.Modelica.Math.Matrices.continuousSylvester

Information

Syntax

         X = Matrices.continuousSylvester(A, B, C);
         X = Matrices.continuousSylvester(A, B, C, AisSchur, BisSchur);

Description

Function continuousSylvester computes the solution X of the continuous-time Sylvester equation

 A*X + X*B = C.

using the Schur method for Sylvester equations proposed by Bartels and Stewart [1].

In a nutshell, the problem is reduced to the corresponding problem

 S*Y + Y*T = D.

with S=U'*A*U is the real Schur of A, T=V'*T*V is the real Schur form of B and D=U'*C*V and Y=U*X*V' are the corresponding transformations of C and X. This problem is solved sequentially by exploiting the block triangular form of S and T. Finally the solution of the original problem is recovered as X=U'*Y*V.
The Boolean inputs "AisSchur" and "BisSchur" indicate to omit one or both of the transformation to Schur in the case that A and/or B have already Schur form.

The function applies LAPACK-routine DTRSYL. See LAPACK.dtrsyl for more information.

References

  [1] Bartels, R.H. and Stewart G.W.
      Algorithm 432: Solution of the matrix equation AX + XB = C.
      Comm. ACM., Vol. 15, pp. 820-826, 1972.

Example

  A = [17.0,   24.0,   1.0,   8.0,   15.0 ;
       23.0,    5.0,   7.0,  14.0,   16.0 ;
        0.0,    6.0,  13.0,  20.0,   22.0;
        0.0,    0.0,  19.0,  21.0,    3.0 ;
        0.0,    0.0,   0.0,   2.0,    9.0];

  B =  [8.0, 1.0, 6.0;
        0.0, 5.0, 7.0;
        0.0, 9.0, 2.0];

  C = [62.0,  -12.0, 26.0;
       59.0,  -10.0, 31.0;
       70.0,  -6.0,   9.0;
       35.0,  31.0,  -7.0;
       36.0, -15.0,   7.0];

  X = continuousSylvester(A, B, C);

  results in:

  X = [0.0,  0.0,  1.0;
       1.0,  0.0,  0.0;
       0.0,  1.0,  0.0;
       1.0,  1.0, -1.0;
       2.0, -2.0,  1.0];

See also

Matrices.discreteSylvester, Matrices.continuousLyapunov

Interface

function continuousSylvester
  extends Modelica.Icons.Function;
  import Modelica.Math.Matrices;
  input Real A[:, :] "Square matrix A";
  input Real B[:, :] "Square matrix B";
  input Real C[size(A, 1), size(B, 2)] "Matrix C";
  input Boolean AisSchur = false "True if A has already real Schur form";
  input Boolean BisSchur = false "True if B has already real Schur form";
  output Real X[size(A, 1), size(B, 2)] "Solution of the continuous Sylvester equation";
end continuousSylvester;

Revisions

• 2010/05/31 by Marcus Baur, DLR-RM