MB04IY - SLICOT Library Routine Documentation

## MB04IY

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

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

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'.

Specification

          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( * )

Arguments

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).

Input/Output Parameters

      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).

Workspace

      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.

Error Indicator

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

Method

      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.

Numerical Aspects

      The implemented method is numerically stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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