MB04IY

Applying the product of elementary reflectors used for QR factorization of a matrix having a lower left zero triangle

Purpose

  To overwrite the real n-by-m matrix  C  with  Q' * C,  Q * C,
  C * Q',  or  C * Q,  according to the following table

                  SIDE = 'L'     SIDE = 'R'
  TRANS = 'N':      Q * C          C * Q
  TRANS = 'T':      Q'* C          C * Q'

  where  Q  is a real orthogonal matrix defined as the product of
  k elementary reflectors

     Q = H(1) H(2) . . . H(k)

  as returned by SLICOT Library routine MB04ID.  Q  is of order n
  if  SIDE = 'L'  and of order m if  SIDE = 'R'.

      SUBROUTINE MB04IY( SIDE, TRANS, N, M, K, P, A, LDA, TAU, C, LDC,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, K, LDA, LDC, LDWORK, M, N, P
      CHARACTER          SIDE, TRANS
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * ), DWORK( * ), TAU( * )

Mode Parameters

  SIDE    CHARACTER*1
          Specify if  Q  or  Q'  is applied from the left or right,
          as follows:
          = 'L':  apply  Q  or  Q'  from the left;
          = 'R':  apply  Q  or  Q'  from the right.

  TRANS   CHARACTER*1
          Specify if  Q  or  Q'  is to be applied, as follows:
          = 'N':  apply  Q   (No transpose);
          = 'T':  apply  Q'  (Transpose).

  N       (input) INTEGER
          The number of rows of the matrix C.  N >= 0.

  M       (input) INTEGER
          The number of columns of the matrix C.  M >= 0.

  K       (input) INTEGER
          The number of elementary reflectors whose product defines
          the matrix Q.
          N >= K >= 0,  if  SIDE = 'L';
          M >= K >= 0,  if  SIDE = 'R'.

  P       (input) INTEGER
          The order of the zero triagle (or the number of rows of
          the zero trapezoid) in the matrix triangularized by SLICOT
          Library routine MB04ID.  P >= 0.

  A       (input) DOUBLE PRECISION array, dimension (LDA,K)
          On input, the elements in the rows  i+1:min(n,n-p-1+i)  of
          the  i-th  column, and  TAU(i),  represent the orthogonal
          reflector  H(i),  so that matrix  Q  is the product of
          elementary reflectors:  Q = H(1) H(2) . . . H(k).
          A is modified by the routine but restored on exit.

  LDA     INTEGER
          The leading dimension of the array  A.
          LDA >= max(1,N),  if  SIDE = 'L';
          LDA >= max(1,M),  if  SIDE = 'R'.

  TAU     (input) DOUBLE PRECISION array, dimension (K)
          The scalar factors of the elementary reflectors.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,M)
          On entry, the leading N-by-M part of this array must
          contain the matrix  C.
          On exit, the leading N-by-M part of this array contains
          the updated matrix C.

  LDC     INTEGER
          The leading dimension of the array  C.  LDC >= max(1,N).

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

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= MAX(1,M),  if  SIDE = 'L';
          LDWORK >= MAX(1,N),  if  SIDE = 'R'.
          For optimum performance LDWORK >= M*NB if SIDE = 'L',
          or LDWORK >= N*NB if SIDE = 'R', where NB is the optimal
          block size.

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

  If  SIDE = 'L',  each elementary reflector  H(i)  modifies
  n-p  elements of each column of  C,  for  i = 1:p+1,  and
  n-i+1  elements, for  i = p+2:k.
  If  SIDE = 'R',  each elementary reflector  H(i)  modifies
  m-p  elements of each row of  C,  for  i = 1:p+1,  and
  m-i+1  elements, for  i = p+2:k.

  The implemented method is numerically stable.

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Program Text

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