MB04QF - SLICOT Library Routine Documentation

## MB04QF

### Forming the triangular block factors of a symplectic block reflector

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

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.

Specification

          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,*)

Arguments

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.

Input/Output Parameters

      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.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (3*K)

References

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

Numerical Aspects

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

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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