## SB03OZ

### Solution of stable continuous- or discrete-time complex Lyapunov equations (Cholesky factor)

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

Purpose

                            H
      To solve for X = op(U) *op(U) either the stable non-negative
      definite continuous-time Lyapunov equation
              H                     2      H
         op(A) *X + X*op(A) = -scale *op(B) *op(B),                  (1)

      or the convergent non-negative definite discrete-time Lyapunov
      equation
              H                     2      H
         op(A) *X*op(A) - X = -scale *op(B) *op(B),                  (2)

      where op(K) = K or K**H  (i.e., the conjugate transpose of the
      matrix K), A is an N-by-N matrix, op(B) is an M-by-N matrix, U is
      an upper triangular matrix containing the Cholesky factor of the
      solution matrix X, and scale is an output scale factor, set less
      than or equal to 1 to avoid overflow in X. If matrix B has full
      rank, then the solution matrix X will be positive definite and
      hence the Cholesky factor U will be nonsingular, but if B is rank
      deficient, then X may be only positive semi-definite and U will be
      singular.

      In the case of equation (1) the matrix A must be stable (that is,
      all the eigenvalues of A must have negative real parts), and for
      equation (2) the matrix A must be convergent (that is, all the
      eigenvalues of A must lie inside the unit circle).

Specification

          SUBROUTINE SB03OZ( DICO, FACT, TRANS, N, M, A, LDA, Q, LDQ, B,
         $                   LDB, SCALE, W, DWORK, ZWORK, LZWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, FACT, TRANS
          INTEGER           INFO, LDA, LDB, LDQ, LZWORK, M, N
          DOUBLE PRECISION  SCALE
    C     .. Array Arguments ..
          COMPLEX*16        A(LDA,*), B(LDB,*), Q(LDQ,*), W(*), ZWORK(*)
          DOUBLE PRECISION  DWORK(*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of Lyapunov equation to be solved, as
              follows:
              = 'C':  Equation (1), continuous-time case;
              = 'D':  Equation (2), discrete-time case.

      FACT    CHARACTER*1
              Specifies whether or not the Schur factorization of the
              matrix A is supplied on entry, as follows:
              = 'F':  On entry, A and Q contain the factors from the
                      Schur factorization of the matrix A;
              = 'N':  The Schur factorization of A will be computed
                      and the factors will be stored in A and Q.

      TRANS   CHARACTER*1
              Specifies the form of op(K) to be used, as follows:
              = 'N':  op(K) = K    (No transpose);
              = 'C':  op(K) = K**H (Conjugate transpose).

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrix A and the number of columns of
              the matrix op(B).  N >= 0.

      M       (input) INTEGER
              The number of rows of the matrix op(B).  M >= 0.
              If M = 0, A is unchanged on exit, and Q and W are not set.

      A       (input/output) COMPLEX*16 array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the matrix A. If FACT = 'F', then A contains
              an upper triangular matrix S in Schur form; the elements
              below the diagonal of the array A are then not referenced.
              On exit, the leading N-by-N upper triangular part of this
              array contains the upper triangle of the matrix S.
              The contents of the array A is not modified if FACT = 'F'.

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

      Q       (input or output) COMPLEX*16 array, dimension (LDQ,N)
              On entry, if FACT = 'F', then the leading N-by-N part of
              this array must contain the unitary matrix Q of the Schur
              factorization of A.
              Otherwise, Q need not be set on entry.
              On exit, the leading N-by-N part of this array contains
              the unitary matrix Q of the Schur factorization of A.
              The contents of the array Q is not modified if FACT = 'F'.

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

      B       (input/output) COMPLEX*16 array, dimension (LDB,N)
              if TRANS = 'N', and dimension (LDB,max(M,N)), if
              TRANS = 'C'.
              On entry, if TRANS = 'N', the leading M-by-N part of this
              array must contain the coefficient matrix B of the
              equation.
              On entry, if TRANS = 'C', the leading N-by-M part of this
              array must contain the coefficient matrix B of the
              equation.
              On exit, the leading N-by-N part of this array contains
              the upper triangular Cholesky factor U of the solution
              matrix X of the problem, X = op(U)**H * op(U).
              If M = 0 and N > 0, then U is set to zero.

      LDB     INTEGER
              The leading dimension of the array B.
              LDB >= MAX(1,N,M), if TRANS = 'N';
              LDB >= MAX(1,N),   if TRANS = 'C'.

      SCALE   (output) DOUBLE PRECISION
              The scale factor, scale, set less than or equal to 1 to
              prevent the solution overflowing.

      W       (output) COMPLEX*16 array, dimension (N)
              If INFO >= 0 and INFO <= 3, W contains the eigenvalues of
              the matrix A.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

      ZWORK   COMPLEX*16 array, dimension (LZWORK)
              On exit, if INFO = 0 or INFO = 1, ZWORK(1) returns the
              optimal value of LZWORK.
              On exit, if INFO = -16, ZWORK(1) returns the minimum value
              of LZWORK.

      LZWORK  INTEGER
              The length of the array ZWORK.
              If M > 0, LZWORK >= MAX(1,2*N+MAX(MIN(N,M)-2,0));
              If M = 0, LZWORK >= 1.
              For optimum performance LZWORK should sometimes be larger.

              If LZWORK = -1, then a workspace query is assumed; the
              routine only calculates the optimal size of the ZWORK
              array, returns this value as the first entry of the ZWORK
              array, and no error message related to LZWORK is issued by
              XERBLA.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  if the Lyapunov equation is (nearly) singular
                    (warning indicator);
                    if DICO = 'C' this means that while the matrix A
                    (or the factor S) has computed eigenvalues with
                    negative real parts, it is only just stable in the
                    sense that small perturbations in A can make one or
                    more of the eigenvalues have a non-negative real
                    part;
                    if DICO = 'D' this means that while the matrix A
                    (or the factor S) has computed eigenvalues inside
                    the unit circle, it is nevertheless only just
                    convergent, in the sense that small perturbations
                    in A can make one or more of the eigenvalues lie
                    outside the unit circle;
                    perturbed values were used to solve the equation;
              = 2:  if FACT = 'N' and DICO = 'C', but the matrix A is
                    not stable (that is, one or more of the eigenvalues
                    of A has a non-negative real part), or DICO = 'D',
                    but the matrix A is not convergent (that is, one or
                    more of the eigenvalues of A lies outside the unit
                    circle); however, A will still have been factored
                    and the eigenvalues of A returned in W;
              = 3:  if FACT = 'F' and DICO = 'C', but the Schur factor S
                    supplied in the array A is not stable (that is, one
                    or more of the eigenvalues of S has a non-negative
                    real part), or DICO = 'D', but the Schur factor S
                    supplied in the array A is not convergent (that is,
                    one or more of the eigenvalues of S lies outside the
                    unit circle); the eigenvalues of A are still
                    returned in W;
              = 6:  if FACT = 'N' and the LAPACK Library routine ZGEES
                    has failed to converge. This failure is not likely
                    to occur. The matrix B will be unaltered but A will
                    be destroyed.

Method

      The method used by the routine is based on the Bartels and Stewart
      method [1], except that it finds the upper triangular matrix U
      directly without first finding X and without the need to form the
      normal matrix op(B)**H * op(B).

      The Schur factorization of a square matrix A is given by
                H
         A = QSQ ,

      where Q is unitary and S is an N-by-N upper triangular matrix.
      If A has already been factored prior to calling the routine, then
      the factors Q and S may be supplied and the initial factorization
      omitted.

      If TRANS = 'N' and 6*M > 7*N, the matrix B is factored as
      (QR factorization)
             _   _
         B = P ( R ),
               ( 0 )
            _                                 _
      where P is an M-by-M unitary matrix and R is a square upper
                                          _   _
      triangular matrix. Then, the matrix B = RQ is factored as
         _
         B = PR.

      If TRANS = 'N' and 6*M <= 7*N, the matrix BQ is factored as

         BQ = P ( R ),   M >= N,   BQ = P ( R  Z ),   M < N.
                ( 0 )

      If TRANS = 'C' and 6*M > 7*N, the matrix B is factored as
      (RQ factorization)
                  _   _
         B = ( 0  R ) P,
            _                                 _
      where P is an M-by-M unitary matrix and R is a square upper
                                          _    H _
      triangular matrix. Then, the matrix B = Q  R is factored as
         _
         B = RP.
                                                 H
      If TRANS = 'C' and 6*M <= 7*N, the matrix Q  B is factored as

          H                              H     ( Z )
         Q  B = ( 0  R ) P,   M >= N,   Q  B = (   ) P,   M < N.
                                               ( R )

      These factorizations are utilised to either transform the
      continuous-time Lyapunov equation to the canonical form
             H      H              H                     2      H
        op(S) *op(V) *op(V) + op(V) *op(V)*op(S) = -scale *op(F) *op(F),

      or the discrete-time Lyapunov equation to the canonical form
             H      H                    H               2      H
        op(S) *op(V) *op(V)*op(S) - op(V) *op(V) = -scale *op(F) *op(F),

      where V and F are upper triangular, and

         F = R,  M >= N,   F = ( R  Z ),  M < N,  if TRANS = 'N';
                               ( 0  0 )

         F = R,  M >= N,   F = ( 0  Z ),  M < N,  if TRANS = 'C'.
                               ( 0  R )

      The transformed equation is then solved for V, from which U is
      obtained via the QR factorization of V*Q**H, if TRANS = 'N', or
      via the RQ factorization of Q*V, if TRANS = 'C'.

References

      [1] Bartels, R.H. and Stewart, G.W.
          Solution of the matrix equation  A'X + XB = C.
          Comm. A.C.M., 15, pp. 820-826, 1972.

      [2] Hammarling, S.J.
          Numerical solution of the stable, non-negative definite
          Lyapunov equation.
          IMA J. Num. Anal., 2, pp. 303-325, 1982.

Numerical Aspects

                                3
      The algorithm requires 0(N ) operations and is backward stable.

Further Comments

      The Lyapunov equation may be very ill-conditioned. In particular,
      if A is only just stable (or convergent) then the Lyapunov
      equation will be ill-conditioned.  A symptom of ill-conditioning
      is "large" elements in U relative to those of A and B, or a
      "small" value for scale.

      SB03OZ routine can be also used for solving "unstable" Lyapunov
      equations, i.e., when matrix A has all eigenvalues with positive
      real parts, if DICO = 'C', or with moduli greater than one,
      if DICO = 'D'. Specifically, one may solve for X = op(U)**H*op(U)
      either the continuous-time Lyapunov equation
              H                    2      H
         op(A) *X + X*op(A) = scale *op(B) *op(B),                   (3)

      or the discrete-time Lyapunov equation
              H                    2      H
         op(A) *X*op(A) - X = scale *op(B) *op(B),                   (4)

      provided, for equation (3), the given matrix A is replaced by -A,
      or, for equation (4), the given matrices A and B are replaced by
      inv(A) and B*inv(A), if TRANS = 'N' (or inv(A)*B, if TRANS = 'C'),
      respectively. Although the inversion generally can rise numerical
      problems, in case of equation (4) it is expected that the matrix A
      is enough well-conditioned, having only eigenvalues with moduli
      greater than 1.

Example

Program Text

      None

Program Data

      None

Program Results

      None

Return to index