linmeq

Sylvester and Lyapunov equations solver

Calling Sequence

[X(,sep)] = linmeq(task,A,(B,)C,flag,trans(,schur))

Arguments

task integer option to determine the equation type:  

:=1 solve the Sylvester equation (1a) or (1b); : :=2 solve the Lyapunov equation (2a) or (2b); : :=3 solve for the Cholesky factor op(X) the Lyapunov equation (3a)

or (3b).

:

: :A real matrix : :B real matrix : :C real matrix : :flag (optional) integer vector of length 3 or 2 containing options.

:task = 1 : flag has length 3

:flag(1) = 0 : solve the continuous-time equation (1a); otherwise,

solve the discrete-time equation (1b).

: :flag(2) = 1 : A is (quasi) upper triangular; : :flag(2) = 2 : A is upper Hessenberg; : :otherwise A is in general form. : :flag(3) = 1 : B is (quasi) upper triangular; : :flag(3) = 2 : B is upper Hessenberg; : :otherwise, B is in general form. :

: :task = 2 : flag has length 2

:flag(1) if 0 solve continuous-time equation (2a), otherwise, solve

discrete-time equation (2b).

: :flag(2) = 1 : A is (quasi) upper triangular otherwise, A is in

general form.

:

: :task = 3 : flag has length 2

:flag(1) = 0 : solve continuous-time equation (3a); otherwise, solve

discrete-time equation (3b).

: :flag(2) = 1 : A is (quasi) upper triangular; otherwise, A is in

general form.

:

:

Default: flag(1) = 0, flag(2) = 0 (, flag(3) = 0). : :trans (optional) integer specifying a transposition option.

:= 0 : solve the equations (1) - (3) with op(M) = M. : := 1 : solve the equations (1) - (3) with op(M) = M’. : := 2 : solve the equations (1) with op(A) = A’; op(B) = B; : := 3 : solve the equations (1) with op(A) = A; op(B) = B’. :

Default: trans = 0. : :schur (optional) integer specifying whether the Hessenberg-Schur or Schur method should be used. Available for task = 1.

:= 1 : Hessenberg-Schur method (one matrix is reduced to Schur form). : := 2 : Schur method (two matrices are reduced to Schur form). :

Default: schur = 1. : :X : :sep (optional) estimator of Sep(op(A),-op(A)’) for (2.a) or

Sepd(A,A’) for (2.b).

:

Description

linmeq function for solving Sylvester and Lyapunov equations using SLICOT routines SB04MD, SB04ND, SB04PD, SB04QD, SB04RD, SB03MD, and SB03OD.

[X] = linmeq(1,A,B,C,flag,`trans`_,`schur`_)
[X,sep] = linmeq(2,A,C,flag,`trans`_)
[X] = linmeq(2,A,C,flag,`trans`_)
[X] = linmeq(3,A,C,flag,`trans`_)

linmeq solves various Sylvester and Lyapunov matrix equations:

op(A)*X + X*op(B) = C,                           (1a)

op(A)*X*op(B) + X = C,                           (1b)

op(A)'*X + X*op(A) = C,                          (2a)

op(A)'*X*op(A) - X = C,                          (2b)

op(A)'*(op(X)'*op(X)) + (op(X)'*op(X))*op(A) =
                      -  op(C)'*op(C),           (3a)

op(A)'*(op(X)'*op(X))*op(A) - op(X)'*op(X) =
                            - op(C)'*op(C),      (3b)

where op(M) = M, or M’.

Comments

:1. For equation (1a) or (1b), when schur = 1, the Hessenberg-Schur

method is used, reducing one matrix to Hessenberg form and the other one to a real Schur form. Otherwise, both matrices are reduced to real Schur forms. If one or both matrices are already reduced to Schur/Hessenberg forms, this could be specified by flag(2) and flag(3). For general matrices, the Hessenberg-Schur method could be significantly more efficient than the Schur method.

: :2. For equation (2a) or (2b), matrix C is assumed symmetric. : :3. For equation (3a) or (3b), matrix A must be stable or

convergent, respectively.

: :4. For equation (3a) or (3b), the computed matrix X is the Cholesky

factor of the solution, i.e., the real solution is op(X)’*op(X), where X is an upper triangular matrix.

:

Revisions

V. Sima, Katholieke Univ. Leuven, Belgium, May 1999, May, Sep. 2000. V. Sima, University of Bucharest, Romania, May 2000.

Examples

//(1a)
n=40;m=30;
A=`rand`_(n,n);C=`rand`_(n,m);B=`rand`_(m,m);
X = linmeq(1,A,B,C);
`norm`_(A*X+X*B-C,1)
//(1b)
flag=[1,0,0]
X = linmeq(1,A,B,C,flag);
`norm`_(A*X*B+X-C,1)
//(2a)
A=`rand`_(n,n);C=`rand`_(A);C=C+C';
X = linmeq(2,A,C);
`norm`_(A'*X + X*A -C,1)
//(2b)
X = linmeq(2,A,C,[1 0]);
`norm`_(A'*X*A -X-C,1)
//(3a)
A=`rand`_(n,n);
A=A-(`max`_(`real`_(`spec`_(A)))+1)*`eye`_(); //shift eigenvalues
C=`rand`_(A);
X=linmeq(3,A,C);
`norm`_(A'*X'*X+X'*X*A +C'*C,1)
//(3b)
A = [-0.02, 0.02,-0.10, 0.02,-0.03, 0.12;
      0.02, 0.14, 0.12,-0.10,-0.02,-0.14;
     -0.10, 0.12, 0.05, 0.03,-0.04,-0.04;
      0.02,-0.10, 0.03,-0.06, 0.08, 0.11;
     -0.03,-0.02,-0.04, 0.08, 0.14,-0.07;
      0.12,-0.14,-0.04, 0.11,-0.07, 0.04]

C=`rand`_(A);
X=linmeq(3,A,C,[1 0]);
`norm`_(A'*X'*X*A - X'*X +C'*C,1)

See Also

• sylv Sylvester equation.
• lyap Lyapunov equation