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linmeq

linmeq — Sylvester and Lyapunov equations solver

Calling sequence

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

Parameters

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;
otherwiseA 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, lyap

Author

H. Xu, TU Chemnitz, FR Germany, Dec. 1998.

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