Scilab Function

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;
- `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) = 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.