## MB04SU

### Symplectic QR decomposition of a real 2M-by-N matrix

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

Purpose

      To compute a symplectic QR decomposition of a real 2M-by-N matrix
      [A; B],

                [ A ]             [ R11  R12 ]
                [   ] = Q * R = Q [          ],
                [ B ]             [ R21  R22 ]

      where Q is a symplectic orthogonal matrix, R11 is upper triangular
      and R21 is strictly upper triangular.
      If [A; B] is symplectic then, theoretically, R21 = 0 and
      R22 = inv(R11)^T. Unblocked version.

Specification

          SUBROUTINE MB04SU( M, N, A, LDA, B, LDB, CS, TAU, DWORK, LDWORK,
         $                   INFO )
    C     .. Scalar Arguments ..
          INTEGER           INFO, LDA, LDB, LDWORK, M, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), CS(*), DWORK(*), TAU(*)

Arguments

Input/Output Parameters

      M       (input) INTEGER
              The number of rows of A and B. M >= 0.

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

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading M-by-N part of this array must
              contain the matrix A.
              On exit, the leading M-by-N part of this array contains
              the matrix [ R11  R12 ] and, in the zero parts of R,
              information about the elementary reflectors used to
              compute the symplectic QR decomposition.

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

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the leading M-by-N part of this array must
              contain the matrix B.
              On exit, the leading M-by-N part of this array contains
              the matrix [ R21  R22 ] and, in the zero parts of B,
              information about the elementary reflectors used to
              compute the symplectic QR decomposition.

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

      CS      (output) DOUBLE PRECISION array, dimension (2 * min(M,N))
              On exit, the first 2*min(M,N) elements of this array
              contain the cosines and sines of the symplectic Givens
              rotations used to compute the symplectic QR decomposition.

      TAU     (output) DOUBLE PRECISION array, dimension (min(M,N))
              On exit, the first min(M,N) elements of this array
              contain the scalar factors of some of the elementary
              reflectors.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0,  DWORK(1)  returns the optimal
              value of LDWORK.
              On exit, if  INFO = -10,  DWORK(1)  returns the minimum
              value of LDWORK.

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

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The matrix Q is represented as a product of symplectic reflectors
      and Givens rotations

      Q = diag( H(1),H(1) ) G(1) diag( F(1),F(1) )
          diag( H(2),H(2) ) G(2) diag( F(2),F(2) )
                            ....
          diag( H(k),H(k) ) G(k) diag( F(k),F(k) ),

      where k = min(m,n).

      Each H(i) has the form

            H(i) = I - tau * w * w'

      where tau is a real scalar, and w is a real vector with
      w(1:i-1) = 0 and w(i) = 1; w(i+1:m) is stored on exit in
      B(i+1:m,i), and tau in B(i,i).

      Each F(i) has the form

            F(i) = I - nu * v * v'

      where nu is a real scalar, and v is a real vector with
      v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
      A(i+1:m,i), and nu in TAU(i).

      Each G(i) is a Givens rotation acting on rows i of A and B,
      where the cosine is stored in CS(2*i-1) and the sine in
      CS(2*i).

References

      [1] Bunse-Gerstner, A.
          Matrix factorizations for symplectic QR-like methods.
          Linear Algebra Appl., 83, pp. 49-77, 1986.

      [2] Byers, R.
          Hamiltonian and Symplectic Algorithms for the Algebraic
          Riccati Equation.
          Ph.D. Dissertation, Center for Applied Mathematics,
          Cornell University, Ithaca, NY, 1983.

Numerical Aspects

      The algorithm requires
         8*M*N*N - 8/3*N*N*N +  2*M*N + 6*N*N + 8/3*N,  if M >= N,
         8*M*M*N - 8/3*M*M*M + 14*M*N - 6*M*M + 8/3*N,  if M <= N,
      floating point operations and is numerically backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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