# 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
- lyap