## MB03ZA

### Reordering a selected cluster of eigenvalues of a given matrix pair in periodic Schur form

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

Purpose

      1. To compute, for a given matrix pair (A,B) in periodic Schur
         form, orthogonal matrices Ur and Vr so that

             T           [ A11  A12 ]     T           [ B11  B12 ]
           Vr * A * Ur = [          ],  Ur * B * Vr = [          ], (1)
                         [  0   A22 ]                 [  0   B22 ]

         is in periodic Schur form, and the eigenvalues of A11*B11
         form a selected cluster of eigenvalues.

      2. To compute an orthogonal matrix W so that

                    T  [  0  -A11 ]       [  R11   R12 ]
                   W * [          ] * W = [            ],           (2)
                       [ B11   0  ]       [   0    R22 ]

         where the eigenvalues of R11 and -R22 coincide and have
         positive real part.

      Optionally, the matrix C is overwritten by Ur'*C*Vr.

      All eigenvalues of A11*B11 must either be complex or real and
      negative.

Specification

          SUBROUTINE MB03ZA( COMPC, COMPU, COMPV, COMPW, WHICH, SELECT, N,
         $                   A, LDA, B, LDB, C, LDC, U1, LDU1, U2, LDU2, V1,
         $                   LDV1, V2, LDV2, W, LDW, WR, WI, M, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         COMPC, COMPU, COMPV, COMPW, WHICH
          INTEGER           INFO, LDA, LDB, LDC, LDU1, LDU2, LDV1, LDV2,
         $                  LDW, LDWORK, M, N
    C     .. Array Arguments ..
          LOGICAL           SELECT(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
         $                  U1(LDU1,*), U2(LDU2,*), V1(LDV1,*), V2(LDV2,*),
         $                  W(LDW,*), WI(*), WR(*)

Arguments

Mode Parameters

      COMPC   CHARACTER*1
              = 'U':  update the matrix C;
              = 'N':  do not update C.

      COMPU   CHARACTER*1
              = 'U':  update the matrices U1 and U2;
              = 'N':  do not update U1 and U2.
              See the description of U1 and U2.

      COMPV   CHARACTER*1
              = 'U':  update the matrices V1 and V2;
              = 'N':  do not update V1 and V2.
              See the description of V1 and V2.

      COMPW   CHARACTER*1
              Indicates whether or not the user wishes to accumulate
              the matrix W as follows:
              = 'N':  the matrix W is not required;
              = 'I':  W is initialized to the unit matrix and the
                      orthogonal transformation matrix W is returned;
              = 'V':  W must contain an orthogonal matrix Q on entry,
                      and the product Q*W is returned.

      WHICH   CHARACTER*1
              = 'A':  select all eigenvalues, this effectively means
                      that Ur and Vr are identity matrices and A11 = A,
                      B11 = B;
              = 'S':  select a cluster of eigenvalues specified by
                      SELECT.

      SELECT  LOGICAL array, dimension (N)
              If WHICH = 'S', then SELECT specifies the eigenvalues of
              A*B in the selected cluster. To select a real eigenvalue
              w(j), SELECT(j) must be set to .TRUE.. To select a complex
              conjugate pair of eigenvalues w(j) and w(j+1),
              corresponding to a 2-by-2 diagonal block in A, both
              SELECT(j) and SELECT(j+1) must be set to .TRUE.; a complex
              conjugate pair of eigenvalues must be either both included
              in the cluster or both excluded.

Input/Output Parameters

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

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the upper quasi-triangular matrix A of the matrix
              pair (A,B) in periodic Schur form.
              On exit, the leading M-by-M part of this array contains
              the matrix R22 in (2).

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the leading N-by-N part of this array must
              contain the upper triangular matrix B of the matrix pair
              (A,B) in periodic Schur form.
              On exit, the leading N-by-N part of this array is
              overwritten.

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

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, if COMPC = 'U', the leading N-by-N part of this
              array must contain a general matrix C.
              On exit, if COMPC = 'U', the leading N-by-N part of this
              array contains the updated matrix Ur'*C*Vr.
              If COMPC = 'N' or WHICH = 'A', this array is not
              referenced.

      LDC     INTEGER
              The leading dimension of the array C.  LDC >= 1.
              LDC >= N,  if COMPC = 'U' and WHICH = 'S'.

      U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,N)
              On entry, if COMPU = 'U' and WHICH = 'S', the leading
              N-by-N part of this array must contain U1, the (1,1)
              block of an orthogonal symplectic matrix
              U = [ U1, U2; -U2, U1 ].
              On exit, if COMPU = 'U' and WHICH = 'S', the leading
              N-by-N part of this array contains U1*Ur.
              If COMPU = 'N' or WHICH = 'A', this array is not
              referenced.

      LDU1    INTEGER
              The leading dimension of the array U1.  LDU1 >= 1.
              LDU1 >= N,  if COMPU = 'U' and WHICH = 'S'.

      U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,N)
              On entry, if COMPU = 'U' and WHICH = 'S', the leading
              N-by-N part of this array must contain U2, the (1,2)
              block of an orthogonal symplectic matrix
              U = [ U1, U2; -U2, U1 ].
              On exit, if COMPU = 'U' and WHICH = 'S', the leading
              N-by-N part of this array contains U2*Ur.
              If COMPU = 'N' or WHICH = 'A', this array is not
              referenced.

      LDU2    INTEGER
              The leading dimension of the array U2.  LDU2 >= 1.
              LDU2 >= N,  if COMPU = 'U' and WHICH = 'S'.

      V1      (input/output) DOUBLE PRECISION array, dimension (LDV1,N)
              On entry, if COMPV = 'U' and WHICH = 'S', the leading
              N-by-N part of this array must contain V1, the (1,1)
              block of an orthogonal symplectic matrix
              V = [ V1, V2; -V2, V1 ].
              On exit, if COMPV = 'U' and WHICH = 'S', the leading
              N-by-N part of this array contains V1*Vr.
              If COMPV = 'N' or WHICH = 'A', this array is not
              referenced.

      LDV1    INTEGER
              The leading dimension of the array V1.  LDV1 >= 1.
              LDV1 >= N,  if COMPV = 'U' and WHICH = 'S'.

      V2      (input/output) DOUBLE PRECISION array, dimension (LDV2,N)
              On entry, if COMPV = 'U' and WHICH = 'S', the leading
              N-by-N part of this array must contain V2, the (1,2)
              block of an orthogonal symplectic matrix
              V = [ V1, V2; -V2, V1 ].
              On exit, if COMPV = 'U' and WHICH = 'S', the leading
              N-by-N part of this array contains V2*Vr.
              If COMPV = 'N' or WHICH = 'A', this array is not
              referenced.

      LDV2    INTEGER
              The leading dimension of the array V2.  LDV2 >= 1.
              LDV2 >= N,  if COMPV = 'U' and WHICH = 'S'.

      W       (input/output) DOUBLE PRECISION array, dimension (LDW,2*M)
              On entry, if COMPW = 'V', then the leading 2*M-by-2*M part
              of this array must contain a matrix W.
              If COMPW = 'I', then W need not be set on entry, W is set
              to the identity matrix.
              On exit, if COMPW = 'I' or 'V' the leading 2*M-by-2*M part
              of this array is post-multiplied by the transformation
              matrix that produced (2).
              If COMPW = 'N', this array is not referenced.

      LDW     INTEGER
              The leading dimension of the array W.  LDW >= 1.
              LDW >= 2*M,  if COMPW = 'I' or COMPW = 'V'.

      WR      (output) DOUBLE PRECISION array, dimension (M)
      WI      (output) DOUBLE PRECISION array, dimension (M)
              The real and imaginary parts, respectively, of the
              eigenvalues of R11. The eigenvalues are stored in the same
              order as on the diagonal of R22, with
              WR(i) = -R22(i,i) and, if R22(i:i+1,i:i+1) is a 2-by-2
              diagonal block, WI(i) > 0 and WI(i+1) = -WI(i).
              In exact arithmetic, these eigenvalue are the positive
              square roots of the selected eigenvalues of the product
              A*B. However, if an eigenvalue is sufficiently
              ill-conditioned, then its value may differ significantly.

      M       (output) INTEGER
              The number of selected eigenvalues.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if  INFO = -28,  DWORK(1)  returns the minimum
              value of LDWORK.

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX( 1, 4*N, 8*M ).

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  reordering of the product A*B in Step 1 failed
                    because some eigenvalues are too close to separate;
              = 2:  reordering of some submatrix in Step 2 failed
                    because some eigenvalues are too close to separate;
              = 3:  the QR algorithm failed to compute the Schur form
                    of some submatrix in Step 2;
              = 4:  the condition that all eigenvalues of A11*B11 must
                    either be complex or real and negative is
                    numerically violated.

Method

      Step 1 is performed using a reordering technique analogous to the
      LAPACK routine DTGSEN for reordering matrix pencils [1,2]. Step 2
      is an implementation of Algorithm 2 in [3]. It requires O(M*N*N)
      floating point operations.

References

      [1] Kagstrom, B.
          A direct method for reordering eigenvalues in the generalized
          real Schur form of a regular matrix pair (A,B), in M.S. Moonen
          et al (eds), Linear Algebra for Large Scale and Real-Time
          Applications, Kluwer Academic Publ., 1993, pp. 195-218.

      [2] Kagstrom, B. and Poromaa P.:
          Computing eigenspaces with specified eigenvalues of a regular
          matrix pair (A, B) and condition estimation: Theory,
          algorithms and software, Numer. Algorithms, 1996, vol. 12,
          pp. 369-407.

      [3] Benner, P., Mehrmann, V., and Xu, H.
          A new method for computing the stable invariant subspace of a
          real Hamiltonian matrix,  J. Comput. Appl. Math., 86,
          pp. 17-43, 1997.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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