MB04QF

Forming the triangular block factors of a symplectic block reflector

Purpose

  To form the triangular block factors R, S and T of a symplectic
  block reflector SH, which is defined as a product of 2k
  concatenated Householder reflectors and k Givens rotations,

      SH = 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) ).

  The upper triangular blocks of the matrices

                              [ S1 ]       [ T11 T12 T13 ]
      R  = [ R1 R2 R3 ],  S = [ S2 ],  T = [ T21 T22 T23 ],
                              [ S3 ]       [ T31 T32 T33 ]

  with R2 unit and S1, R3, T21, T31, T32 strictly upper triangular,
  are stored rowwise in the arrays RS and T, respectively.

      SUBROUTINE MB04QF( DIRECT, STOREV, STOREW, N, K, V, LDV, W, LDW,
     $                   CS, TAU, RS, LDRS, T, LDT, DWORK )
C     .. Scalar Arguments ..
      CHARACTER         DIRECT, STOREV, STOREW
      INTEGER           K, LDRS, LDT, LDV, LDW, N
C     .. Array Arguments ..
      DOUBLE PRECISION  CS(*), DWORK(*), RS(LDRS,*), T(LDT,*),
     $                  TAU(*), V(LDV,*), W(LDW,*)

Mode Parameters

  DIRECT  CHARACTER*1
          This is a dummy argument, which is reserved for future
          extensions of this subroutine. Not referenced.

  STOREV  CHARACTER*1
          Specifies how the vectors which define the concatenated
          Householder F(i) reflectors are stored:
          = 'C':  columnwise;
          = 'R':  rowwise.

  STOREW  CHARACTER*1
          Specifies how the vectors which define the concatenated
          Householder H(i) reflectors are stored:
          = 'C':  columnwise;
          = 'R':  rowwise.

  N       (input) INTEGER
          The order of the Householder reflectors F(i) and H(i).
          N >= 0.

  K       (input) INTEGER
          The number of Givens rotations.  K >= 1.

  V       (input) DOUBLE PRECISION array, dimension
                  (LDV,K) if STOREV = 'C',
                  (LDV,N) if STOREV = 'R'
          On entry with STOREV = 'C', the leading N-by-K part of
          this array must contain in its i-th column the vector
          which defines the elementary reflector F(i).
          On entry with STOREV = 'R', the leading K-by-N part of
          this array must contain in its i-th row the vector
          which defines the elementary reflector F(i).

  LDV     INTEGER
          The leading dimension of the array V.
          LDV >= MAX(1,N),  if STOREV = 'C';
          LDV >= K,         if STOREV = 'R'.

  W       (input) DOUBLE PRECISION array, dimension
                  (LDW,K) if STOREW = 'C',
                  (LDW,N) if STOREW = 'R'
          On entry with STOREW = 'C', the leading N-by-K part of
          this array must contain in its i-th column the vector
          which defines the elementary reflector H(i).
          On entry with STOREV = 'R', the leading K-by-N part of
          this array must contain in its i-th row the vector
          which defines the elementary reflector H(i).

  LDW     INTEGER
          The leading dimension of the array W.
          LDW >= MAX(1,N),  if STOREW = 'C';
          LDW >= K,         if STOREW = 'R'.

  CS      (input) DOUBLE PRECISION array, dimension (2*K)
          On entry, the first 2*K elements of this array must
          contain the cosines and sines of the symplectic Givens
          rotations G(i).

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          On entry, the first K elements of this array must
          contain the scalar factors of the elementary reflectors
          F(i).

  RS      (output) DOUBLE PRECISION array, dimension (K,6*K)
          On exit, the leading K-by-6*K part of this array contains
          the upper triangular matrices defining the factors R and
          S of the symplectic block reflector SH. The (strictly)
          lower portions of this array are not used.

  LDRS    INTEGER
          The leading dimension of the array RS.  LDRS >= K.

  T       (output) DOUBLE PRECISION array, dimension (K,9*K)
          On exit, the leading K-by-9*K part of this array contains
          the upper triangular matrices defining the factor T of the
          symplectic block reflector SH. The (strictly) lower
          portions of this array are not used.

  LDT     INTEGER
          The leading dimension of the array T.  LDT >= K.

  DWORK   DOUBLE PRECISION array, dimension (3*K)

  [1] Kressner, D.
      Block algorithms for orthogonal symplectic factorizations.
      BIT, 43 (4), pp. 775-790, 2003.

  The algorithm requires ( 4*K - 2 )*K*N + 19/3*K*K*K + 1/2*K*K
  + 43/6*K - 4 floating point operations.

  None

Program Text

  None

  None

  None