## SG03BU

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

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

Purpose

      To compute the Cholesky factor U of the matrix X, X = U**T * U or
      X = U * U**T, which is the solution of the generalized d-stable
      discrete-time Lyapunov equation

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

      or the transposed equation

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

      respectively, where A, E, B, and U are real 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 generalized Schur
      form ( A upper quasitriangular, E upper triangular ). Moreover, it
      must be d-stable, i.e., the moduli of its eigenvalues must be less
      than one. B must be an upper triangular matrix with 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 SG03BU( TRANS, N, A, LDA, E, LDE, B, LDB, SCALE, DWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         TRANS
          DOUBLE PRECISION  SCALE
          INTEGER           INFO, LDA, LDB, LDE, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), DWORK(*), E(LDE,*)

Arguments

Mode Parameters

      TRANS   CHARACTER*1
              Specifies whether equation (1) or equation (2) is to be
              solved:
              = 'N':  Solve equation (1);
              = 'T':  Solve equation (2).

Input/Output Parameters

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

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N upper Hessenberg part of this array
              must contain the quasitriangular matrix A. The elements
              below the upper Hessenberg part are not referenced.

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

      E       (input) DOUBLE PRECISION array, dimension (LDE,N)
              The leading N-by-N upper triangular part of this array
              must contain the triangular matrix E. The elements below
              the main diagonal are not referenced.

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

      B       (input/output) DOUBLE PRECISION 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. The elements below
              the main diagonal are not referenced.

      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 (6*N-6)

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the generalized Sylvester equation to be solved in
                    step II (see METHOD) is (nearly) singular to working
                    precision;  perturbed values were used to solve the
                    equation (but the matrices A and E are unchanged);
              = 2:  the generalized Schur form of the pencil
                    A - lambda * E contains a 2-by-2 main diagonal block
                    whose eigenvalues are not a pair of complex
                    conjugate numbers;
              = 3:  the pencil A - lambda * E is not d-stable, i.e.,
                    there are eigenvalues outside the open unit circle;
              = 4:  the LAPACK routine DSYEVX utilized to factorize M3
                    failed to converge. This error is unlikely to occur.

Method

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

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

      The matrix A is an upper quasitriangular matrix, i.e., it is a
      block triangular matrix with square blocks on the main diagonal
      and the block order at most 2. We use the following partitioning
      for the matrices A, E, B and the solution matrix U

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

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

      The size of the (1,1)-blocks is 1-by-1 (iff A(2,1) = 0.0) or
      2-by-2.

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

      Step I:

         From (1) and (3) we get the 1-by-1 or 2-by-2 equation

                 T      T                  T      T
              A11  * U11  * U11 * A11 - E11  * U11  * U11 * E11

                     T
              = - 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 6).

         Furthermore, the auxiliary matrices M1 and M2 defined as
         follows

                                -1      -1
            M1 = U11 * A11 * E11   * U11  ,

                          -1      -1
            M2 = B11 * E11   * U11  ,

         are computed in a numerically reliable way.

      Step II:

         We solve for U12**T the generalized Sylvester equation

               T      T           T      T
            A22  * U12  * M1 - E22  * U12

                   T           T      T      T      T
            = - B12  * M2 + E12  * U11  - A12  * U11  * M1.

      Step III:

         One can show that

               T      T                  T      T
            A22  * U22  * U22 * A22 - E22  * U22  * U22 * E22  =

                 T              T
            - B22  * B22 - y * y                                     (4)

         holds, where y is defined as follows

                   T      T      T      T
            w = A12  * U11  + A22  * U12 ,

                     T
            y = ( B12   w ) * M3EV,

         where M3EV is a matrix which fulfils

                 ( I-M2*M2**T   -M2*M1**T )              T
            M3 = (                        ) = M3EV * M3EV .
                 (  -M1*M2**T  I-M1*M1**T )

         M3 is positive semidefinite and its rank is equal to the size
         of U11. Therefore, a matrix M3EV can be found by solving the
         symmetric eigenvalue problem for M3 such that y consists of
         either 1 or 2 rows.

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

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

         then

                 T              T                T
            - B22  * B22 - y * y   =  - B22_tilde  * B22_tilde.

         Replacing the right hand side in (4) by the term
         - B22_tilde**T * B22_tilde leads to a generalized Lyapunov
         equation of lower dimension compared to (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 reciprocal
      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.
      When setting the error indicator INFO, the routine does not test
      for near instability in the equation but only for exact
      instability.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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