvester

Information

Syntax

X = Matrices.dsylvester(A, B, C);
   or
X = Matrices.dsylvester(A, B, C, AisHess, BTisSchur, sgn, eps);

Description

Function dsylvester 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 sequently by exploiting the specific forms of S and H. Finally, the solution of the 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.

References

 [1] Golub, G.H., Nash, S. and Van Loan, C.F. (1979):

A Hessenberg-Schur method for the problem AX + XB = C. IEEE Transaction on Automatic Control, AC-24, no. 6, pp. 909-913.
 

Example

  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];

Interface

function dsylvester
  import MatricesMSL = Modelica.Math.Matrices;
  import Modelica_LinearSystems2.Math.Matrices;
  input Real A[:, size(A, 1)] "Square matrix A in A*X*B + sgn*X = C";
  input Real B[:, size(B, 1)] "Square matrix B in A*X*B + sgn*X = C";
  input Real C[size(A, 2), size(B, 1)] "Rectangular matrix C in A*X*B + sgn*X = C";
  input Boolean AisHess = false "True if A has already Hessenberg form";
  input Boolean BTisSchur = false "True if B' has already real Schur form";
  input Integer sgn = 1 "Specifies the sign in A*X*B + sgn*X = C";
  input Real eps = MatricesMSL.norm(A, 1)*10*Modelica.Constants.eps "Tolerance";
  output Real X[size(A, 2), size(B, 1)] "solution of the discrete Sylvester equation A*X*B + sgn*X = C";
end dsylvester;

Revisions