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    1: *> \brief \b DLATRZ factors an upper trapezoidal matrix by means of orthogonal transformations.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at 
    6: *            http://www.netlib.org/lapack/explore-html/ 
    7: *
    8: *> \htmlonly
    9: *> Download DLATRZ + dependencies 
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrz.f"> 
   11: *> [TGZ]</a> 
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrz.f"> 
   13: *> [ZIP]</a> 
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrz.f"> 
   15: *> [TXT]</a>
   16: *> \endhtmlonly 
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
   22: * 
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            L, LDA, M, N
   25: *       ..
   26: *       .. Array Arguments ..
   27: *       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
   28: *       ..
   29: *  
   30: *
   31: *> \par Purpose:
   32: *  =============
   33: *>
   34: *> \verbatim
   35: *>
   36: *> DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix
   37: *> [ A1 A2 ] = [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R  0 ) * Z, by means
   38: *> of orthogonal transformations.  Z is an (M+L)-by-(M+L) orthogonal
   39: *> matrix and, R and A1 are M-by-M upper triangular matrices.
   40: *> \endverbatim
   41: *
   42: *  Arguments:
   43: *  ==========
   44: *
   45: *> \param[in] M
   46: *> \verbatim
   47: *>          M is INTEGER
   48: *>          The number of rows of the matrix A.  M >= 0.
   49: *> \endverbatim
   50: *>
   51: *> \param[in] N
   52: *> \verbatim
   53: *>          N is INTEGER
   54: *>          The number of columns of the matrix A.  N >= 0.
   55: *> \endverbatim
   56: *>
   57: *> \param[in] L
   58: *> \verbatim
   59: *>          L is INTEGER
   60: *>          The number of columns of the matrix A containing the
   61: *>          meaningful part of the Householder vectors. N-M >= L >= 0.
   62: *> \endverbatim
   63: *>
   64: *> \param[in,out] A
   65: *> \verbatim
   66: *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   67: *>          On entry, the leading M-by-N upper trapezoidal part of the
   68: *>          array A must contain the matrix to be factorized.
   69: *>          On exit, the leading M-by-M upper triangular part of A
   70: *>          contains the upper triangular matrix R, and elements N-L+1 to
   71: *>          N of the first M rows of A, with the array TAU, represent the
   72: *>          orthogonal matrix Z as a product of M elementary reflectors.
   73: *> \endverbatim
   74: *>
   75: *> \param[in] LDA
   76: *> \verbatim
   77: *>          LDA is INTEGER
   78: *>          The leading dimension of the array A.  LDA >= max(1,M).
   79: *> \endverbatim
   80: *>
   81: *> \param[out] TAU
   82: *> \verbatim
   83: *>          TAU is DOUBLE PRECISION array, dimension (M)
   84: *>          The scalar factors of the elementary reflectors.
   85: *> \endverbatim
   86: *>
   87: *> \param[out] WORK
   88: *> \verbatim
   89: *>          WORK is DOUBLE PRECISION array, dimension (M)
   90: *> \endverbatim
   91: *
   92: *  Authors:
   93: *  ========
   94: *
   95: *> \author Univ. of Tennessee 
   96: *> \author Univ. of California Berkeley 
   97: *> \author Univ. of Colorado Denver 
   98: *> \author NAG Ltd. 
   99: *
  100: *> \date September 2012
  101: *
  102: *> \ingroup doubleOTHERcomputational
  103: *
  104: *> \par Contributors:
  105: *  ==================
  106: *>
  107: *>    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
  108: *
  109: *> \par Further Details:
  110: *  =====================
  111: *>
  112: *> \verbatim
  113: *>
  114: *>  The factorization is obtained by Householder's method.  The kth
  115: *>  transformation matrix, Z( k ), which is used to introduce zeros into
  116: *>  the ( m - k + 1 )th row of A, is given in the form
  117: *>
  118: *>     Z( k ) = ( I     0   ),
  119: *>              ( 0  T( k ) )
  120: *>
  121: *>  where
  122: *>
  123: *>     T( k ) = I - tau*u( k )*u( k )**T,   u( k ) = (   1    ),
  124: *>                                                 (   0    )
  125: *>                                                 ( z( k ) )
  126: *>
  127: *>  tau is a scalar and z( k ) is an l element vector. tau and z( k )
  128: *>  are chosen to annihilate the elements of the kth row of A2.
  129: *>
  130: *>  The scalar tau is returned in the kth element of TAU and the vector
  131: *>  u( k ) in the kth row of A2, such that the elements of z( k ) are
  132: *>  in  a( k, l + 1 ), ..., a( k, n ). The elements of R are returned in
  133: *>  the upper triangular part of A1.
  134: *>
  135: *>  Z is given by
  136: *>
  137: *>     Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).
  138: *> \endverbatim
  139: *>
  140: *  =====================================================================
  141:       SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )
  142: *
  143: *  -- LAPACK computational routine (version 3.4.2) --
  144: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  145: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  146: *     September 2012
  147: *
  148: *     .. Scalar Arguments ..
  149:       INTEGER            L, LDA, M, N
  150: *     ..
  151: *     .. Array Arguments ..
  152:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
  153: *     ..
  154: *
  155: *  =====================================================================
  156: *
  157: *     .. Parameters ..
  158:       DOUBLE PRECISION   ZERO
  159:       PARAMETER          ( ZERO = 0.0D+0 )
  160: *     ..
  161: *     .. Local Scalars ..
  162:       INTEGER            I
  163: *     ..
  164: *     .. External Subroutines ..
  165:       EXTERNAL           DLARFG, DLARZ
  166: *     ..
  167: *     .. Executable Statements ..
  168: *
  169: *     Test the input arguments
  170: *
  171: *     Quick return if possible
  172: *
  173:       IF( M.EQ.0 ) THEN
  174:          RETURN
  175:       ELSE IF( M.EQ.N ) THEN
  176:          DO 10 I = 1, N
  177:             TAU( I ) = ZERO
  178:    10    CONTINUE
  179:          RETURN
  180:       END IF
  181: *
  182:       DO 20 I = M, 1, -1
  183: *
  184: *        Generate elementary reflector H(i) to annihilate
  185: *        [ A(i,i) A(i,n-l+1:n) ]
  186: *
  187:          CALL DLARFG( L+1, A( I, I ), A( I, N-L+1 ), LDA, TAU( I ) )
  188: *
  189: *        Apply H(i) to A(1:i-1,i:n) from the right
  190: *
  191:          CALL DLARZ( 'Right', I-1, N-I+1, L, A( I, N-L+1 ), LDA,
  192:      $               TAU( I ), A( 1, I ), LDA, WORK )
  193: *
  194:    20 CONTINUE
  195: *
  196:       RETURN
  197: *
  198: *     End of DLATRZ
  199: *
  200:       END