functiondiscreteSylvester

Return solution of the discrete-time Sylvester equation A*X*B + sgn*X = C

Extends from Modelica.Icons.Function (Icon for functions).

Information

Syntax

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

Description

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.

References

[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.

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];

Inputs

Type	Name	Default	Description
Real[:,size(A, 1)]	A		Square matrix A in AXB + sgn*X = C
Real[:,size(B, 1)]	B		Square matrix B in AXB + sgn*X = C
Real[size(A, 2),size(B, 1)]	C		Rectangular matrix C in AXB + sgn*X = C
Boolean	AisHess	false	= true, if A has already Hessenberg form
Boolean	BTisSchur	false	= true, if B' has already real Schur form
Integer	sgn	1	Specifies the sign in AXB + sgn*X = C
Real	eps	Matrices.norm(A, 1)10Modelica.Constants.eps	Tolerance

Outputs

Type	Name	Default	Description
Real[size(A, 2),size(B, 1)]	X		solution of the discrete Sylvester equation AXB + sgn*X = C

Revisions

2010/05/31 by Marcus Baur, DLR-RM