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    1: *> \brief \b DGERQF
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DGERQF + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgerqf.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgerqf.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgerqf.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
   22: *
   23: *       .. Scalar Arguments ..
   24: *       INTEGER            INFO, LDA, LWORK, 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: *> DGERQF computes an RQ factorization of a real M-by-N matrix A:
   37: *> A = R * Q.
   38: *> \endverbatim
   39: *
   40: *  Arguments:
   41: *  ==========
   42: *
   43: *> \param[in] M
   44: *> \verbatim
   45: *>          M is INTEGER
   46: *>          The number of rows of the matrix A.  M >= 0.
   47: *> \endverbatim
   48: *>
   49: *> \param[in] N
   50: *> \verbatim
   51: *>          N is INTEGER
   52: *>          The number of columns of the matrix A.  N >= 0.
   53: *> \endverbatim
   54: *>
   55: *> \param[in,out] A
   56: *> \verbatim
   57: *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   58: *>          On entry, the M-by-N matrix A.
   59: *>          On exit,
   60: *>          if m <= n, the upper triangle of the subarray
   61: *>          A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
   62: *>          if m >= n, the elements on and above the (m-n)-th subdiagonal
   63: *>          contain the M-by-N upper trapezoidal matrix R;
   64: *>          the remaining elements, with the array TAU, represent the
   65: *>          orthogonal matrix Q as a product of min(m,n) elementary
   66: *>          reflectors (see Further Details).
   67: *> \endverbatim
   68: *>
   69: *> \param[in] LDA
   70: *> \verbatim
   71: *>          LDA is INTEGER
   72: *>          The leading dimension of the array A.  LDA >= max(1,M).
   73: *> \endverbatim
   74: *>
   75: *> \param[out] TAU
   76: *> \verbatim
   77: *>          TAU is DOUBLE PRECISION array, dimension (min(M,N))
   78: *>          The scalar factors of the elementary reflectors (see Further
   79: *>          Details).
   80: *> \endverbatim
   81: *>
   82: *> \param[out] WORK
   83: *> \verbatim
   84: *>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
   85: *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
   86: *> \endverbatim
   87: *>
   88: *> \param[in] LWORK
   89: *> \verbatim
   90: *>          LWORK is INTEGER
   91: *>          The dimension of the array WORK.
   92: *>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= M, otherwise.
   93: *>          For optimum performance LWORK >= M*NB, where NB is
   94: *>          the optimal blocksize.
   95: *>
   96: *>          If LWORK = -1, then a workspace query is assumed; the routine
   97: *>          only calculates the optimal size of the WORK array, returns
   98: *>          this value as the first entry of the WORK array, and no error
   99: *>          message related to LWORK is issued by XERBLA.
  100: *> \endverbatim
  101: *>
  102: *> \param[out] INFO
  103: *> \verbatim
  104: *>          INFO is INTEGER
  105: *>          = 0:  successful exit
  106: *>          < 0:  if INFO = -i, the i-th argument had an illegal value
  107: *> \endverbatim
  108: *
  109: *  Authors:
  110: *  ========
  111: *
  112: *> \author Univ. of Tennessee
  113: *> \author Univ. of California Berkeley
  114: *> \author Univ. of Colorado Denver
  115: *> \author NAG Ltd.
  116: *
  117: *> \ingroup doubleGEcomputational
  118: *
  119: *> \par Further Details:
  120: *  =====================
  121: *>
  122: *> \verbatim
  123: *>
  124: *>  The matrix Q is represented as a product of elementary reflectors
  125: *>
  126: *>     Q = H(1) H(2) . . . H(k), where k = min(m,n).
  127: *>
  128: *>  Each H(i) has the form
  129: *>
  130: *>     H(i) = I - tau * v * v**T
  131: *>
  132: *>  where tau is a real scalar, and v is a real vector with
  133: *>  v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
  134: *>  A(m-k+i,1:n-k+i-1), and tau in TAU(i).
  135: *> \endverbatim
  136: *>
  137: *  =====================================================================
  138:       SUBROUTINE DGERQF( M, N, A, LDA, TAU, WORK, LWORK, INFO )
  139: *
  140: *  -- LAPACK computational routine --
  141: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  142: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  143: *
  144: *     .. Scalar Arguments ..
  145:       INTEGER            INFO, LDA, LWORK, M, N
  146: *     ..
  147: *     .. Array Arguments ..
  148:       DOUBLE PRECISION   A( LDA, * ), TAU( * ), WORK( * )
  149: *     ..
  150: *
  151: *  =====================================================================
  152: *
  153: *     .. Local Scalars ..
  154:       LOGICAL            LQUERY
  155:       INTEGER            I, IB, IINFO, IWS, K, KI, KK, LDWORK, LWKOPT,
  156:      $                   MU, NB, NBMIN, NU, NX
  157: *     ..
  158: *     .. External Subroutines ..
  159:       EXTERNAL           DGERQ2, DLARFB, DLARFT, XERBLA
  160: *     ..
  161: *     .. Intrinsic Functions ..
  162:       INTRINSIC          MAX, MIN
  163: *     ..
  164: *     .. External Functions ..
  165:       INTEGER            ILAENV
  166:       EXTERNAL           ILAENV
  167: *     ..
  168: *     .. Executable Statements ..
  169: *
  170: *     Test the input arguments
  171: *
  172:       INFO = 0
  173:       LQUERY = ( LWORK.EQ.-1 )
  174:       IF( M.LT.0 ) THEN
  175:          INFO = -1
  176:       ELSE IF( N.LT.0 ) THEN
  177:          INFO = -2
  178:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
  179:          INFO = -4
  180:       END IF
  181: *
  182:       IF( INFO.EQ.0 ) THEN
  183:          K = MIN( M, N )
  184:          IF( K.EQ.0 ) THEN
  185:             LWKOPT = 1
  186:          ELSE
  187:             NB = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )
  188:             LWKOPT = M*NB
  189:          END IF
  190:          WORK( 1 ) = LWKOPT
  191: *
  192:          IF ( .NOT.LQUERY ) THEN
  193:             IF( LWORK.LE.0 .OR. ( N.GT.0 .AND. LWORK.LT.MAX( 1, M ) ) )
  194:      $         INFO = -7
  195:          END IF
  196:       END IF
  197: *
  198:       IF( INFO.NE.0 ) THEN
  199:          CALL XERBLA( 'DGERQF', -INFO )
  200:          RETURN
  201:       ELSE IF( LQUERY ) THEN
  202:          RETURN
  203:       END IF
  204: *
  205: *     Quick return if possible
  206: *
  207:       IF( K.EQ.0 ) THEN
  208:          RETURN
  209:       END IF
  210: *
  211:       NBMIN = 2
  212:       NX = 1
  213:       IWS = M
  214:       IF( NB.GT.1 .AND. NB.LT.K ) THEN
  215: *
  216: *        Determine when to cross over from blocked to unblocked code.
  217: *
  218:          NX = MAX( 0, ILAENV( 3, 'DGERQF', ' ', M, N, -1, -1 ) )
  219:          IF( NX.LT.K ) THEN
  220: *
  221: *           Determine if workspace is large enough for blocked code.
  222: *
  223:             LDWORK = M
  224:             IWS = LDWORK*NB
  225:             IF( LWORK.LT.IWS ) THEN
  226: *
  227: *              Not enough workspace to use optimal NB:  reduce NB and
  228: *              determine the minimum value of NB.
  229: *
  230:                NB = LWORK / LDWORK
  231:                NBMIN = MAX( 2, ILAENV( 2, 'DGERQF', ' ', M, N, -1,
  232:      $                 -1 ) )
  233:             END IF
  234:          END IF
  235:       END IF
  236: *
  237:       IF( NB.GE.NBMIN .AND. NB.LT.K .AND. NX.LT.K ) THEN
  238: *
  239: *        Use blocked code initially.
  240: *        The last kk rows are handled by the block method.
  241: *
  242:          KI = ( ( K-NX-1 ) / NB )*NB
  243:          KK = MIN( K, KI+NB )
  244: *
  245:          DO 10 I = K - KK + KI + 1, K - KK + 1, -NB
  246:             IB = MIN( K-I+1, NB )
  247: *
  248: *           Compute the RQ factorization of the current block
  249: *           A(m-k+i:m-k+i+ib-1,1:n-k+i+ib-1)
  250: *
  251:             CALL DGERQ2( IB, N-K+I+IB-1, A( M-K+I, 1 ), LDA, TAU( I ),
  252:      $                   WORK, IINFO )
  253:             IF( M-K+I.GT.1 ) THEN
  254: *
  255: *              Form the triangular factor of the block reflector
  256: *              H = H(i+ib-1) . . . H(i+1) H(i)
  257: *
  258:                CALL DLARFT( 'Backward', 'Rowwise', N-K+I+IB-1, IB,
  259:      $                      A( M-K+I, 1 ), LDA, TAU( I ), WORK, LDWORK )
  260: *
  261: *              Apply H to A(1:m-k+i-1,1:n-k+i+ib-1) from the right
  262: *
  263:                CALL DLARFB( 'Right', 'No transpose', 'Backward',
  264:      $                      'Rowwise', M-K+I-1, N-K+I+IB-1, IB,
  265:      $                      A( M-K+I, 1 ), LDA, WORK, LDWORK, A, LDA,
  266:      $                      WORK( IB+1 ), LDWORK )
  267:             END IF
  268:    10    CONTINUE
  269:          MU = M - K + I + NB - 1
  270:          NU = N - K + I + NB - 1
  271:       ELSE
  272:          MU = M
  273:          NU = N
  274:       END IF
  275: *
  276: *     Use unblocked code to factor the last or only block
  277: *
  278:       IF( MU.GT.0 .AND. NU.GT.0 )
  279:      $   CALL DGERQ2( MU, NU, A, LDA, TAU, WORK, IINFO )
  280: *
  281:       WORK( 1 ) = IWS
  282:       RETURN
  283: *
  284: *     End of DGERQF
  285: *
  286:       END