## SG03BT

### Solving (for Cholesky factor) stable generalized continuous-time complex Lyapunov equations, with A, E, and B upper triangular

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute the Cholesky factor U of the matrix X, X = U**H * U or
      X = U * U**H, which is the solution of the generalized c-stable
      continuous-time Lyapunov equation

          H            H                  2    H
         A  * X * E + E  * X * A = - SCALE  * B  * B,                (1)

      or the conjugate transposed equation

                  H            H          2        H
         A * X * E  + E * X * A  = - SCALE  * B * B ,                (2)

      respectively, where A, E, B, and U are complex N-by-N matrices.
      The Cholesky factor U of the solution is computed without first
      finding X. The pencil A - lambda * E must be in complex
      generalized Schur form (A and E are upper triangular and the
      diagonal elements of E are non-negative real numbers). Moreover,
      it must be c-stable, i.e., its eigenvalues must have negative real
      parts. B must be an upper triangular matrix with real non-negative
      entries on its main diagonal.

      The resulting matrix U is upper triangular. The entries on its
      main diagonal are non-negative. SCALE is an output scale factor
      set to avoid overflow in U.

Specification

          SUBROUTINE SG03BT( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
         $                   ZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TRANS
          DOUBLE PRECISION  SCALE
          INTEGER           INFO, LDA, LDB, LDE, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  DWORK(*)
          COMPLEX*16        A(LDA,*), B(LDB,*), E(LDE,*), ZWORK(*)

Arguments

Mode Parameters

      TRANS   CHARACTER*1
              Specifies whether equation (1) or equation (2) is to be
              solved:
              = 'N':  Solve equation (1);
              = 'C':  Solve equation (2).

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices.  N >= 0.

      A       (input/workspace) COMPLEX*16 array, dimension (LDA,N)
              The leading N-by-N upper triangular part of this array
              must contain the triangular matrix A. The lower triangular
              part is used as workspace, but the diagonal is restored.

      LDA     INTEGER
              The leading dimension of the array A.  LDA >= MAX(1,N).

      E       (input/workspace) COMPLEX*16 array, dimension (LDE,N)
              The leading N-by-N upper triangular part of this array
              must contain the triangular matrix E. If TRANS = 'N', the
              strictly lower triangular part is used as workspace.

      LDE     INTEGER
              The leading dimension of the array E.  LDE >= MAX(1,N).

      B       (input/output) COMPLEX*16 array, dimension (LDB,N)
              On entry, the leading N-by-N upper triangular part of this
              array must contain the matrix B.
              On exit, the leading N-by-N upper triangular part of this
              array contains the solution matrix U.

      LDB     INTEGER
              The leading dimension of the array B.  LDB >= MAX(1,N).

      SCALE   (output) DOUBLE PRECISION
              The scale factor set to avoid overflow in U.
              0 < SCALE <= 1.

Workspace

      DWORK   DOUBLE PRECISION array, dimension MAX(N-1,0)

      ZWORK   COMPLEX*16, dimension MAX(3*N-3,0)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 3:  the pencil A - lambda * E is not stable, i.e., there
                    is an eigenvalue without a negative real part.

Method

      The method used by the routine is an extension of Hammarling's
      algorithm [1] to generalized Lyapunov equations. The real case is
      described in [2].

      We present the method for solving equation (1). Equation (2) can
      be treated in a similar fashion. For simplicity, assume SCALE = 1.

      Since all matrices A, E, B, and U are upper triangular, we use the
      following partitioning

                ( A11   A12 )        ( E11   E12 )
            A = (           ),   E = (           ),
                (   0   A22 )        (   0   E22 )

                ( B11   B12 )        ( U11   U12 )
            B = (           ),   U = (           ),                  (3)
                (   0   B22 )        (   0   U22 )

      where the size of the (1,1)-blocks is 1-by-1.

      We compute U11, U12**H and U22 in three steps.

      Step I:

         From (1) and (3) we get the 1-by-1 equation

               H     H                 H     H                  H
            A11 * U11 * U11 * E11 + E11 * U11 * U11 * A11 = -B11 * B11.

         For brevity, details are omitted here. The technique for
         computing U11 is similar to those applied to standard Lyapunov
         equations in Hammarling's algorithm ([1], section 5).

         Furthermore, the auxiliary scalars M1 and M2 defined as follows

            M1 = A11 / E11 ,   M2 = B11 / E11 / U11 ,

         are computed in a numerically reliable way.

      Step II:

         We solve for U12**H the linear system of equations, with
         scaling to prevent overflow,

                 H           H      H
            ( A22  + M1 * E22  ) U12  =

                        H              H           H
            = - M2 * B12  - U11 * ( A12  + M1 * E12  ) .

      Step III:

         One can show that

               H      H                  H      H
            A22  * U22  * U22 * E22 + E22  * U22  * U22 * A22  =

                 H              H
            - B22  * B22 - y * y                                     (4)

         holds, where y is defined as follows

                   H                   H      H      H
            y = B12  - M2 * ( U11 * E12  + E22  * U12  ).

         If B22_tilde is the square triangular matrix arising from the
         QR-factorization

                ( B22_tilde )     ( B22  )
            Q * (           )  =  (      ),
                (     0     )     ( y**H )

         then

                 H              H                H
            - B22  * B22 - y * y   =  - B22_tilde  * B22_tilde.

         Replacing the right hand side in (4) by the term
         - B22_tilde**H * B22_tilde leads to a generalized Lyapunov
         equation like (1), but of dimension N-1.

      The solution U of the equation (1) can be obtained by recursive
      application of the steps I to III.

References

      [1] Hammarling, S.J.
          Numerical solution of the stable, non-negative definite
          Lyapunov equation.
          IMA J. Num. Anal., 2, pp. 303-323, 1982.

      [2] Penzl, T.
          Numerical solution of generalized Lyapunov equations.
          Advances in Comp. Math., vol. 8, pp. 33-48, 1998.

Numerical Aspects

      The routine requires 2*N**3 flops. Note that we count a single
      floating point arithmetic operation as one flop.

Further Comments

      The Lyapunov equation may be very ill-conditioned. In particular,
      if the pencil A - lambda * E has a pair of almost degenerate
      eigenvalues, then the Lyapunov equation will be ill-conditioned.
      Perturbed values were used to solve the equation.
      A condition estimate can be obtained from the routine SG03AD.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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