MB3LZP

Eigenvalues and right deflating subspace of a complex skew-Hamiltonian/Hamiltonian pencil (applying transformations on panels of columns)

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

Purpose

  To compute the eigenvalues of a complex N-by-N skew-Hamiltonian/
  Hamiltonian pencil aS - bH, with

        (  A  D  )         (  B  F  )
    S = (        ) and H = (        ).                           (1)
        (  E  A' )         (  G -B' )

  The structured Schur form of the embedded real skew-Hamiltonian/
  skew-Hamiltonian pencil aB_S - bB_T, defined as

          (  Re(A)  -Im(A)  |  Re(D)  -Im(D)  )
          (                 |                 )
          (  Im(A)   Re(A)  |  Im(D)   Re(D)  )
          (                 |                 )
    B_S = (-----------------+-----------------) , and
          (                 |                 )
          (  Re(E)  -Im(E)  |  Re(A')  Im(A') )
          (                 |                 )
          (  Im(E)   Re(E)  | -Im(A')  Re(A') )
                                                                 (2)
          ( -Im(B)  -Re(B)  | -Im(F)  -Re(F)  )
          (                 |                 )
          (  Re(B)  -Im(B)  |  Re(F)  -Im(F)  )
          (                 |                 )
    B_T = (-----------------+-----------------) ,  T = i*H,
          (                 |                 )
          ( -Im(G)  -Re(G)  | -Im(B')  Re(B') )
          (                 |                 )
          (  Re(G)  -Im(G)  | -Re(B') -Im(B') )

  is determined and used to compute the eigenvalues. The notation M'
  denotes the conjugate transpose of the matrix M. Optionally,
  if COMPQ = 'C', an orthonormal basis of the right deflating
  subspace of the pencil aS - bH, corresponding to the eigenvalues
  with strictly negative real part, is computed. Namely, after
  transforming aB_S - bB_H by unitary matrices, we have

             ( BA  BD  )              ( BB  BF  )
    B_Sout = (         ) and B_Hout = (         ),               (3)
             (  0  BA' )              (  0 -BB' )

  and the eigenvalues with strictly negative real part of the
  complex pencil aB_Sout - bB_Hout are moved to the top. The
  embedding doubles the multiplicities of the eigenvalues of the
  pencil aS - bH.

Specification
      SUBROUTINE MB3LZP( COMPQ, ORTH, N, A, LDA, DE, LDDE, B, LDB, FG,
     $                   LDFG, NEIG, Q, LDQ, ALPHAR, ALPHAI, BETA,
     $                   IWORK, DWORK, LDWORK, ZWORK, LZWORK, BWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER          COMPQ, ORTH
      INTEGER            INFO, LDA, LDB, LDDE, LDFG, LDQ, LDWORK,
     $                   LZWORK, N, NEIG
C     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   ALPHAI( * ), ALPHAR( * ), BETA( * ), DWORK( * )
      COMPLEX*16         A( LDA, * ), B( LDB, * ), DE( LDDE, * ),
     $                   FG( LDFG, * ), Q( LDQ, * ), ZWORK( * )

Arguments

Mode Parameters

  COMPQ   CHARACTER*1
          Specifies whether to compute the deflating subspace
          corresponding to the eigenvalues of aS - bH with strictly
          negative real part.
          = 'N': do not compute the deflating subspace; compute the
                 eigenvalues only;
          = 'C': compute the deflating subspace and store it in the
                 leading subarray of Q.

  ORTH    CHARACTER*1
          If COMPQ = 'C', specifies the technique for computing an
          orthonormal basis of the deflating subspace, as follows:
          = 'P':  QR factorization with column pivoting;
          = 'S':  singular value decomposition.
          If COMPQ = 'N', the ORTH value is not used.

Input/Output Parameters
  N       (input) INTEGER
          The order of the pencil aS - bH.  N >= 0, even.

  A       (input/output) COMPLEX*16 array, dimension (LDA, N)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the matrix A.
          On exit, if COMPQ = 'C', the leading N-by-N part of this
          array contains the upper triangular matrix BA in (3) (see
          also METHOD). The strictly lower triangular part is not
          zeroed; it is preserved in the leading N/2-by-N/2 part.
          If COMPQ = 'N', this array is unchanged on exit.

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

  DE      (input/output) COMPLEX*16 array, dimension (LDDE, N)
          On entry, the leading N/2-by-N/2 lower triangular part of
          this array must contain the lower triangular part of the
          skew-Hermitian matrix E, and the N/2-by-N/2 upper
          triangular part of the submatrix in the columns 2 to N/2+1
          of this array must contain the upper triangular part of
          the skew-Hermitian matrix D.
          On exit, if COMPQ = 'C', the leading N-by-N part of this
          array contains the skew-Hermitian matrix BD in (3) (see
          also METHOD). The strictly lower triangular part of the
          input matrix is preserved.
          If COMPQ = 'N', this array is unchanged on exit.

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

  B       (input/output) COMPLEX*16 array, dimension (LDB, N)
          On entry, the leading N/2-by-N/2 part of this array must
          contain the matrix B.
          On exit, if COMPQ = 'C', the leading N-by-N part of this
          array contains the upper triangular matrix BB in (3) (see
          also METHOD). The strictly lower triangular part is not
          zeroed; the elements below the first subdiagonal of the
          input matrix are preserved.
          If COMPQ = 'N', this array is unchanged on exit.

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

  FG      (input/output) COMPLEX*16 array, dimension (LDFG, N)
          On entry, the leading N/2-by-N/2 lower triangular part of
          this array must contain the lower triangular part of the
          Hermitian matrix G, and the N/2-by-N/2 upper triangular
          part of the submatrix in the columns 2 to N/2+1 of this
          array must contain the upper triangular part of the
          Hermitian matrix F.
          On exit, if COMPQ = 'C', the leading N-by-N part of this
          array contains the Hermitian matrix BF in (3) (see also
          METHOD). The strictly lower triangular part of the input
          matrix is preserved. The diagonal elements might have tiny
          imaginary parts.
          If COMPQ = 'N', this array is unchanged on exit.

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

  NEIG    (output) INTEGER
          If COMPQ = 'C', the number of eigenvalues in aS - bH with
          strictly negative real part.

  Q       (output) COMPLEX*16 array, dimension (LDQ, 2*N)
          On exit, if COMPQ = 'C', the leading N-by-NEIG part of
          this array contains an orthonormal basis of the right
          deflating subspace corresponding to the eigenvalues of the
          pencil aS - bH with strictly negative real part.
          The remaining entries are meaningless.
          If COMPQ = 'N', this array is not referenced.

  LDQ     INTEGER
          The leading dimension of the array Q.
          LDQ >= 1,           if COMPQ = 'N';
          LDQ >= MAX(1, 2*N), if COMPQ = 'C'.

  ALPHAR  (output) DOUBLE PRECISION array, dimension (N)
          The real parts of each scalar alpha defining an eigenvalue
          of the pencil aS - bH.

  ALPHAI  (output) DOUBLE PRECISION array, dimension (N)
          The imaginary parts of each scalar alpha defining an
          eigenvalue of the pencil aS - bH.
          If ALPHAI(j) is zero, then the j-th eigenvalue is real.

  BETA    (output) DOUBLE PRECISION array, dimension (N)
          The scalars beta that define the eigenvalues of the pencil
          aS - bH.
          Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
          beta = BETA(j) represent the j-th eigenvalue of the pencil
          aS - bH, in the form lambda = alpha/beta. Since lambda may
          overflow, the ratios should not, in general, be computed.

Workspace
  IWORK   INTEGER array, dimension (N+1)

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal LDWORK.
          On exit, if INFO = -20, DWORK(1) returns the minimum value
          of LDWORK.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= MAX( 4*N*N + 2*N + MAX(3,N) ), if COMPQ = 'N';
          LDWORK >= MAX( 1, 11*N*N + 2*N ),        if COMPQ = 'C'.
          For good performance LDWORK should be generally larger.

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

  ZWORK   COMPLEX*16 array, dimension (LZWORK)
          On exit, if INFO = 0, ZWORK(1) returns the optimal LZWORK.
          On exit, if INFO = -22, ZWORK(1) returns the minimum
          value of LZWORK.

  LZWORK  INTEGER
          The dimension of the array ZWORK.
          LZWORK >= 1,       if COMPQ = 'N';
          LZWORK >= 8*N + 4, if COMPQ = 'C'.
          For good performance LZWORK should be generally 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.

  BWORK   LOGICAL array, dimension (LBWORK)
          LBWORK >= 0, if COMPQ = 'N';
          LBWORK >= N, if COMPQ = 'C'.

Error Indicator
  INFO    INTEGER
          = 0: succesful exit;
          < 0: if INFO = -i, the i-th argument had an illegal value;
          = 1: QZ iteration failed in the SLICOT Library routine
               MB04FP (QZ iteration did not converge or computation
               of the shifts failed);
          = 2: QZ iteration failed in the LAPACK routine ZHGEQZ when
               trying to triangularize the 2-by-2 blocks;
          = 3: the singular value decomposition failed in the LAPACK
               routine ZGESVD (for ORTH = 'S');
          = 4: warning: the pencil is numerically singular.

Method
  First, T = i*H is set. Then, the embeddings, B_S and B_T, of the
  matrices S and T, are determined and, subsequently, the SLICOT
  Library routine MB04FP is applied to compute the structured Schur
  form, i.e., the factorizations

  ~                     (  S11  S12  )
  B_S = J Q' J' B_S Q = (            ) and
                        (   0   S11' )

  ~                     (  T11  T12  )           (  0  I  )
  B_T = J Q' J' B_T Q = (            ), with J = (        ),
                        (   0   T11' )           ( -I  0  )

  where Q is real orthogonal, S11 is upper triangular, and T11 is
  upper quasi-triangular.

  Second, the SLICOT Library routine MB3JZP is applied, to compute a
                 ~
  unitary matrix Q, such that

                     ~    ~
    ~     ~   ~   (  S11  S12  )
  J Q' J' B_S Q = (       ~    ) =: B_Sout,
                  (   0   S11' )

    ~        ~    ~   (  H11  H12  )
  J Q' J'(-i*B_T) Q = (            ) =: B_Hout,
                      (   0  -H11' )
       ~                                               ~       ~
  with S11, H11 upper triangular, and such that Spec_-(B_S, -i*B_T)
  is contained in the spectrum of the 2*NEIG-by-2*NEIG leading
                       ~
  principal subpencil aS11 - bH11.

  Finally, the right deflating subspace is computed.
  See also page 22 in [1] for more details.

References
  [1] Benner, P., Byers, R., Mehrmann, V. and Xu, H.
      Numerical Computation of Deflating Subspaces of Embedded
      Hamiltonian Pencils.
      Tech. Rep. SFB393/99-15, Technical University Chemnitz,
      Germany, June 1999.

Numerical Aspects
                                                            3
  The algorithm is numerically backward stable and needs O(N )
  complex floating point operations.

Further Comments
  This routine does not perform any scaling of the matrices. Scaling
  might sometimes be useful, and it should be done externally.
  For large values of N, the routine applies the transformations on
  panels of columns. The user may specify in INFO the desired number
  of columns. If on entry INFO <= 0, then the routine estimates a
  suitable value of this number.

Example

Program Text

  None
Program Data
  None
Program Results
  None

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